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Aggregate load modeling for Demand Response via theMinkowski sum

by

Suhail Barot

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of Electrical EngineeringUniversity of Toronto

c© Copyright 2017 by Suhail Barot

Abstract

Aggregate load modeling for Demand Response via the Minkowski sum

Suhail Barot

Doctor of Philosophy

Graduate Department of Electrical Engineering

University of Toronto

2017

Aggregations of flexible loads can provide several power system services through de-

mand response programs, for example load shifting and curtailment. The capabilities

of demand response should therefore be represented in system operators’ planning and

operational routines. However, incorporating models of every load in an aggregation into

these routines could compromise their tractability by adding exorbitant numbers of new

variables and constraints.

We propose a novel approximation for concisely representing the capabilities of a het-

erogeneous aggregation of flexible loads. We assume that each load is mathematically

described by a convex polytope, i.e., a set of linear constraints. We discuss the polytopic

formulation of many classes of loads including deferrable loads, thermostatically con-

trolled loads, and generic energy storage. The set-wise sum of the loads is the Minkowski

sum, which is in general computationally intractable. In this thesis, we develop a new

outer approximation of the Minkowski sum.

The new approximation is applicable for linear constraints, is easily computable, and

only uses one variable per time period corresponding to the aggregation’s net power usage.

We prove that the approximation is exact when applied to deferrable loads without power

constraints and loads modelled for two time periods only. Additionally, numerical results

indicate that the approximation is accurate for broad classes of loads. We also develop a

tightening procedure to further improve upon the accuracy of the approximation.

ii

Following, we extend the above approximation to semidefinite constraints and second-

order cone constraints. The approximation is extended to loads that do not have matching

constraints via the use of the Gershgorin circle theorem. This extension allows for the

modelling of loads with apparent power constraints or for stochastic loads modelled

with chance constraints. Numerical results are shown for the case where load charging

efficiency and total energy demand are taken as stochastic quantities.

Finally, we consider the problem of finding inner approximation to load models. We

use an ellipsoidal projection technique to find the maximum inscribed ellipsoid for the

Minkowski sum of several loads. We perform numerical simulations to illustrate the

technique.

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Contents

1 Introduction 1

1.1 Electricity and Demand Response . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Problem Statement for Load Aggregation . . . . . . . . . . . . . . . . . . 4

1.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3.1 Research on Demand Response implementation . . . . . . . . . . 7

1.3.2 Research on Load Aggregation . . . . . . . . . . . . . . . . . . . . 8

1.3.3 Research on Minkowski sum computation . . . . . . . . . . . . . . 10

1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.5 Our Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Background 14

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.1 Converting between polytope representations . . . . . . . . . . . . 17

2.4 Semidefinite and Second-Order Cone Constraints . . . . . . . . . . . . . 18

2.5 Minkowski Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.6 Outer and Inner approximations . . . . . . . . . . . . . . . . . . . . . . . 21

2.7 Role within power system operations . . . . . . . . . . . . . . . . . . . . 21

2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

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3 Load Models for Demand Response 25

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Storage-like Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 Thermostatic loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.4 Deferrable loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4.1 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.5 Differential power constraints . . . . . . . . . . . . . . . . . . . . . . . . 31

3.6 Apparent power constraints . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.7 Non-convex loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4 Deterministic Load Aggregation 34

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.2 Approximate Load Aggregation . . . . . . . . . . . . . . . . . . . . . . . 35

4.2.1 Polytopes with the same A-matrix . . . . . . . . . . . . . . . . . 35

4.2.2 Extension to general polytopes . . . . . . . . . . . . . . . . . . . 37

4.3 Preconditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.4 Load aggregation algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.4.1 Recovering a feasible solution . . . . . . . . . . . . . . . . . . . . 49

4.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.5.1 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.5.2 Analytical Results . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5 Stochastic Load Aggregation 65

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.2 Outer Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

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5.2.1 Semidefinite constraints . . . . . . . . . . . . . . . . . . . . . . . 67

5.2.2 Second-order cone constraints . . . . . . . . . . . . . . . . . . . . 70

5.3 Demand response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.3.1 Polytope electric vehicle model . . . . . . . . . . . . . . . . . . . 73

5.3.2 Apparent power constraints . . . . . . . . . . . . . . . . . . . . . 73

5.3.3 Chance-constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.4.1 Analytical Example . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.4.2 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6 Inner Approximations 83

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.2 Ellipsoidal approximations of convex sets . . . . . . . . . . . . . . . . . . 84

6.3 Ellipsoidal inner bounds of the Minkowski sum . . . . . . . . . . . . . . . 86

6.4 Finding an MIE over the Minkowski sum from the MIE of the combined

load and sum spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6.4.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.5 Finding an MIE using a Linear Decision Rule . . . . . . . . . . . . . . . 90

6.6 Numerical Results for DR Loads . . . . . . . . . . . . . . . . . . . . . . . 91

6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

7 Conclusion 94

7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

7.2 Challenges for Implementation . . . . . . . . . . . . . . . . . . . . . . . . 97

7.3 Recommendations for Future Work . . . . . . . . . . . . . . . . . . . . . 98

Bibliography 100

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Chapter 1

Introduction

1.1 Electricity and Demand Response

Electricity is one of the most fundamental innovations underlying our modern existence.

The National Academy of Engineering considers the electric power grid to be the greatest

engineering achievement of the 20th century [21]. Electricity is generated from a variety of

sources with renewable sources, hydroelectricity or nuclear plants providing carbon-free

electricity, as well as various fossil-fuelled plants.

While we expect electricity to be immediately and reliably available to us, this requires

a great deal of complex coordination between generators and other entities in the power

system, all managed by an independent system operator. Part of the system operator’s

job is to balance demand and supply instantaneously in order to maintain the frequency

and voltage stability of the power system. This role has become increasingly challenging

with the increase in renewable energy generation, which is intermittent, uncontrollable

and somewhat unpredictable.

In order to balance demand and supply, the system operator conventionally controls

the output of resources on the supply-side of the power system (i.e. flexible generators).

It is however, also possible to do so by the control of resources on the demand-side of the

1

Chapter 1. Introduction 2

power system. As renewable energy sources (which are less controllable and predictable)

increase in quantity, the ability of the demand-side of the power system to help stabilize

the grid will become increasingly important.

We refer to the coordinated control of flexible loads, for the benefit of the power

system, as Demand Response (DR). Some advantages of DR include increasing spatial

diversity of grid resources, providing increased reliability (as a larger number of small

resources will experience milder failures than a small number of large resources), reduced

emissions (if DR can displace inefficient ramping generators) and fast reaction times. DR

may also be used for primary or secondary control for the power grid, though such is not

the focus of our work and is not discussed in this dissertation. DR resources generally

receive a financial incentive for being available for DR purposes and / or for participating

in a DR event. While DR is currently mostly engaged in for demand reduction in order

to avoid the need for expensive generation, in the future it will also entail increasing

demand when generation is overabundant.

Loads suitable for DR are flexible; i.e. they have a variable power profile that can be

deferred, brought forward or otherwise adjusted; examples include residential and com-

mercial air conditioning and heating loads, intelligent lighting systems, dishwashers and

such task loads, electric vehicles and other storage loads, and so on. For example, an

air-conditioning system can pre-cool a space to the lower end of an allowed temperature

range, in order to reduce its power consumption during a subsequent time period. Simi-

larly, a dishwasher can be set up to operate any time over a long time window, wherein

the exact time it runs is dictated based on the availability of surplus grid power. The Fed-

eral Energy Regulatory Commission (FERC) recognizes DR as an essential new source

of flexibility for renewable integration [26]. DR activities are now widely engaged in by

third party companies, utilities, and system operators. It is important to note that there

are only three possible ways that a load can change its consumption of energy for the

purposes of DR: reducing / curtailing their consumption, time-shifting their consumption


or making use of on-site standby generation [58].

To a significant degree, DR activities today are the preserve of large, generally in-

dustrial and commercial loads, and are carried out on an ad-hoc basis. Loads are often

required to have a minimum curtailment volume (expressed in kWh or kW). However,

with the increasing proliferation of smart loads (loads with two-way communication ca-

pabilities and intelligent controllers), and the Internet of Things (IoT), DR capabilities

will be available from smaller, consumer scale loads as well.

We refer to the entity controlling a collection of such loads as the load aggregator.

The load aggregator stands between individual loads and the power system operator; it

communicates the total combined capabilities of its loads to the power system operator,

receives dispatched instructions and then provides appropriate controls to its loads to

accomplish the given change in power consumption. Examples of a load aggregators ac-

tive in Ontario include the EnerNOC corporation, the Enershift corporation and Energy

Curtailment Specialists, Inc.

Load aggregators need to communicate with loads in order to provide them with

instructions. The frequency of communication between the aggregator depends on the

purpose of the aggregation; for example, an aggregator may provide the system operator

with a description of the aggregated load once per day for use in day ahead planning,

and then receive instructions for dispatch that must be communicated to the loads every

hour. While only one-way communication is needed, two-way communication allows the

aggregator to verify that the change in load has taken place as expected. For utilities do-

ing their own load aggregation, such communication may make use of advanced metering

infrastructure, else take place over a customer’s broadband internet connection.

In Ontario, the Independent Electricity System Operator (IESO) procures DR re-

sources, with a minimum participation size of 1 MW, thus smaller entities may only

participate in DR as part of a load aggregation [35]. During 2016, approximately 400

MW of DR capability was procured by the IESO.


Other advantages of load aggregation are that DR capabilities can be offered to a

utility or system operator with higher levels of reliability. This is because an aggregator

can make use of multiple resources to fulfil an instruction from the system operator, and

can meet its obligations even in the case that one of its resources is unavailable. This

increased reliability generally brings additional financial benefits to DR participants.

1.2 Problem Statement for Load Aggregation

The objective of this thesis is to generate concise and convex representations of DR

aggregations, that are compatible with power system optimization. System operators

must integrate DR into their operational routines to fully leverage its capabilities. This

requires that the capabilities of DR resources be expressed in the form of convex load

models that are compatible with power system operational routines such as multi-period

optimal power flow, unit commitment and longer-term planning. Multi-period optimal

power flow is the standard optimization problem solved in the dispatch of power system

resources with dynamic constraints, such as ramping limits, storage capacities and mini-

mum up-time requirements. It fundamentally involves constraints on power consumption

and generation in addition to constraints enforcing line and voltage limits and the lin-

earised power flow between nodes. More information about multi-period optimal power

flow can be found in [60]. These routines can then be used to perform load-shifting using

DR alongside energy storage [40,59].

This is challenging because the loads in DR programs are often small, diverse, and

numerous. However the power system, as currently designed, is not equipped to control

and dispatch large numbers of small, heterogeneous loads. Exactly representing the

loads of multiple DR aggregations within multi-period optimal power flow could add

millions of new variables and constraints, making it computationally intractable [32, 51,

55]. Moreover, the individual load models may be known only to the load aggregator but


not the system operator. A conceptual representation of the place of an aggregator in

the power system can be seen in Figure 1.1.

Figure 1.1: The position of an Aggregator in the power system

To overcome these difficulties, load aggregators need concise models of their loads’

aggregate characteristics, thus enabling them to share their capabilities with the system

operator without describing every load individually. System operators can then straight-

forwardly incorporate such a model into tasks like economic dispatch, multi-period opti-

mal power flow or unit commitment as they would a conventional resource like grid-scale

storage [19]. Because the model is concise, i.e., consisting of a small number of variables

and constraints, it does not increase the difficulty of the system operator’s tasks. We

further discuss the formulation of multi-period optimal power flow as well as the role of

DR and concise modelling within multi-period optimal power flow in Section 2.7.

Once a representation of a load aggregation exists, it can be used by the system

operator, for a variety of optimization problems in power systems. The information

about the load aggregation is communicated to the system operator in the form of a list

of convex constraints, to be used in multi-period optimal power flow in a similar way


to storage. In particular, the system operator may use this model in hourly dispatch,

day-ahead scheduling, for unit commitment and longer term generation and transmission

planning. Different load aggregation models at different time-scales can be used to solve

these problems (e.g. hourly, multi-day).

We make the following assumptions about loads and the overall power system in this

thesis:

• We assume two-way communication between loads and the aggregator. Loads are

able to communicate a list of linear or second-order cone constraints to the aggre-

gator (e.g. on a daily basis) and the aggregator is able to communicate control

instructions to the loads (e.g. on an hourly basis). While loads can be dispatched

more frequently, for shorter term control needs for the grid, we do not consider this.

• We assume that all loads to be aggregated are present at a single node in the

system, or alternatively at multiple nodes connected by uncongested lines, which

allows power to freely flow between them.

• We assume that loads can provide different set of constraints for use at different

time-scales, for use by the aggregator. The aggregator can then produce different

models for use in different problems, such as day-ahead scheduling, unit commit-

ment or longer term planning.

• We assume that the system operator will make use of these aggregate models in

multi-period linear or convex relaxations of power flow. As the use of DR resources

generally implies temporal shifting of the use of energy, it is necessary that the sys-

tem operator make use of multi-period optimal power flow solutions in dispatching

these resources.

• We assume loads are convex and model them in a way that is compatible with

optimal power flow; i.e. loads will adjust themselves to accept whatever level


of power they are dispatched to. This assumption is normal in the solution of

power flow which only models power exchange and not further electrical physics.

When loads are non-convex (such as discretized loads) we assume reasonable convex

relaxation models of the loads exist.

• We model only PQ loads, i.e. loads whose limits are specified in terms of real

and reactive power. Impedance loads are not handled by our framework. This

represents a potential area for future work.

• We acknowledge that the outer approximation will likely contain infeasible solu-

tions, i.e., power consumption trajectories that are beyond a load aggregation’s

actual flexibility. This can be remedied by making use of the nearest feasible solu-

tion to solutions obtained using an outer approximation, (which may be found via

a convex optimization problem solved by the aggregator) [10].

1.3 Literature Review

1.3.1 Research on Demand Response implementation

Several comprehensive surveys on DR exist, for example [17,54,58]. Callaway and Hiskens

in [17] describe ideal loads for DR as being “fully responsive” (i.e. it is capable of

high-resolution control across different time-scales) and capable of “non-disruptive per-

formance” (i.e. that the use of the resource for DR does not degrade its performance for

its primary function). They also argue that price-response is an inadequate mechanism

for DR, and that direct load control is necessary.

Callaway and Hiskens also discuss how DR takes place in the context of economic dis-

patch (decisions on which generators to use, and to what extent) and unit commitment

(decisions on which generators to make available for a given period). They describe how

most current implementations of DR are disruptive, and take place by the use of relays


to interrupt power to loads, and that most associated control schemes for loads are open

loop. They also describe newly available technologies such as “Open ADR”, an auto-

mated framework for DR, the potential from communications infrastructure associated

with advanced metering as well as the increasing role of aggregators as intermediaries

for DR [17]. Palensky and Dietrich explain “Open ADR” as a distributed, server-client

setup for DR, operating over the internet and capable of interfacing with building energy

management systems [54]. Standards such as this offer the potential for lower costs,

increased interoperability and reliability and enhanced flexibility. Other potentially rele-

vant standards for DR include IEC 61850, Zigbee Smart Energy Profile, BACnet by the

American Society of heating, refrigeration and air-conditioning engineers, and KNIVES

(a Japanese home energy management standard) [54].

In [54], Palensky and Dietrich also discusses the information security needs for DR

programs, listing the following needs: confidentiality, authenticity, integrity, availability,

access-control and non-repudiation. He comments interestingly that the conventionally

most important information security need of confidentiality may be the least important

need for DR programs. We are also reminded of the importance of energy efficiency as a

prerequisite to DR program implementation.

Kwac and Rajgopal in [43] discuss how the best loads for DR can be found using big-

data techniques, once high-resolution metering data is available. [54, 58] also list several

examples of existing DR programs and test-beds.

1.3.2 Research on Load Aggregation

A number of existing papers describe techniques for concisely modelling large collections

of loads, which we now summarize. Work on this topic has been ongoing since the 1980s

beginning with Malhame and Chong who first considered using thermostatic loads for

DR (via load-shedding) [46]. They developed a partial differential equation-based model

for the probability distribution of a Markov system, which is used to model the switch-


ing dynamics of a collection of thermostatic loads. More recently, Callaway proposed a

control scheme for thermostatic loads to respond to variability associated with renewable

energy generation, via manipulation of load set-point temperatures [18]. This is note-

worthy as it is among the first papers that takes advantage of the inherent flexibility of

the load for DR instead of resorting to load shedding; an idea we build upon.

Work by Molina-Garcia makes use of stochastic modelling techniques to simulate

the trajectories of individual load temperatures for large numbers of loads [50]. They

use these load trajectories to find probability density functions for power demand and

indoor temperatures using coupled Fokker-Planck partial differential equations as well as

discrete approximation methods, and compare their effectiveness. In contrast, Bashash

and Fathy develop a Lyapunov-stable controller for the management of aggregated air-

conditioning loads [11]. Thermostatic loads are a particular focus area within DR work as

they represent almost 20% of load in industrialized countries such as the U.S, according

to U.S Department of Energy [53]. A unique approach is taken by Perfumo et. all, where

the authors model the control of a collection of thermostatic loads using a second-order

LTI system and design a controller to achieve desired power outputs and then return

the aggregate system to steady-state [56]. Our approach to thermostatic loads may be

seen in Section 3.3; we makes use of linear ”dead-band” constraints on the temperature

associated with the load, building on work in [32].

Our work is closely related to several recent papers that approximate a collection of

loads as generalized energy storage. In [51], charging electric vehicles are modelled as

deferrable loads, and analytical generalized storage expressions for their aggregate ca-

pabilities are obtained; our work reproduces a central analytical result of theirs. In [3],

many types of loads are clustered and aggregated using generalized battery models whose

parameters are found by summing over the loads in a cluster. The cluster based aggre-

gation bears much theoretical resemblance to our aggregations of loads with the same

A-matrices (see section 4.2) Load aggregations are approximated as time-varying ther-


mal batteries in [48, 49] and as generalized batteries in [32]; the latter derives inner and

outer generalized battery models to represent a collection of thermostatic loads and also

develops a control scheme for them. The storage models obtained in these papers consist

of linear constraints, similar to the polytope-based framework employed in this paper.

In [63], the exact Minkowski sum is identified as a measure of a power systems total

flexibility, and is used to quantify and visualize flexibility.

1.3.3 Research on Minkowski sum computation

Exact Minkowski Sums are an NP-hard problem [62], but can be carried out in low

dimensions using several methods; these methods either involve pivoting and depth-

first search, or a “double description method” involving a hyperplane by hyperplane

reconstruction of a polytope, to find its vertices [7].

Approximate Minkowski sums are an active research area, but most work focuses on

the calculation of two and three-dimensional sums of highly complex polytopes as in [2],

[64] and [31]. Gouveia et. all also describe how to lift complicated polytopes to higher-

dimensional second-order cone representations, which may be relevant for polytopes that

don’t work well with our outer approximation, but which may be more tractable when

their second-order cone representations are added [29].

1.4 Outline

In this dissertation, we develop concise, approximate representation for aggregations of

loads, both deterministic and stochastic. For deterministic loads, we primarily consider

convex polytopic models, i.e., sets of linear constraints. Later, we consider broader

classes of convex sets, that help us model loads with uncertainty. Since we only deal with

convex polytopes, we will henceforth omit the term ‘convex’ and simply write ‘polytope’.

In order to develop representations of load models, we make use of the Minkowski sum,


which is the set-wise sum of two sets. We present background material on polytopes and

the Minkowski sum for the interested reader in Chapter 2.

In Chapter 3, we present polytopic load models for various types of loads, that may

be used within our Minkowski sum aggregation framework. As observed in [3, 32], the

flexibility of an aggregation of polytopic loads is captured by the Minkowski sum. Un-

fortunately, exact calculation of the Minkowski sum is computationally intractable even

for polytopes. We further discuss the role of the Minkowski sum in load aggregation in

Section 4.1.

In Chapter 4, Section 4.2, we develop a novel outer approximation of the Minkowski

sum, which is easily computable in polynomial-time. Our method is generally applicable

regardless of the number of dimensions of the polytope, and also results in a polytope

with the same dimensionality as the original polytopes. For the load aggregation problem,

the number of dimensions in RD, is equivalent to the number of time periods that the

loads are modelled for. This makes it easy to incorporate into optimization routines

for power system operations without sacrificing tractability. Later in the same chapter,

we present theoretical and numerical results demonstrating the efficacy of this approach.

The material in this chapter has been presented at the CIGRE 2015 Conference [8] as well

as published in the International Journal of Electrical Power and Energy Systems [10].

Next, in Chapter 5, we extend our outer approximation to convex conic sets defined

by second-order cone and semidefinite constraints. In Section 5.3, we apply the extension

to two types of loads that no prior approach can accommodate: (i) loads that provide

reactive power support through inverters, which are subject to convex quadratic appar-

ent power constraints, and (ii) uncertain loads modelled by chance constraints, which

in special cases can be converted to second-order cone constraints. We illustrate our

approach with an electric vehicle model and provide numerical performance results in

Section 5.4.2. The material in this chapter has been published in the proceedings of the

2016 IEEE Conference on Decision and Control [9].


In the final chapter of this dissertation, we describe ellipsoidal inner approximations

to polytopic loads. Inner approximations are useful in conjunction with outer approxima-

tions, in providing a lower bound on the size of the polytope, and hence the capabilities

of an aggregation. We conduct a numerical study of these inner approximations for the

same cases as for the polytopic outer approximation.

1.5 Our Contributions

The novel contributions of this dissertation are as follows:

• We develop a novel outer approximation for the Minkowski sum of flexible loads

described by polytopes of the same shape in Section 4.2.1.

• We extend the outer approximation to the Minkowski sum of general polytopes in

Section 4.2.2, and give a procedure for tightening the approximation. Our work is

more generally applicable than existing work because it can handle arbitrary loads

represented by polytopes, in a way that is compatible with power system planning.

• We describe an algorithm for the aggregation of a large number of loads in Sec-

tion 4.4. The algorithm works by generating a list of all unique constraints from all

loads in an aggregation, and then generation of the appropriate ”redundant con-

straints” to fill in those missing in the description of individual loads. Finally, the

constraints from individual loads are added together to obtain the constraints on

the aggregation.

• We show analytically that the approximation is exact for special classes of loads in

Section 4.5.2.

• We extend our outer approximation to convex conic sets defined by second-order

cone and semidefinite constraints in Chapter 5. No existing work considers loads

described by second-order cone or semidefinite constraints.


• We numerically characterize the performance of the approximations in Section 4.5.1

and Section 5.4.2.

• We describe an inner approximation scheme for polytopic loads and their aggrega-

tions in Chapter 6.

Chapter 2

Background

2.1 Introduction

In this chapter, we first introduce the notation used in this thesis to describe vectors,

matrices, sets and other mathematical constructs. We also introduce the reader to ideas

relating to polytopes and other convex sets as well as the Minkowski sum, that will be

necessary in understanding subsequent chapters. Finally, we explain how polytopes and

other convex sets can be used in multi-period optimal power flow, the primary application

for this work.

2.2 Notation

We denote vectors (or points represented through position vectors) using lowercase itali-

cized letters, e.g., x, y. Countable sets of vectors (or points) are denoted using uppercase

letters with a bar, e.g., X, Y . General convex sets are represented using calligraphic

letters while polytopes are represented using bold script. We use uppercase letters to

represent matrices e.g. X, Y

We use subscripts to matrices, vectors and sets of points to indicate association with

a particular polytope (defined below) and superscripts denote elements of vectors and

14

Chapter 2. Background 15

matrices.

In our work, we seek general convex representations of load aggregations having a

small number of constraints (which we quantify in Section 4.4). We work with linear

constraints of the form P = {x | Ax ≤ b} in the next chapter, and follow with a treat-

ment of second-order cone and semidefinite constraints, which allow for consideration

of stochastic loads. Here, x ∈ RD is the vector of power injections into the aggrega-

tion through time (“D” time periods), and the set P represents the “flexibility” of the

aggregation.

2.3 Polytopes

A polytope is a set in RD whose boundary is composed of flat surfaces called facets [68].

These facets are derived from hyperplanes and are sets in RD−1. We denote polytopes

using bold script, e.g., P,Q. We restrict our attention to polytopes that are convex,

closed and bounded, i.e., compact. Convexity means that any line drawn between two

points in the set lies completely within the set. It is reasonable to restrict our attention

to bounded sets, because unboundedness in DR corresponds to the physically impossible

case of infinite power consumption.

For such polytopes, the interior points within the polytope can be represented as con-

vex combinations of the extreme points of the polytope [16]. Thus, the set of such extreme

points (vertices) of such polytopes then forms a minimal unique (up to ordering) repre-

sentation for a polytope. Such a representation is referred to as the Vertex-representation

or V-representation of a polytope.

An alternate representation for a polytope is as the intersection of a collection of half-

spaces. A half-space is defined by a hyperplane (which divides the space into two) and a

surface normal. We refer to such a representation as the Half-space or H-representation

of a polytope. In the H-representation, each irredundant half-space generates a facet


of the polytope and is represented as a linear inequality, e.g., aTx ≤ b. A minimal

H-representation contains only inequalities corresponding to facets of the polytope with

non-zero area, and is unique up to ordering and scaling. The H-representation is generally

preferred to the V-representation for DR because it is the form of almost all load models.

Both V-representations and H-representations may contain redundant information. In

the V-representation, this implies the inclusion of points lying inside, or on the bound-

ary of the polytope. In the H-representation, this implies the inclusion of non-binding

inequalities (i.e. inequalities that do not generate a facet of the polytope as their asso-

ciated hyperplanes either lie outside the polytope or are tangent to it at a single point).

Testing a component of either representation for redundancy can be done with linear

programming [28].

We may write the H-representation of a polytope in matrix form as APx ≤ bP , and

denote it compactly by the matrix-vector pair (AP , bP ). The polytope can also be written

explicitly as P = {x |A1x ≤ b1} We use the term A-matrix to refer to the matrix AP of

a polytope in H-representation.

Example 1: Consider a triangle in R2. In, V-representation, we may denote it by

its set of vertices as XP = {(0, 0), (1, 0), (0, 1)}. In H-representation, we may denote it

by the matrix-vector pair (AP , bP ), where


AP =

−1 0

0 −1

1 1

, bP =

0

0

1

.Note how in this case, the vertices of the polytope are generated by solving the equalities

associated with each inequality. In general, the vertices of a polytope will be generated

from the solution of equalities associated with adjacent facets.

2.3.1 Converting between polytope representations

V-Representations and H-representations of a polytope can be derived from each other.

Conversion from the H-representation to the V-representation is known as vertex enu-

meration; the reverse problem is known as facet enumeration.

In general, vertex enumeration in RD requires the solution of (n−D)! subsets of D

equations, where n is the total number of equations in the H-representation, to generate

a point and then evaluation of all of the remaining equations with the point, to check if

the point is feasible and irredundant.

Facet enumeration, in turn, generates the equation of a hyperplane from D vertices

of the set, and then evaluating whether the hyperplane forms a boundary of the set (i.e.

if all the vertices of the polytope lie only on one side of the hyperplane) .

Unfortunately both of the above problems are, in general, NP-hard [15]. For polytopes

that are bounded, the complexity of vertex and facet enumeration remains an open

problem [38]. No tractable solutions to these problems are currently known for general

polytopes. In addition to the number of computations, another limitation in vertex or

facet enumeration is the amount of memory needed for an algorithm to run.

Additionally, the exact calculation of the volume of a convex polytope is also an

NP-hard problem [37].


2.4 Semidefinite and Second-Order Cone Constraints

A real-valued semidefinite constraint requires that a symmetric matrix function has non-

negative eigenvalues. It is written

A(x) = A0 +n∑i=1

xiAi � 0,

where each Ai is real and square. It is the generalization of linear programming where

the set is defined by the intersection of cones of positive definite matrices as opposed to

the intersection of hyperplanes. A second-order cone constraint is written

‖Ax+ b‖ ≤ cTx+ d,

where A ∈ Rm×n, b ∈ Rm, c ∈ Rn, and d ∈ R. A second-order cone constraint is a special

case of a semidefinite constraint. Specifically, the constraint

(cTx+ d)I Ax+ b

(Ax+ b)T cTx+ d

� 0 (2.1)

implies the above second-order cone constraint, which can be shown using the Schur

complement. Here, I is the identity matrix. Similarly, a linear constraint can be thought

of as a semidefinite constraint where all the Ai’s are real constants times the identity

matrix.

A linear constraint is a special case of a second-order cone constraint when its matrix

A and vector b are zero. Sets defined by second-order cone and semidefinite constraints

do not have V-representations. We refer the reader to [16] for further reading on convex

conic sets.


2.5 Minkowski Sum

The Minkowski sum is a special way of adding together sets, wherein the result is com-

prised of all elements formed by taking any member of the first set and adding it to any

member of the second set. This is easily generalized to the sum of an arbitrary number of

sets. We are interested in the Minkowski sum as when loads are modelled using convex

sets, the Minkowski sum represents the load aggregation.

The Minkowski sum of two polytopes in RD, P1 and P2, is itself a polytope defined

by

P3 = {z | z = x+ y, x ∈ P1, y ∈ P2}. (2.2)

We denote the Minkowski sum using the ⊕ symbol, i.e. P3 = P1 ⊕ P2. We see

that each element in the sum is composed of an element in the first set and an element

of the second set (and can be so de-constructed). This idea may be straightforwardly

generalized to large numbers of arbitrary sets. Should the sets not exist in the same

dimensional spaces, they can be lifted to higher dimensional spaces. It is important to

note that the Minkowski sum is an operation that preserves convexity. Should we have

n polytopes in in RD, P1 . . .Pn we may define the sum M as

M = {y | y =n∑i=1

xi, xi ∈ Pi}. (2.3)

Returning to the two polytope case, if the polytopes have V-Representations X and

Y respectively, then the V-representation of the Minkowski sum can be found by taking

the sum of each vertex pair {x + y | x ∈ X, y ∈ Y }, and removing all redundant points

from the result. This requires O(n2) linear programs, where n is the order of the number

of vertices of each set. Better algorithms take advantage of the fact that the extreme

points of the Minkowski sum in any given direction d, are the sum of the extreme points

of the constituent polytopes in the same direction d [13]. It is also worth noting that the

problem of testing whether a point lies inside the Minkowski sum of two polytopes is a


linear program.

However, if the polytopes are specified in H-representation, the above method is com-

putationally intractable for non-trivial polytopes. This is because it requires performing

the vertex enumeration operation for both polytopes - and performing an additional fi-

nal facet enumeration if the result is desired in the H-representation. As discussed in

Section 2.3, no known polynomial time algorithm exists for vertex enumeration [15,38].

2 3 4 5 6 7

0.01

1

100

Time needed for Vertex Enumeration

Number of dimensions of polytope

Tim

e (s

econ

ds)

Polytope with Random ConstraintsPolytope with only Power ConstraintsPolytope with Power and Energy Constraints

Figure 2.1: The Minkowski sum of two triangles.

For polytopes of practical interest, representing loads with power constraints, power

and energy constraints and polytopes with randomly generated constraints, the (average)

time needed to carry out vertex enumeration on a laptop, using the MPT toolbox [42]

may be seen in Figure 2.1. We see that the time needed for the vertex enumeration op-

eration rises exponentially with dimension. We also see that the more complex (random)

polytopes require more time to be vertex enumerated than the polytopes with only power

constraints (which are simple hypercubes).

As sets defined by second-order cone and semidefinite constraints do not have V-

representations, their exact Minkowski sum cannot be calculated. However they are


still defined by the element-wise sum of every possible pair of elements drawn from the

associated sets. And since they are successive generalizations of linear constraints, the

complexities of their Minkowski sums are at least as hard as the polytope case.

2.6 Outer and Inner approximations

An outer approximation of the set S is any set T for which S ⊂ T ; i.e. the set is

contained inside the approximation. It is also referred to as a necessary approximation.

Equivalently, an inner approximation of the set S is any set T for which T ⊂ S; the

set contains the approximation. It is also referred to as a sufficient approximation.

We will primarily be concerned with developing outer and inner approximations to

the Minkowski sum of various sets. We will generally use the volume ratio of an ap-

proximation to a set to quantify the quality of an approximation. It is worth noting

that as volumes increase exponentially with dimension, volume ratios will also increase

exponentially with dimension. So a good algorithm must show lesser than exponential

growth in error.

2.7 Role within power system operations

Large aggregations of flexible loads are valuable resources for power system operators

and hence should be represented in power system dispatch routines. Multi-period optimal

power flow is a standard approach to dispatching power systems with dynamic constraints

such as ramping and storage capacity limits [60]. Since DR also has dynamic constraints

such as keeping the temperature of a building within a fixed range (a thermostatic load)

and arrival and departure times (an electric vehicle), DR should also be represented

within multi-period optimal power flow.


A simple instance of multi-period optimal power flow is given by

minimizep,θ

F (p)

such that pi(t) ≤ pi(t) ≤ pi(t),

pi(t) =N∑j=1

bij(θi(t)− θj(t)),

i = 1, ..., N, t = 1, ..., D

where N is the number of nodes, D the number of time periods, pi(t) the real power

at node i and time t, and θi(t) the voltage angle at node i and time t. The objective,

F (p), is the total cost of generation over a sequence of time periods, which we assume to

be convex. The first set of constraints limits the power produced or consumed at every

node, and the second set of constraints enforces nodal power balances and the linearized

power flow. Examples of the former are generation limits or load levels. Because the

above optimization has linear constraints and a convex objective, it is easy to solve at

realistic scales encountered in power systems.

A number of studies have recently developed high fidelity representations of flexible

load aggregations in the form of storage with time-varying parameters. For example, [51]

identifies effective storage models for deferrable load aggregations. Lossless storage with

only energy constraints is represented by

ei(t+ 1) = ei(t) + ui(t)

0 ≤ ei(t) ≤ Si(t)

where ei(t) is the state of charge, ui(t) the power injection or extraction, and Si(t) the

energy capacity at storage i and time t. Observe that this storage models fits seamlessly

within multi-period optimally power flow because the constraints are linear.

A number of DR resources could also be represented in optimal power flow this way.


For example, any of the polytope models of Chapter 3 could be straightforwardly inserted

into a multi-period optimal power flow. However, such an approach could introduce

millions of new variables and constraints, which would be unwieldy for system operators

to manage and difficult for load aggregators to communicate to system operators, e.g.,

as part of a bidding process. This is the motivation for representing load aggregations

as generalized storage in [3,48,49,51]. However, this approach is also restrictive because

aggregations of some load types may not be well represented as storage.

Since the load aggregation P is also a (small) polytope in the case of linear constraints

(and otherwise a small convex set), it can be straightforwardly added to the above multi-

period optimal power flow without adding a large number of variables and constraints,

thus preserving its computational tractability.

2.8 Conclusion

Polytopes are geometric objects with several useful properties; most notably that convex

polytopes can be represented via a list of linear inequalities. Polytopes and other convex

sets can be included in multi-period optimal power flow in a manner similar to storage.

Other types of convex constraints include second-order cone constraints and semidefinite

constraints which are successive generalizations of linear constraints seen in polytopes

An alternate representation for polytopes exists as the convex combination of a set

of extreme points (vertices), but the interconversion between these representations is an

np-hard problem. We will make use of polytopes in modelling resources available for DR,

in Chapter 3

The Minkowski sum is a way of adding polytopes (and other sets) together, that

we will make use of for the purpose of load aggregation. While the Minkowski sum is

straightforwardly computed when polytopes are described in terms of their vertices, it

is intractable when they are represented via linear constraints, second-order cone con-


straints or semidefinite constraints. This motivates the search for approximate methods

to find the Minkowski sum.

Chapter 3

Load Models for Demand Response

3.1 Introduction

In this chapter, we survey commonly known H-representations of polytope or second-

order cone descriptions for examples of several standard loads. We do not seek to be

exhaustive, rather to provide the reader with sufficient background to develop their own

load model relevant to their application. Note that some of these models overlap, and that

our intent is to give models of standard load types like electric vehicles and thermostatic

loads that are often discussed in the literature [3, 32,51].

As mentioned earlier, we confine ourselves to compact sets because loads cannot

consume infinite power over a finite number of time periods. For simplicity of exposition,

we assume that the duration of each time period is one. The (constant) power use by a

load over D time periods is represented as a vector of power injections x ∈ RD.

We now define some basic quantities that appear in multiple load types. Denote

(time-varying) maximum and minimum power limits as Pmax(t) and Pmin(t). We use S

to represent the maximum energy usable by the load (or energy storable by the load), and

S0 to represent the initial energy stored by the load. We define a dissipation constant α

to model losses of stored energy over one time period. Finally, we make use of input and

25

Chapter 3. Load Models for Demand Response 26

output efficiencies ηin and ηout. These efficiencies may represent losses between a load

and the electric grid, e.g., AC to DC conversion losses during electric vehicle charging.

We will present examples of three load models that are commonly seen in the litera-

ture: storage-like loads (like electric vehicles), thermostatic loads (like air conditioners)

and deferrable loads , though the constraint types used in them can also be used in de-

scribing other load models. We also describe differential power constraints and apparent

power constraints, which may be added to any of the above load models.

3.2 Storage-like Loads

We first consider loads modelled by storage that have energy and power limits, leak-

age losses, and conversion inefficiencies, for instance, a charging electric vehicle (see,

e.g., [60]). We break the power flow x into the components xin and xout which are power

flows into and out of the load, respectively. The energy constraint is written as:

0 ≤ αjS0 +

j∑t=1

αj−tηinxin(t) +

j∑t=1

αj−tηoutxout(t) ≤ S,

1 ≤ j ≤ D. The power constraints are simply:

0 ≤ xin(t) ≤ Pmax(t) and Pmin(t) ≤ xout(t) ≤ 0

Define the matrix

Γ =

1 0 0 . . . 0

α 1 0 . . . 0

α2 α 1 . . . 0

......

αD−1 αD−2 αD−3 . . . 1


Then, the polytope is defined by the matrices

A1 =

I 0

−I 0

0 I

0 −I

ηinΓ ηoutΓ

−ηinΓ −ηoutΓ

and b1 =

Pmax

0

0

−Pmin

S − αS0

...

S − αDS0

αS0

...

αDS0

.

In this case, we explicitly write the polytope as

P1 =

xinxout

∣∣∣∣∣∣∣A1

xinxout

≤ b1

.

These expressions simplify considerably under perfect efficiencies, i.e., when α = ηin =

ηout = 1. Let L be a D×D lower triangular matrix of 1’s. Then the resulting expressions

are:

A1 =

I

−I

L

−L

and b1 =

Pmax

−Pmin

S − S0

S0

;

P1 = {[x] |A1 [x] ≤ b1} .


3.2.1 Example 1

Consider a lossy storage load in R2, with Pmax = 10, Pmin = −10, α = 0.95, S = 20,

S0 = 6. It can be represented, after simplification, by the Polytope P1 = {x |A1 x ≤ b1},

with A and b as follows:

A1 =

1 0

0 1

−1 0

0 −1

0.95 1

−0.95 −1

and b1 =

10

10

6

10

14

6

We plot the resultant polytope in Figure 3.1.

Figure 3.1: Plot of Storage load polytope


3.3 Thermostatic loads

Thermostatically-controlled loads (TCLs) refer to loads which are operated to maintain

a certain temperature range. Examples of such loads include air-conditioners, electric

heaters, heat pumps and refrigerators. These loads may share constraint types with the

storage-like loads described in the previous section, or with deferrable loads discussed in

the next section.

These types of loads are modelled in [32], which shows how to map parameters associ-

ated with TCLs to those associated with generalized loads. The authors specify TCLs in

terms of a set of parameters χk = (a, b, θa, θr,∆, Pm), where a = 1RC

, b = ηC

, R is thermal

resistance (the rate at which thermal energy is transferred from the controlled space to

the outside), C is thermal capacitance (effective thermal energy storage capacity), Pm is

rated electrical power, η is coefficient of performance (the ratio of thermal energy moved

to the electrical energy needed to do so), θa is ambient temperature, θr is the set-point

temperature, and ∆ is the dead-band.

For a TCL, the dynamics are written in terms of the temperature θ(t) as follows:

θ(t+ 1) = (1− a)θ(t) + aθa − bx(t).

We can expand this equation as:

θ(j) = (1− a)jθ0 + a

j∑t=1

θa(t)(1− a)j−t − bj∑t=1

(1− a)j−tx(j).

The temperature deadband constraint is then given by:

θr −∆ ≤ θ(t) ≤ θr + ∆, 1 ≤ t ≤ D.

Let us denote (1− a)jθ0 + a∑j−1

t=0 θa(t)(1− a)t as θj. Then, the deadband constraint


is equivalently stated as:

θr −∆− θjb

≤ −j∑t=1

(1− a)j−tx(j) ≤ θr + ∆− θjb

, 1 ≤ j ≤ D.

The above inequality is similar to the energy constraint of a generalized storage load,

and can be similarly written in H-representation.

3.4 Deferrable loads

Deferrable loads, like electric vehicles are essentially storage-like loads with arrival and

departure times. Thermostatic loads discussed in the previous section, as well as other

loads, can be made to be deferrable loads with the addition of arrival / departure con-

straints. Some household appliances like washing machines, dishwashers or dryers may

also be well represented by such a model. In this example, we present perfectly effi-

cient deferrable loads, which are restricted to be charging only, have a maximum power

constraint and a single equality energy constraint [51]. We denote the total energy re-

quirement of the load by E.

The constraints for a deferrable load may be written as:

0 ≤ x(t) ≤ Pmax(t), 1 ≤ t ≤ D andD∑i=1

x(t) = E.

The associated matrix representation is:

A1 =

I

−I

1 . . . 1

−1 . . . −1

and b1 =

Pmax

0

E

−E


Arrival and departure constraints are encoded in the vector Pmax by setting

Pmax(t) = 0 for t < ta or t ≥ td,

where ta is the arrival time and td the departure time. Because these constraints are

affine and hence describe polytopes, they may be straightforwardly arranged in H-

representation.

3.4.1 Example 2

Consider a deferrable load in R2, with Pmax = 10, Pmin = −10 and E = 15. We assume

the load is present for both time periods. It can be represented, after simplification, by

the Polytope P2 = {x |A2 x ≤ b2}, with A and b as follows:

A2 =

1 0

0 1

−1 0

0 −1

1 1

−1 −1

and b2 =

10

10

0

0

15

−15

We plot the resultant polytope in Figure 3.2. Note that the equality constraint reduces

the dimension of the polytope hence the result is a line segment.

3.5 Differential power constraints

Differential power constraints can be used to prevent large changes in the power consump-

tion or supply of a load, and are commonly encountered when dealing with industrial

equipment or standby generators. They may be added into any of the above load mod-

els. We denote the maximum allowed bi-directional difference between the power used


Figure 3.2: Plot of Deferrable load polytope

in a period and the power used is a subsequent period as δ > 0. The differential power

constraints may be written as below, and a matrix formulation is easily derived.

−δ ≤ x(t+ 1)− x(t) ≤ δ for 1 ≤ t ≤ D − 1.

3.6 Apparent power constraints

Loads that interface with the power system through inverters, such as electric vehi-

cles [65], can also provide reactive power support. In this case, the charging limit is

replaced by the inverter’s apparent power limit,

xt2i + qt2i ≤ Qt2i , t = 1, ..., T,

where qti is the reactive power provided or consumed by the inverter at time t. This is a

convex quadratic constraint ( a special case of a second-order cone constraint), which is


written in second-order cone form as∥∥∥∥∥∥∥ xti

qti

∥∥∥∥∥∥∥ ≤ Qt

i, t = 1, ..., T.

3.7 Non-convex loads

Finally, it is worth discussing a type of load that does not have a convex formulation.

Consider a load which must use 100 kW of power for a one hour period during a spec-

ified three-hour window. We can represent this load as the union of three points in

R3: {(100, 0, 0), (0, 100, 0), (0, 0, 100)}. Obviously, the resultant set is non-convex, and

would typically be represented with integer constraints. Such a load cannot be simply

aggregated with other polytopes using our subsequent approach. However, polytopic or

other convex relaxations of such load models can often be constructed. For instance, the

above example can be relaxed to∑3

t=1 x(t) = 100, 0 ≤ x(t) ≤ 100 for t = 1, 2, 3, which

is a polytope.

3.8 Conclusion

Polytopes are able to model examples of several commonly seen load models. Examples

of suitable constraint types include constraints on power, energy as well as dead-band

constraints on temperature for thermostatic loads. Polytopic models are able to handle

input / output efficiencies, dissipation and arrival / departure times. Apparent power can

be modeled using second-order cone constraints which are a generalization of the linear

constraints used in polytopes but are also convex. All of the above constraint types are

deterministic; stochastic constraints are discussed in Section 5.3.3.

Chapter 4

Deterministic Load Aggregation

4.1 Introduction

Individual loads in DR programs are generally small compared to the size of resources

normally dispatched by system operators. As discussed in the Introduction and Sec-

tion 2.7, adding a large number of small loads to the scope of their responsibilities is

undesirable. Aggregators act as intermediaries, finding a single compact representation

of these loads for the system operator and then controlling the loads in response to the

system operator’s instructions. For loads specified as polytopes, their aggregate capabil-

ity is exactly described by the Minkowski sum, as observed in [32]; in [3], this quantity

is referred to as the plasticity of the aggregation.

In words, if P1 and P2 are the sets of feasible power profiles of two loads, P3 = P1⊕P2

is the set of feasible power profiles of the aggregation of the two loads.

In this chapter, we develop a novel technique for the (approximate) aggregation of con-

vex, heterogeneous polytopic loads [10]. As discussed in Chapter 2, the exact Minkowski

sum is not, in general, a tractable problem. We will extend this technique to second-order

cone and semidefinite constraints in the following chapter. Our framework will make use

of an approximation to the Minkowski sum of polytopes specified in the H-representation.

34

Chapter 4. Deterministic Load Aggregation 35

The approach is powerful because it captures a wide range of load types, is computa-

tionally tractable because it relies only on linear algebra and linear programming, and

theoretically and empirically accurate in scenarios of practical interest.

We will first present the approximation for homogeneous loads and extend it to het-

erogeneous loads. We will explain a preconditioning procedure that improves accuracy

and develop an algorithm to implement the approximate load aggregation for many loads.

We then discuss how to recover a feasible solution from the result of the aggregation.

Finally we present numerical results to evaluate the accuracy of the approximation and

some theoretical results to describe when it is exact. In particular, we make use of [32] as

a comparison for the performance of our technique on aggregating together large numbers

of thermostatic loads. We also use Monte Carlo methods in evaluating the accuracy of

our load aggregations relative to the exact results, and look at the impact of including a

load aggregation of electric vehicles in multi-period optimal power flow.

4.2 Approximate Load Aggregation

In this section, we present the main technical contribution of this thesis. We develop an

outer approximation of the Minkowski sum of two polytopes, in their H-representation.

We first consider the case of similar polytopes followed later by the case of generic poly-

topes.

4.2.1 Polytopes with the same A-matrix

Consider the following pair of polytopes in H-representation:

P1 = {x |A1x ≤ b1} and P2 = {y |A2y ≤ b2}.

We would like to find an approximate representation for the polytope P3 = P1⊕P2 =

{z | z = x+ y, x ∈ P1, y ∈ P2}, the Minkowski sum of P1 and P2.


Proposition 1 (Outer Approximation). Consider two polytopes with the same A-matrices,

i.e. A1 = A2 = A. The polytope P4 = {z |Az ≤ b1+b2} ⊂ RD is an outer approximation

to P3, the Minkowski sum of P1 and P2.

Proof. We must show P1 ⊕ P2 ⊆ P4. Let z ∈ P1 ⊕ P2. Then there exists x1 ∈ P1

and x2 ∈ P2 such that z = x1 + x2. Then we have Az = Ax1 + Ax2 ≤ b1 + b2. Hence

z ∈ P4.

We will refer to the polytope P4 = {z |Az ≤ (b1 + b2)} as the outer Minkowski ap-

proximation. We remark that the outer Minkowski approximation could also be referred

to as a relaxation of the exact Minkowski sum.

Example 1: Consider the polytopes P1, P2 and P3 shown in Figure 4.1, where

P3 is the Minkowski Sum of P1 and P2. All are triangles in R2. In, V-representation,

X1 = {(1, 1), (2, 1), (1, 2)}, X2 = {(2, 1), (4, 1), (2, 3)} and X3 = {(3, 2), (6, 2), (3, 5)}.

The reader can see that the vertices of P3 are the sum of vertices of P1 and P2, and

that other points generated by the sum of points inside P1 and P2 lie within P3. The

Figure 4.1: The Minkowski sum of two triangles.


H-representations of P1 and P2 are:

A1 =

−1 0

0 −1

1 1

, b1 =

−1

−1

3

,

A2 =

−1 0

0 −1

1 1

, b2 =

−2

−1

5

.Since A1 = A2 = A, Proposition 1 may be used to find the outer approximation, which

we denote P4 and is given in H-representation by

A =

−1 0

0 −1

1 1

, b4 =

−3

−2

8

.

It can be verified that this is the exact Minkowski sum.

Remark 1 (Significance of Result). While the above result is very simple, we have been

unable to find reference to it in the literature. We posit this is due to research interest in

polytopes being generally confined to low-dimensional polytopes in the V-representation.

In Section 4.2.2, we will extend this to general polytopes, and in Chapter 5, we will extend

this to sets containing second-order cone and semidefinite constraints. This is the first

presentation of a technique capable of finding an approximate Minkowski sum for sets

containing generic second-order cone and semidefinite constraints.

4.2.2 Extension to general polytopes

The approximation developed in Section 4.2.1 is limited to Minkowski Sums of polytopes

that have the same A-matrices, which restricts its applicability to aggregations of loads


of the same type. We now extend this formulation to arbitrary polytopes in RD, which

broadens its applicability to aggregations containing many different types of loads.

Consider two polytopes in their minimum H-Representation, P1 and P2, described

by the matrix-vector pairs (A1, b1) and (A2, b2). An exact, alternate H-representation for

P1 and P2 can be constructed in terms of the matrix-vector pairs (A′, b′1) and (A′, b′2),

where A′, b′1, and b′2 are new matrices which we describe below.

Observe that P1 can be described in set notation as an intersection of half-spaces,

each of which is defined by a linear inequality:

P1 =N⋂i=1

{x | a1(i)Tx ≤ b1(i)},

where

A1 =

a1(1)T

...

a1(N)T

and b1 =

b1(1)

...

b1(N)

.Here, a1(1)T . . . a1(N)T are row-vectors in RD and b1(1) . . . b1(N) are scalars. From

this expression, we see that:

• The rows of the matrix-vector pair (A1, b1) can be arbitrarily reordered without

changing the polytope.

• We can add an additional linear constraint to the polytope (i.e., an additional row

to the matrix-vector pair), a(N + 1)Tx ≤ b(N + 1), provided that the following

inclusion is satisfied:

{x |A1x ≤ b1} ⊆ {x | a(N + 1)Tx ≤ b(N + 1)} (4.1)

Equation (4.1) states that the polytope P1 lies inside the half-space defined by a(N +

1)Tx ≤ b(N+1). We refer to such inequality constraints as redundant constraints because

they can be added to or eliminated from a polytope without changing it [6].


Our subsequent approximation attains the highest accuracy when redundant con-

straints with the smallest possible b1(N + 1) are used. For an arbitrary row-vector,

a(N + 1)T , we can find the smallest constant b1(N + 1)∗ that satisfies Equation (4.1) by

solving the linear program:

b1(N + 1)∗ = maximizex

a(N + 1)Tx

subject to A1x ≤ b1.(4.2)

For this choice of b(N + 1)∗, the equality a(N + 1)Tx = b(N + 1)∗ describes a hyperplane

that touches P1. This idea is illustrated in Fig 4.2.

Figure 4.2: Placement of a hyperplane to touch a convex polytope

Thus, if a constraint a2(M)Tx ≤ b2(M) is present in the H-representation of polytope

P2 but not P1, we can add it as the N + 1th row in A1, and find the associated scalar

b1(N + 1)∗ using Equation (4.1) (or vice-versa). This constraint will then touch P1.

By adding redundant constraints as described above and reordering, we construct

alternate representations for polytopes P1 and P2 as the matrix-vector pairs (A′, b′1) and


(A′, b′2). These representations have the same A-matrices, and therefore we can obtain

their outer Minkowski approximation via Proposition 1. Consider the case where the two

polytopes have no constraints in common. Then, the Minkowski Sum S is given by:

s ∈ S

∣∣∣∣∣∣∣A1

A2

x ≤b1 + b∗2

b∗1 + b2

. (4.3)

where the entries of b∗1 and b∗2 are found by solving linear programs over P2 and P1

respectively, using Equation 4.2.

It should be noted that if polytopes P1 and P2 have m1 and m2 constraints, respec-

tively, with c constraints in common, then the outer Minkowski approximation will have

m1 +m2 − c constraints; i.e. it’s A-matrix will have m1 +m2 − c rows.

Example 2: Suppose we have two loads with A-matrices

A1 =

AaT1

and A2 =

AaT2

,and P1 = {x |A1x ≤ b1} and P2 = {x |A2x ≤ b2}. Suppose further that b1 ∈ RN and

b2 ∈ RN , and define

b′1(N + 1)∗ = maximizex

aT2x and

subject to A1x ≤ b1

b′2(N)∗ = maximizex

aT1x

subject to A2x ≤ b2


b2(N + 1)∗ is defined analogously. Let

A′ =

A

aT1

aT2

, b′1 =

b1(1)

...

b1(N)

b′1(N + 1)∗

, b′2 =

b2(1)

...

b2(N − 1)

b′2(N)∗

b2(N)

.

Then P1 = {x |A′x ≤ b′1} and P2 = {x |A′x ≤ b′2}. Using Proposition 1, we obtain

P4 = {x |A′x ≤ b′1 + b′2} as an outer approximation of the Minkowski sum of P1 and P2.

Example 3: Consider now the Minkowski sum of a storage load and a deferrable

load. For convenience, we make use of the loads described in Section 3.2.1 and Section

3.4.1.

We may note that while they share certain constraints, they will each need con-

straints to be added in, using the techniques of this section. We also will need to adjust

certain b-vector values in accordance with the Preconditioning procedures described in

the following section. On performing the necessary linear programs, we obtain that:

A1 =

1 0

0 1

−1 0

0 −1

0.95 1

−0.95 −1

11

−1− 1

and b1 =

10

10

6

10

14

6

14.5

6.3


A2 =

1 0

0 1

−1 0

0 −1

0.95 1

−0.95 −1

1 1

−1 −1

and b2 =

10

10

−5

−5

14.75

−14.5

15

−15

On adding the associated b-vectors, we obtain the H-Representation of the Outer

Minkowski approximation, which we denote as R. Then R = {x|AR x ≤ bR}

AR =

1 0

0 1

−1 0

0 −1

0.95 1

−0.95 −1

1 1

−1 −1

and bR =

20

20

1

5

28.75

−8.5

29.5

−8.7

We plot the result in Figure 4.3. Note that by Corollary 1, the approximation is exact

for this case as it is in R2.

4.3 Preconditioning

In Section 4.2.1, we first dealt with the approximate Minkowski sum of polytopes having

irredundant H-representations but identical A-matrices. Next, in Section 4.2.2, we ex-


Figure 4.3: Plot of Minkowski sum of Deferrable load and Storage load

tended this to polytopes having irredundant H-representations but arbitrary A-matrices,

by adding new, redundant constraints.

In this section, we consider polytopes that have arbitrary redundant H-representations,

i.e. at least one row in the matrix-vector pair (A, b) is a redundant constraint. We retain

the requirement that the polytopes in question be closed and convex.

Given a polytope P1, with H-representation (A1, b1), we can test the kth constraint

(the kth row) for redundancy by solving the following linear program:

maximize a(k)Tx

subject to A1x ≤ b1.(4.4)

If the solution to this linear program is strictly less than b(k), then a constraint is

redundant and slack (not touching the polytope). Such a constraint can be dropped from

the H-representation without changing the polytope. Alternately, it can be left in if b(k)

is replaced with the result of the linear program, which eliminates the slack.

The importance of this test arises because the presence of slack constraints in the

description of polytope can reduce the accuracy of the outer Minkowski approximation.


We illustrate this by means of the following example in R2.

Example 4: Consider the polytopes P1 and P2 shown in Figure 4, both in R2. In, V-

representation, we may denote P1 by its set of vertices as X1 = {(0, 0), (2, 0), (0, 2)}, and

P2 by its set of vertices as X2 = {(0, 0), (1, 0), (0, 1), (1, 1)}. The polytopes are plotted in

Figure 4.4.In H-representation, we may denote them by the matrix-vector pairs (A1, b1)

and (A2, b2), where:

A1 =

−1 0

0 −1

1 1

, b1 =

0

0

2

,and P1 = {x |A1x ≤ b1},

A2 =

−1 0

0 −1

1 0

0 1

, b2 =

0

0

1

1

,

and P2 = {x |A2x ≤ b2}.

We may find an alternate representation of these polytopes, as described in as de-

scribed in section 4.2.2, and compute the outer Minkowski approximation. The alternate

H-representations of P1 and P2 and the H-representation of the outer Minkowski ap-

proximation, P3 are:


Figure 4.4: Polytopes P1 (in light gray) and P2 (in dark gray) for Example 2.

A∗1 =

−1 0

0 −1

1 1

1 0

0 1

, b1 =

0

0

2

2

2

,

and P1 = {x |A∗1x ≤ b1},

A∗2 =

−1 0

0 −1

1 1

1 0

0 1

, b2 =

0

0

2

1

1

,

and P2 = {x |A∗2x ≤ b2},


A3 =

−1 0

0 −1

1 1

1 0

0 1

, b3 =

0

0

4

3

3

,

and P3 = {x |A3x ≤ b3}.

In this case, the outer Minkowski approximation is exact, as the reader may ver-

ify. The polytope P3 is plotted in Figure 4.5. The fourth and fifth rows in the H-

representation of P1 and the third row of P2 correspond to redundant constraints that

are touching their respective polytopes.

Now consider the polytopes P4 and P5 with the following H-representations:

A4 =

−1 0

0 −1

1 1

1 0

0 1

, b4 =

0

0

2

10

10

,

and P4 = {x |A4x ≤ b4},

A5 =

−1 0

0 −1

1 1

1 0

0 1

, b5 =

0

0

12

1

1

,

and P5 = {x |A5x ≤ b5}.


P4 and P5 are identical to P1 and P2 respectively, but they have redundant con-

straints. However unlike the redundant constraints in alternate representations of P1

and P2, these constraints are no longer touching the polytopes. If the outer Minkowski

approximation of the Minkowski sum of P4 and P5 is directly computed, without pre-

conditioning, we obtain P6 whose H-representation is:

A6 =

−1 0

0 −1

1 1

1 0

0 1

, b6 =

0

0

14

11

11

,

and P6 = {x |A6x ≤ b6}.

In Figure 4.5, we see the approximation P6 with the preconditioned approximation

P3 (which also happens to be the exact result).

Figure 4.5: Polytopes P3 (in dark gray) and P6 (in light gray) for Example 2.

Here we see that the approximate Minkowski sum (without preconditioning) of the

two polytopes is much larger than the exact sum, whereas implementation of the precon-


ditioning step results in the approximation being exact. While the improvement in the

general case is unlikely to be as dramatic, a performance gain should be seen by using

this technique.

4.4 Load aggregation algorithm

We now present our procedure as an algorithm for approximately representing aggrega-

tions of loads described by polytopes. As a reminder, a conceptual representation of the

place of an aggregator in the power system can be seen in Figure 1.1.

1. Input: N load polytopes overD time periods in H-representation: P1,P2, . . . ,PN ∈

RD. Each polytope is described by an arbitrary number of constraints.

2. Search through the A-matrices of all N polytopes and make a list of every unique

row. This is a polynomial-time sorting operation. The A′ matrix consists of all

unique rows, and is in Rc×N .

3. For all N polytopes and all c unique constraints, run linear programs to find facets

adjacent to the polytope, and construct the vectors b′1, ..., b′N. The total number of

linear programs run is upper bounded by cN , and can be substantially less if the

A-matrices contain many common rows.

4. Output: By Proposition 1, the polytope {x |A′x ≤∑N

i=1 b′i} is an outer approxima-

tion of the Minkowski sum of the N polytopic loads.

Linear programs have polynomial time complexity [12]. As our algorithm invokes a

polynomial number of LPs, its complexity also grows polynomially with the number of

loads and dimensions.


4.4.1 Recovering a feasible solution

The algorithm given in Section 4.4 is a tractable approach to generating low-dimensional

approximations of Minkowski sums. The resulting approximation can be used in system

operator routines such as multi-period optimal power flow, as described in Section 2.7.

Unfortunately, because of the nature of outer approximations, the resulting solution may

be infeasible for the load aggregation. We now give a simple method for recovering

feasible solutions.

Suppose that the load aggregation is requested to provide z ∈ RD, where z is a vector

containing the power injection or extraction for each time period, t = 1, ..., D. The below

quadratic program produces a feasible power vector for each load in the population.

minimizexi, i=1,...,N

∥∥∥∥∥z −N∑i=1

xi

∥∥∥∥∥subject to Aixi ≤ bi, i = 1, ..., N.

The resulting solution, xi, i = 1, ..., N , is feasible by virtue of the problem’s constraints.

Although N may be very large, this is a linearly-constrained quadratic program and

hence can be solved at large scales [16]. Note that the load aggregator and not the

system operator would solve this problem.

4.5 Examples

4.5.1 Numerical Examples

In this subsection, we numerically evaluate the accuracy of the outer Minkowski ap-

proximation for two general classes of loads, thermostatic loads and generalized energy

storage. As our load aggregations are closed polytopes, they can be characterized by

volumes. The outer Minkowski approximation contains the exact Minkowski sum and


therefore always has larger volume; when their volumes are identical, the approximation

is exact. The ratio of volumes of two polytopes hence measures absolute accuracy when

one polytope is the exact Minkowski sum, and relative accuracy when both polytopes

are approximations. We thus use such volume ratios to describe the error in the outer

Minkowski approximation.

However, the exact computation of volume of a high-dimensional polytope is an NP-

hard problem [24]. We thus make use of a Monte-Carlo method for volume estimation.

For the polytopes in question, we define a bounding box of known volume in RD and

uniformly sample this box. The fraction of points inside the polytope yields an estimate of

the volume. We perform these, and other simulations in this dissertation, in Matlab [61].

Thermostatic Loads

Models for thermostatic loads were described in Section 3.3. We generate sets of random-

ized parameters to describe 1000 distinct loads; the mean values (µ) of the parameters

varied are: the thermal capacitance (2 kWh/◦C), the thermal resistance (2◦C/kW ), the

rated electrical power (5.6 kW), the coefficient of performance (2.5), the temperature set-

point (22.5◦C) and the temperature deadband (0.3◦C), which are taken from [32]. Each

of the load parameters are drawn from a uniform distribution from between 0.9µ− 1.1µ

for a low heterogeneity scenario, and from between 0.8µ− 1.2µ for a high heterogeneity

scenario. Additionally the starting temperature of each load is drawn from a uniform

distribution over the deadband.

We consider a 1-hour time period and look at the performance of the approximation

as the interval of discretization is varied, e.g. two, 30-minute slots, four, 15-minute slots,

etc. When computing the outer Minkowski approximation (denoted as OM), each load

is first approximated by an equivalent load whose dissipation constant is the mean of the

set; this approximation is computed as an outer (necessary) approximation; this produces

a more concise approximation but does add error.


We also compute an outer generalized battery approximation (a necessary approxima-

tion) that we denote as GB-N, and an inner, maximum charging rate sufficient (denoted

as GB-S) generalized battery approximations for the aggregation of these loads, using

the methods described in [32]. Note that [32] also addresses the control of a collection

of thermostatic loads for regulation on very fast timescales, which is beyond our scope;

however the aggregate models developed in it are useful for comparison given that exact

results cannot be obtained, but are intended for a different purpose than our approach.

These battery approximations are modelled as polytopes, as explained in Section 3.2, af-

ter which their volumes are found. One billion points are generated for each Monte-Carlo

volume estimation case.

In Figure 4.6, we plot the volume ratios: OM / GB-N (our outer approximation /

their outer approximation) and GB-S / GB-N (their inner approximation / their outer

approximation) as a function of the number of slots used for discretisation of the 1-hour

period, for both low (Low-h) and high (High-h) heterogeneity scenarios.

2 4 6 8 10 12 14 160

0.2

0.4

0.6

0.8

1

Number of slots in 1−hr period, used for discretization

Vol

ume

Rat

io o

f App

roxi

mat

ions

OM / GB−N, Low−hGB−S / GB−N, Low−hOM / GB−N, High−hGB−S / GB−N, High−h

Figure 4.6: Volume comparison of thermostatic load aggregations.

We find that for both scenarios, the size of the OM approximation is smaller than


the GB-N from [32] (and, by construction, larger than the GB-S). Hence the OM ap-

proximation is more accurate than the GB-N approximation, by approximately a factor

of 1.5− 2 depending on the amount of load heterogeneity. Additionally, we see that the

performance of the OM approximation improves vis-a-vis the GB-N approximation in

the higher heterogeneity scenario.

We now comment on computation time. The computation times for the general-

ized batteries in [32] are effectively zero because they are analytically determined. Our

approximations required no more than 0.2 seconds to compute, with the bulk of the

computation time taken by linear programs.

Storage-like Loads

Models for storage-like loads such as electric vehicles were described in Section 3.2. Here,

we focus on non-dissipative storage-like loads that are fully present for the period of

aggregation and have input/output efficiencies of unity.

We take randomized parameters for 2000 loads, and use them to compute 1000 pair-

wise sums (and/or approximations). We carry out this process for dimensions from R2

to R20 by instantiating loads for time intervals of D = {2, ..., 20} hours, with hourly

slots. The loads have power limits uniformly distributed between 30 and 70, and energy

capacities that are uniformly distributed with between 120 and 280; finally, the initial

states of charge are uniformly distributed from 0 to the energy capacity.

We consider two cases: a population with only storage-like loads, and a population

with half storage-like and half thermostatic loads. We use MPT [42] to compute the

volumes of the approximate and exact pairwise sums up to R6 in the first case and R5

in the latter, beyond which the computations become intractable. We then compute the

average over the 1000 cases of the ratio of the exact volume and that obtained by the

OM approximation. In the first case, we also use a Monte-Carlo method to estimate

the volume of the approximation up to R20, which we use to validate the results from


MPT and to examine the behavior of the approximation with increase in dimension. We

comment that approximately 2.74% of the computed data had to be thrown out because

of numerical errors in computations by the MPT toolbox. We plot the results in Figures

4.7 and 4.8.

5 10 15 2010

0

1010

1020

1030

1040

1050

Dimension

Mea

n P

olyt

ope

Vol

ume

Figure 4.7: Approximation volume for aggregations of storage loads, up to R20.

As observed in Figure 4.7, the mean volume of the approximate aggregation scales

exponentially with dimension as expected; this appears as linear on a semilogarithmic

plot.

The error (defined as the ratio of the volume of the approximation to the volume

of the exact result)is computable only up to R6 in the first case and R5 in the second.

We see in the top subplot of Figure 4.8 that the error remains below 0.7% for those

dimensions, and grows sub-linearly, indicating that the OM approximation continues to

achieve low errors in higher dimensions. In the bottom subplot, the error grows more

quickly, and is approximately 13% in dimension five. This confirms intuition that the

approximation performs better for loads that are more nearly uniform.


2 3 4 5 61

1.005

1.01

Dimension

Mea

n V

olum

e E

rror

2 3 4 5 61

1.1

1.2

Dimension

Mea

n V

olum

eE

rror

Storage Load + Thermostatic Load

Storage Load + Storage Load

10% Error

No Error

No Error

1% Error

Figure 4.8: Approximation error for aggregations of storage loads, up to R6.

Optimal power flow

In this section, we implement our approximation within multi-period optimal power flow,

as discussed earlier in Section 2.7. We consider the examples utilizing the IEEE 9-bus,

30-bus and 39-bus systems.

IEEE 9-bus system

There are D = 12 time periods. Line susceptances, generation costs, and the load at

buses five, seven, and nine are taken from the MATPOWER implementation of the IEEE

9-bus test case [69]. These parameters are constant over all time periods. We add an

additional load at node eight, which is equal to 5t, where t ∈ {1, ..., 12} is the current

time period. This encourages the optimization to use the vehicle flexibility to shift load

toward earlier time periods.

A collection of n = 1, 000 electric vehicles charges at node four. Each vehicle has an

arrival time, ai, which is sampled uniformly from the set {1, ..., D − 1}, and a departure

time, di, that is sampled uniformly from the set {ai + 1, D}. Each vehicle has an energy


need, Ei that is drawn uniformly from[0, E

]and that must be satisfied by di. Each

vehicle has a maximum charging limit, given by max {Ui, Ei/(di − ai + 1)}, where Ui is

drawn uniformly at random from the interval [0, Ei]. This ensures that all individual

vehicles can be feasibly charged.

We compare three cases:

1. (Outer) The outer approximation of the Minkowski sum represents the vehicle

aggregation at node four. In this case, there are 602 variables and 252 equality

constraints (in standard form). This may result in an infeasible solution.

2. (Exact) The exact representation of every electric vehicle in the aggregation is

included. In this case, there are 26,576 variables and 12,240 equality constraints

(in standard form). This attains the true optimal solution.

3. (Nearest feasible solution) The nearest feasible solution to the solution from

the outer approximation, in the vehicles’ feasible set, is found as described in Sec-

tion 4.4.1. The power extraction at node four is then held fixed at this feasible

aggregate charging trajectory. This results in a suboptimal, feasible solution.

We use CVX [30] with the solver Mosek [1] to solve each optimization problem.

In Figure 4.9, we plot the average multiperiod optimal power flow objective as func-

tions of E. Averages are computed over 100 trials. Note that the large number of vehicles

within each trial reduces the variance as well, and is the reason for not using a larger

number of trials.

We observe from the results in Fig. 4.9 that the three approaches perform nearly

identically. Table 4.1 shows that the objectives resulting from each approach are indeed

very similar, indicating that the outer approximation achieves less than one percent error

even in twelve dimensions.


0 0.5 1 1.5 27

8

9

10

11x 104

E

Obj

ectiv

e

OuterExactNearest feasible solution

Figure 4.9: The multiperiod optimal power flow objective attained using the outer ap-proximation, exactly representing all loads, and using the nearest feasible solution to theouter approximation in the load feasible set for the IEEE 9-bus system.

Table 4.1: Multiperiod optimal power flow objectives resulting from the outer approxi-mation, exactly incorporating each load, and minimizing the distance between the outerapproximation solution and the feasible set of the loads for the IEEE 9-bus system.

E Outer Exact Nearest feasible solution1 86,333 86,427 86,4642 101,722 101,852 101,917

IEEE 30-bus system

There are D = 12 time periods. Line susceptances, generation costs, and all loads are

taken from the MATPOWER implementation of the IEEE 30-bus test case [69]. These

parameters are constant over all time periods. We add an additional load at node nine,

which is equal to 5t, where t ∈ {1, ..., 12} is the current time period. This encourages the

optimization to use the vehicle flexibility to shift load toward earlier time periods.

A collection of n = 1, 000 electric vehicles charges at node six. Each vehicle has an











aggregation at node four. In this case, there are 2,162 variables and 936 equality







tion 4.4.1. The power extraction at node six is then held fixed at this feasible






number of trials.






0 0.5 1 1.5 27000

8000

9000

10000

11000

12000

E

Obj

ectiv

e


Figure 4.10: The multiperiod optimal power flow objective attained using the outerapproximation, exactly representing all loads, and using the nearest feasible solution tothe outer approximation in the load feasible set for the IEEE 30-bus system.



IEEE 39-bus system

There are D = 12 time periods. Line susceptances, generation costs, and the loads at

all buses except bus five and bus ten are taken from the MATPOWER implementation

of the IEEE 39-bus test case [69]. These parameters are constant over all time periods.

We add an additional load at node ten, which is equal to 10t, where t ∈ {1, ..., 12} is the

current time period. This encourages the optimization to use the vehicle flexibility to

shift load toward earlier time periods.

A collection of n = 1, 000 electric vehicles charges at node five. Each vehicle has an











aggregation at node four. In this case, there are 2,594 variables and 1,104 equality







tion 4.4.1. The power extraction at node five is then held fixed at this feasible






number of trials.






0 2 4 6 8 103.4

3.6

3.8

4

4.2x 105

E

Obj

ectiv

e


Figure 4.11: The multiperiod optimal power flow objective attained using the outerapproximation, exactly representing all loads, and using the nearest feasible solution tothe outer approximation in the load feasible set for the IEEE 39-bus system.



4.5.2 Analytical Results

In this section we present three useful analytical results regarding the exactness of the

outer Minkowski approximation when applied to specific load classes.

Exactness in R2

Consider two arbitrary convex polytopes in R2. We now show that the exact Minkowski

sum and approximate Minkowski sum are identical. To do so we make use of the fact

that a unique ordering exists for facets (and hence vertices), and that every vertex in the

sum can hence be mapped to the sum of vertices in the constituent polytopes.

Corollary 1. Let A = {x | Hx ≤ b} ⊂ R2 and B = {x | Hx ≤ d} ⊂ R2. Then


C = {x | Hx ≤ b+ d} is equivalent to the Minkowski sum, A⊕B.

Proof. We know from Proposition 1 that A ⊕ B ⊆ C. We now show that A ⊕ B ⊇ C

by showing that any vertex of C is also a vertex of A ⊕ B. Because C and A ⊕ B are

polytopes in R2, their vertices only occur at the intersections of adjacent pairs of half-

planes. Order the half-planes of C clockwise i = 1, ..., n, where we take n+ 1 to refer to

1. Then the ith vertex of C is given by the intersection of the hyperplanes H ix = bi + di

and H i+1x = bi+1 + di+1. We can write this vertex

H i

H i+1

−1 bi + di

bi+1 + di+1

=

H i

H i+1

−1 bi

bi+1

+

H i

H i+1

−1 di

di+1

.The right hand side of the equality is the sum of a vertex in A with one from B, which

must be in A⊕B.

The presence of “tight” redundant constraints means simply that the number of

(unique) vertices is now less than the number of facets. The proof follows as above, by

taking the facet associated with the redundant constraint to be of infinitesimal length.

Constraints may be tightened without changing the polytope. Note that Corollary 1 also

applies to polytopes with different A-matrices via the procedure given in Section 4.2.2.

Loads with only power constraints

Let us consider loads with power limit vectors in RD, Ph and Pl for all D time periods,

such that Pl ≤ x ≤ Ph (different power limits for each time period). They may be


represented by the following simple D-dimensional hypercube:

I

−I

x ≤ Ph−Pl

.

Corollary 2 (Exactness of outer approximation for hypercubes). Consider two hypercube

loads defined by power limit vectors, Ph1 and Pl1 for the first and Ph2 and Pl2 for the

second. The outer Minkowski approximation to the Minkowski sum of these loads is

exact, and is given by: I

−I

x ≤Ph1 + Ph2

Pl1 + Pl2

.Proof. The exact Minkowski sum of two hypercubes can be computed by taking the

convex hull of the sums of all vertex pairs. Straightforward calculation gives the vertex

set⋃Di=1(Pl(i) +Ph(i)). The outer Minkowski approximation is the same hypercube.

Loads with single equality energy constraint and positive power constraints

Let us consider loads present for all D time periods, such that 0 ≤ x (power constrained

to be positive) and with a single equality energy constraint∑D

t=0 xt = E. They may be

represented by a simplex facet of dimension D − 1:

−I

1 1 . . . 1

−1 −1 · · · − 1

x ≤

0

E

−E

.

Corollary 3 (Exactness of outer approximation for simplex facets). Consider two loads

constrained to have positive power, with energy requirements E1 and E2 respectively. The

outer Minkowski approximation to the Minkowski sum of these loads is exact, and is given


by: −I

1 1 . . . 1

−1 −1 · · · − 1

x ≤

0

E1 + E2

−E1 − E2

.Proof. The exact Minkowski sum of two simplex facets can be computed by taking the

convex hull of the sums of all vertex pairs. Straightforward calculation gives the vertex set

(E1+E2, . . . 0), (0, E1+E2, . . . 0) . . . (0, . . . E1+E2). The outer Minkowski approximation

is the same simplex facet.

4.6 Conclusion

In this chapter, our main results on the approximate aggregation of polytopic sets are

presented. Polytopic sets can be used in describing loads for demand resources, as seen

in Chapter 3. We recall that this type of formulation allows for a variety of linear

constraints, including those on power, energy, arrival/departure and more. We consider

load aggregation through the Minkowski sum, as described in Chapter 2.

Our main result is for polytopes in the H-representation, i.e. sets that obey a linear

relation Ax ≤ b. We find that for polytopes with the same A-matrices, an outer approx-

imation to the Minkowski sum may be obtained by adding together the b-vectors of the

constituent polytopes. We also find a technique to extend this to arbitrary polytopes by

generating equivalent representations that have the same A-matrices. This is done by

adding redundant constraints to each polytope, which are placed touching the polytope

via maximization of the vector associated with the constraint over the polytope. To

generate a redundant constraint on a polytope requires the solution of a linear program.

Following this, we describe a preconditioning procedure where we check whether all

the existing constraints in the polytope are irredundant, and hence touching it. Checking

constraints for redundancy may also be done by solving a linear program. We show how


the use of this process can lead to improved accuracy of the approximation, though an

example. We then develop a general algorithm for polytopic aggregation, that we present

in Section 4.4.

One concern that arises in the use of an outer approximation is that a result obtained

via the use of the approximation may not actually be feasible on the actual Minkowski

sum. We describe how the nearest feasible solution inside the actual polytope of the

Minkowski sum, can be found via the solution of a quadratic program. This is a step

that can be carried out by the load aggregator.

We perform three numerical studies to examine the performance of the approximation.

In the numerical studies, we use the volume of the aggregation as the metric to describe

its relative size. First, we look at the aggregation of a large population of thermostatic

loads and compare this to a result in [32]. We see that our result lies within their

inner and outer approximations (as expected), and is smaller in size than their outer

approximation. Second, we consider the size of pairwise sums of homogeneous loads

(storage loads), and heterogeneous loads (storage and thermostatic loads). We see good

performance for error in both cases, though with greater error for the case when the loads

are heterogeneous. Finally, we consider the use of an aggregation of electric vehicles inside

of multi-period optimal power flow and see that the result of an optimization are almost

identical when the approximation is used, in comparison to both the exact result and a

recovered, feasible solution.

We also examine two theoretical cases where the approximation is exact. We are

able to show that our approximation is identical to the exact sum in the cases where the

polytopes are hypercubes, as well as when the polytopes are sets in R2. Future work on

this subject could look for additional cases of exactness as well as the development of

bounds on the size of the approximation.

Chapter 5

Stochastic Load Aggregation

5.1 Introduction

We have thus far only considered deterministic models of loads. However, considerable

uncertainty exists in regards to the capabilities of these devices. Such uncertainty may

arise from inadequate characterization of devices, lack of knowledge regarding device

scheduling, external factors such as weather, and more. This uncertainty reduces the

utility of deterministic DR aggregations. In the absence of good probabilistic models to

represent the capabilities of DR aggregations, they are underutilized so as to ensure that

they are not over-committed.

Various approaches exist toward incorporating uncertainty into this problem. One

method is to assign a simple probability of use to each time period that a resource may

be available. In such a case, the expected value of power consumption may be obtained

by summing the expected power consumption of each resource; this approach is taken

in [36].

An alternate approach is to develop probabilistic models of each individual resource,

as in [47], who do this for thermostatic loads and electric vehicles. One may then for-

mulate various stochastic optimization problems using these models or representations

65

Chapter 5. Stochastic Load Aggregation 66

of aggregations of these models. [47] does so using a scenario approach. Such models

are also seen in [57] which is concerned with control at the individual residence level via

home automation systems.

One may also avoid modelling resource-level uncertainty completely and instead con-

sider this through uncertainty associated with the price-elasticity of demand. Here, it is

assumed that resources will act in response to changes in price. Using this, the overall

capabilities of a DR set may be included into a robust optimization problem as in [66]

or in [20].

Finally, [44] incorporates this uncertainty into a reliability model, where resources are

represented via Markov chains. Transition probabilities may be assigned to the likelihood

that a resource responds as expected. This method also allows for a grid operator to

consider the overall capabilities of a set of resources.

In this chapter, we seek tractable outer approximations of the Minkowski sums of sets

defined by second-order cone, and semidefinite constraints, for the use in stochastic load

aggregation [9]. We make use of chance constraints in representing the uncertainties we

wish to model. Chance constraints have already been utilized in the analysis of power

systems, as in [14]. To our knowledge, they have not been used for aggregating DR

resources.

Sets defined by second-order cone and semidefinite constraints do not have V-representations,

hence no simple algorithm exists to find their Minkowski sum. Since they are successive

generalizations of linear constraints, the complexities of their Minkowski sums are at

least as hard as the polytope case.


5.2 Outer Approximations

5.2.1 Semidefinite constraints

Consider two sets defined by the semidefinite constraints: A = {x | A(x) � 0} ⊂ Rn and

B = {x | B(x) � 0} ⊂ Rn, where A(x) = A0 +∑n

i=1 xiAi and B(x) = B0 +

∑ni=1 x

iBi.

We assume that all matrices have the same dimension, which can be imposed with zero

padding and hence does not reduce generality. We have the following original result:

Proposition 2. Suppose Ai = Bi for i = 1, ..., n. Then the set

C =

{x

∣∣∣∣∣ A0 +B0 +n∑i=1

xiAi � 0

}⊂ Rn

is an outer approximation of A⊕ B.

Proof. Suppose x ∈ A and y ∈ B. Since the sum of two semidefinite matrices is semidef-

inite,

A0 +B0 +n∑i=1

(xi + yi)Ai � 0.

By definition, there exists z = x+ y ∈ C. Hence, any element in A⊕ B is in C.

Proposition 2 gives an outer approximation when all matrices except for A0 and B0

are identical. We now extend Proposition 2 to the case when all matrices are distinct.

Let λmin(A) denote the smallest eigenvalue of A. Define the matrices

Ci =

Ai 0

0 Bi

for i = 1, ..., n. Also define

A′0 =

A0 0

0 B0

, B′0 =

A0 0

0 B0

,


where B0 satisfies

minx∈A

λmin

(B0 +

n∑i=1

xiBi

)≥ 0, (5.1)

and A0 satisfies

minx∈B

λmin

(A0 +

n∑i=1

xiAi

)≥ 0. (5.2)

We can then write A equivalently as

A =

{x

∣∣∣∣∣ A′0 +n∑i=1

xiCi � 0

},

and B equivalently as

B =

{x

∣∣∣∣∣ B′0 +n∑i=1

xiCi � 0

}.

Conditions (5.1) and (5.2) ensure that the new block constraints in the above represen-

tations are redundant, i.e., never bind for any value of x.

We now have representations of A and B in which all matrices are identical except

A′0 and B′0. The outer approximation in Proposition 2 takes the form

C =

{x

∣∣∣∣∣ A′0 +B′0 +n∑i=1

xiCi � 0

}.

We would like choose the matrices A0 and B0 to be as ‘small’ as possible, so that

the resulting outer approximation is close to the true Minkowski sum. Unfortunately, we

do not have a procedure for obtaining the optimal choices, as it would likely involve a

constraint on the minimum of a matrix’s smallest eigenvalue. Instead, we give a heuris-

tic using the Gershgorin Circle Theorem [34]. We restate it below for real, symmetric

matrices.


Theorem 1 (Gershgorin Circle Theorem [34]). Suppose A ∈ Rn×n is symmetric. Then

each eigenvalue of A lies within the interval

[Aii −

∑j 6=i

∣∣Aij∣∣ , Aii +∑j 6=i

∣∣Aij∣∣]

for at least one i.

Define A(x) = A(x)− A0 and B(x) = B(x)−B0, and let

a = mini

minx∈B

Aii(x)−∑j 6=i

∣∣∣Aij(x)∣∣∣ , (5.3)

b = mini

minx∈A

Bii(x)−∑j 6=i

∣∣∣Bij(x)∣∣∣ . (5.4)

We have the following original result:

Lemma 1. The matrices A0 = −aI and B0 = −bI respectively satisfy conditions (5.2)

and (5.1).

Proof. By the Gershgorin Circle Theorem, a and b respectively lower bound the small-

est eigenvalues of A(x) and B(x) for all x. Therefore, λmin

(−aI + A(x)

)≥ 0 and

λmin

(−bI + B(x)

)≥ 0 for any x, which respectively imply conditions (5.2) and (5.1).

The inner minimizations in (5.3) and (5.4) are concave objectives over convex sets.

They can be recast as mixed-binary semidefinite programs by using binary variables to

distinguish whether Aij(x) is positive or negative. We do not pursue this here because

this optimization becomes tractable in the second-order cone and linear cases, which are

more relevant to our application.


5.2.2 Second-order cone constraints

We proceed by specializing Proposition 2 and Lemma 1 to the matrix in (2.1). Consider

two sets defined by second-order cone constraints:

A ={x∣∣ ‖Aix+ bi‖ ≤ cTi x+ di, i = 1, ...,mA

}⊂ Rn,

B ={x∣∣ ‖Eix+ fi‖ ≤ gTi x+ hi, i = 1, ...,mB

}⊂ Rn.

We have the following original result:

Lemma 2. Suppose Ai = Ei and ci = gi for i = 1, ...,mA, and mA = mB. Then the set

C ={x∣∣ ‖Aix+ bi + fi‖ ≤ cTi x+ di + hi, i = 1, ...,mA

}is an outer approximation of A⊕ B.

Proof. From (2.1), we write each pair of second-order cone constraints as the pair of

semidefinite constraints (cTi x+ di)I Aix+ bi

(Aix+ bi)T cTi x+ di

� 0 (5.5)

and (cTi x+ hi)I Aix+ fi

(Aix+ fi)T cTi x+ hi

� 0. (5.6)

By Proposition 2, the semidefinite constraint

(cTi x+ di + hi)I Aix+ bi + fi

(Aix+ bi + fi)T cTi x+ di + hi

� 0

is an outer approximation of the Minkowski sum of the feasible sets defined by (5.5) and


(5.6). Taking the Schur complement, we obtain the second-order cone constraint

‖Aix+ bi + fi‖ ≤ cTi x+ di + hi.

Repeating this argument for each pair i = 1, ...,mA, we obtain the result.

We now extend Lemma 2 to the case when all matrices are different and A and B

have different numbers of constraints. Since second-order cone constraints are written in

scalar form, the procedure is simpler than in the semidefinite case. For each constraint

‖Aix+ bi‖ ≤ cTi x+ di

in A for which there is no constraint in B with the same Ai and ci, add the redundant

constraint ∥∥∥Aix+ bi

∥∥∥ ≤ cTi x+ di (5.7)

to B. Similarly, for each constraint

‖Eix+ fi‖ ≤ gTi x+ hi

in B for which there is no constraint in A with the same Ei and gi, add the redundant

constraint ∥∥∥Eix+ fi

∥∥∥ ≤ gTi x+ hi (5.8)

to A. A and B now have the same number of constraints, and for each constraint in

A (B), there is a constraint in B (A) with the same Ai and ci (Ei and gi) matrices.

Lemma 2 may now be applied directly.

We must choose bi, di, fi, and hi so that (5.7) is satisfied for any x ∈ B and (5.8) is


satisfied for any x ∈ A. This is equivalent to enforcing

minx∈B

λmin

(cTi x+ di

)I Aix+ bi

(Aix+ bi)T cTi x+ di

≥ 0 (5.9)

and

minx∈A

λmin

(gTi x+ hi

)I Eix+ fi

(Eix+ fi)T gTi x+ hi

≥ 0. (5.10)

The parameters bi, di, fi, and hi are analogous to A0 and B0 in Section 5.2.1, which we

had set to be diagonal matrices. This corresponds to setting bi = fi = 0. Applying the

Gershgorin Circle Theorem [34], we obtain the following minimizations for di and hi:

di = −minx∈B

cTi x−∑j

∣∣Ajix∣∣ , (5.11)

hi = −minx∈A

gTi x−∑j

∣∣Eji x∣∣ , (5.12)

where Aji denotes the jth row of Ai. Observe that unlike in (5.3) and (5.4), we only need

consider the Gershgorin circle associated with the bottom row of the matrix

cTi xI Aix

(Aix)T cTi x

because

∑j

∣∣Ajix∣∣ ≥ ∣∣Aki x∣∣ for all k. It is a corollary of Lemma 1 that setting bi = fi = 0

and di and hi as in (5.11) and (5.12) satisfy conditions (5.9) and (5.10). The optimizations

(5.11) and (5.12) are mixed-binary second-order cone programs, which can be solved

efficiently [5] using commercial software [1] .


5.3 Demand response

In this section, we use a collection of electric vehicles with real-valued charging levels to

illustrate the use our outer approximation.

5.3.1 Polytope electric vehicle model

Each electric vehicle i = 1, ..., L is characterized by the following parameters: Ei ≥ 0, its

energy need, ai and di, its arrival and departure times which satisfy 1 ≤ ai ≤ di ≤ T ,

Pi ≥ 0, its maximum charging rate, and 0 ≤ ηi ≤ 1, its charging inefficiency. The vector

x ∈ RT represents the vehicles energy usage in each time period t = 1, ..., T .

The constraints are as follows. We assume that each vehicle can only charge, so that

x ≥ 0. If t ∈ {ai, di}, xt ≤ Pi. Otherwise, xt ≤ 0. The charging trajectory over all

periods must add to the vehicles total energy need: ηi∑T

t=1 xt = Ei.

The matrix A and vector b for vehicle i are

Ai =

−I

I

[ηi, ..., ηi]

− [ηi, ..., ηi]

, bi =

0

Qi

Ei

−Ei

,

where Qti = Pi if t ∈ {ai, di} and 0 otherwise.

5.3.2 Apparent power constraints

We now apply the outer approximation to a collection of electric vehicles with apparent

power limits. For concision, we assume that ηi = η for all i = 1, ..., L. Invoking Lemma 2,


the outer approximation of the flexibility of the electric vehicle aggregation is

(x, q)

∣∣∣∣∣∣∣∣∣∣

−I

[η, ..., η]

−[η, ..., η]

x ≤L∑i=1

0

Ei

−Ei

,∥∥∥∥∥∥∥ xt

qt

∥∥∥∥∥∥∥ ≤

L∑i=1

Qti, t = 1, ..., T

}.

5.3.3 Chance-constraints

We now use the results in Section 5.2.2 to derive outer approximations of the Minkowski

sum of uncertain loads modelled by chance constraints. Simply stated, a chance con-

straint is a constraint of the type:

P (h(x, ξ) ≥ 0) ≥ p | (p ∈ (0, 1))

Here, h(x, ξ) ≥ 0 is a stochastic system of inequalities (which is nonlinear in general), P is

a probability measure, x is a decision vector and ξ is a random vector. In general, chance

constraints are nonlinear constraints and the feasible set may not be convex, polytopic

or even connected [33]. Convex approximations to these constraints may be found as

in [52]. Additionally, certain chance constraints may be inherently linear. For example,

let the system h(x, ξ) ≥ 0 be simplified to the linear inequality aTx ≤ b suppose the

vector a and scalar b are described by the normal distribution N([a, b]T,Σ)

, and let Φ

denote the zero mean, unit variance normal cumulative distribution function. The scalar

chance constraint

Prob(aTx ≤ b

)≥ p


may be equivalently written as the second-order cone constraint

aTx+ Φ−1(p)

∥∥∥∥∥∥∥√

Σ

x

1

∥∥∥∥∥∥∥ ≤ b

when p ≥ 0. We refer the reader to Ch. 4.4.2 in [16] for a derivation.

We now add uncertainty to our electric vehicle model and use the above technique

to compute the outer approximation. Assume that the charging efficiency and en-

ergy requirement of each electric vehicle is uncertain and described by the distribution

N([η, Ei

]T,Σ)

, where

Σ =

ση σηE

σηE σE

.We replace the vehicles exact energy requirement with the constraint

Prob

(η

T∑t=1

xti ≥ Ei

)≥ p. (5.13)

This requires that the probability any vehicle i = 1, ..., L receives at least Ei is p. We

remark that, because we do not permit vehicles to discharge power, overcharging is

unlikely and would only occur in rare scenarios like negative nodal pricing. Further, as

we discuss in Remark 2, the outer approximation of chance constrained load models is

not intended for real-time operation.

Let 1 be an n× n matrix of all ones, and define

Σ′ =

ση1 σηE

σηE σE

.Each vehicle’s chance constraint is equivalently written as the second-order cone con-


straint

Φ−1(p)

∥∥∥∥∥∥∥√

Σ′

xi

1

∥∥∥∥∥∥∥ ≤ η

T∑t=1

xti − Ei. (5.14)

We remark that (5.13) could also be written as a linear constraint by dividing through by

η and inverting the CDF of the distribution of Ei/η. Here we give the above second-order

cone formulation to demonstrate our approach.

The outer approximation of the Minkowski sum is obtained by again applying Lemma 2.

We have

{x

∣∣∣∣∣ 0 ≤ xt ≤L∑i=1

Qti, t = 1, ..., T,

Φ−1(p)

∥∥∥∥∥∥∥√

Σ′

x

L

∥∥∥∥∥∥∥ ≤ η

T∑t=1

xt −L∑i=1

Ei

.

Remark 2 (Interpretation of chance constraints). The chance constraint models a ve-

hicle’s uncertainty prior to its arrival at time t = ai. This is because once a vehicle

has arrived, its energy need, Ei, and inefficiency, ηi, become known, and thus are not

random throughout charging. For this reason, our outer approximation for the flexibility

of a vehicle aggregation is intended for use prior to the start of operation. For instance,

it could be used to represent an aggregation’s flexibility inside of an hourly optimal power

flow routine or in day-ahead unit commitment.

5.4 Examples

5.4.1 Analytical Example

We now consider an analytical example of SOC constraint aggregation for apparent power

constraints, for a load available in two time periods. As each time period requires vari-


ables for both real and reactive power, the resultant set will be in R4. We use the model

described in Section 5.3.2 except we specify the energy as an inequality instead of an

equality. We consider two loads constrained to have positive real power and charging

efficiencies of unity. The first load P has an energy requirement E > 50 and apparent

power limit of 40. The second load Q has an energy requirement E > 75 and apparent

power limit of 50. We thus have the following constraints:

P = (xP, qP)

∣∣∣∣∣∣∣ −I

−[1, ..., 1]

xP ≤ 0

−50

,∥∥∥∥∥∥∥ x1P

q1P

∥∥∥∥∥∥∥ ≤ 40, and

∥∥∥∥∥∥∥ x2P

q2P

∥∥∥∥∥∥∥ ≤ 40,

}.

Q = (xQ, qQ)

∣∣∣∣∣∣∣ −I

−[1, ..., 1]

xQ ≤ 0

−75

,∥∥∥∥∥∥∥ x1Q

q1Q

∥∥∥∥∥∥∥ ≤ 50, and

∥∥∥∥∥∥∥ x2Q

q2Q

∥∥∥∥∥∥∥ ≤ 50,

}.

Then the outer Minkowski approximation S in this case is

S = (xS, qS)

∣∣∣∣∣∣∣ −I

−[1, ..., 1]

xS ≤ 0

−125

,∥∥∥∥∥∥∥ x1S

q1S

∥∥∥∥∥∥∥ ≤ 90, and

∥∥∥∥∥∥∥ x2S

q2S

∥∥∥∥∥∥∥ ≤ 90,

}.

In the above figure, we plot, using the MPT toolbox in Matlab [61], the convex set


Figure 5.1: Convex Set S plotted for the case where q2S = 0.

of the outer Minkowski approximation for the case where q2S = 0. Note that the reactive

power in time period 1 q1S, plotted on the z-axis, and it varies parabolically against p1S,

the real power in time period 1, in accordance with the second-order cone constraint on

apparent power. Correspondingly, other cross-sections of the set are triangular as they

are derived from the other linear constraints.

5.4.2 Numerical example

We now numerically examine the uncertain electric vehicle aggregation described in Sec-

tion 5.3.3. We refer the reader to [10] for simulations of the polytope case.

The parameters of the simulation are as follows. Each vehicle’s mean energy need, Ei,

is drawn from the uniform distribution over the interval [2, 5]. The variance of the energy

need is the same for all vehicles, σE = 0.1. All vehicles have the same mean charging

inefficiency, η = 0.8, and charging inefficiency variance, ση = 0.01. The covariance of the

energy need and charging inefficiency is σEη = 0.01.

The arrival time of each vehicle, ai, is sampled uniformly from the set {1, ..., T − 1},

and its departure time, di, is sampled uniformly from the set {ai + 1, T}. Each vehicle’s


charging limit is given by TEi

2max {Ui, 1/(di − ai + 1)}, where Ui is drawn uniformly at

random from the unit interval. The probability of a vehicles energy need being satisfied

is p = 0.9.

We quantify performance as the ratio of the volumes of the outer approximation

to the volume of the exact Minkowski sum, Vouter/Vexact. Hence, large ratios indicate

poor performance, and a ratio of one means the approximation is exact. Computing the

volume of a polytope is computationally hard [23]. Thus, so is computing the volumes of

sets defined by second-order cone constraints. For this reason, we approximate volumes

using Monte Carlo. Specifically, uniformly random points are generated in a hypercube

containing the exact and approximate Minkowski sums. We count the number of feasible

points in the outer approximation by checking its constraints. We count the number of

feasible points in the exact Minkowski sum by solving a second-order cone program to

determine if the vehicle collection can feasibly produce the aggregate power profile. We

use CVX [30] with the solver Mosek [1] to solve the second-order cone programs.

We consider L = 100 electric vehicles over T = 5 time periods. Note that intractability

makes it difficult to accurately compute volumes in higher dimensions. We compare the

volume of the outer approximation to the Minkowski sum as the variance of the vehicles’

energy needs is increased from σE = 0.02, ..., 0.1, which tightens constraint (5.14). The

volumes were computed with N = 10, 000 Monte Carlo points.

Figure 5.2 shows the ratio of the volume of the outer approximation to the volume

of the inner approximation. We see that the error increases from approximately 20% to

24% as the variance of the vehicles’ energy needs increases. We interpret this outcome as

follows. Intuitively, the outer approximation performs best when the individual load sets

are similar. Tightening constraint (5.14) makes differences between the vehicles more

important, thus decreasing the accuracy of the outer approximation.


5.5 Conclusion

In this chapter, we concern ourselves with stochastic loads. This topic is of importance

because loads are not deterministic. We derive a formulation for modelling uncertainty

in loads using chance constraints, which are constraints wherein the probability that an

inequality with random variables is satisfied must be greater than a given probability.

We show how certain types of chance constraints can be converted into linear or second-

order cone constraints [16]. While we have seen in Section 4.2 how to aggregate linear

constraints, this motivates a method to aggregate second-order cone constraints. We add

that constraints on apparent power of loads can also be expressed using second-order

cone constraints.

To find a method of aggregating second-order cone constraints, we first consider the

more general problem of aggregating together semidefinite constraints. We find a method

to do so by adding together the various A-matrices of the semidefinite constraints that

define the semidefinite constraint. When a matching constraint does not exist, one may

be generated by an optimization over the eigenvalues of a constraint with similar form.

Due to the intractability of this optimization problem, we present a heuristic that makes

use of the Gershgorin Circle Theorem that provides a solution. Finally, we apply the

result for the aggregation and the heuristic to second-order cone constraints and find

simplified versions for both.

These results allow for the approximate aggregation of loads with second-order cone

and semidefinite constraints. We believe this work to be the first (approximate) ana-

lytical calculation of the Minkowski sum for such constraints. We note that analytical

expressions do exist for the Minkowski sum of ellipsoids (a sub-class of second-order cone

constraints) [41], but not in the general case.

Finally we show, via numerical simulations, that the approximation that we have

developed has good performance for electric vehicles with random energy needs. Poten-

tial future work on this topic may include additional numerical characterization of the


approximation, application to other types of load uncertainty, and finding of bounds on

the size of the approximation.


σE

0.02 0.04 0.06 0.08 0.1

VO

ute

r /

VE

xa

ct

1.19

1.195

1.2

1.205

1.21

1.215

1.22

1.225

1.23

1.235

Figure 5.2: The ratio of the volume of outer approximation to the exact Minkowski sumas a function of the variance of the vehicles’ energy needs.

Chapter 6

Inner Approximations

6.1 Introduction

In addition to the earlier outer approximations, it is useful to find an inner approximation

for the Minkowski Sum of convex sets. This would allow us to have both necessary and

sufficient conditions for membership in the Minkowski sum.

Given a general convex set, the problem of finding an inner approximation is well un-

derstood. Approaches range from generating polytopic approximations [27], approaches

based on triangulation [25] as well as the finding of nonlinear representations such as

ellipsoids [41], [22], [16]. In this chapter, we apply ellipsoidal techniques to obtain inner

approximations of Minkowski sums of polytopic sets.

We use the approach of finding the maximum volume inscribed ellipsoid (MIE) as an

inner approximation, which is a semidefinite programming problem. MIE’s to polytopic

sets are commonly used in linear programming algorithms. The computational complex-

ity of this problem has been studied by Khachiyan in [39] and later by Anstreicher in [4].

Anstreicher finds the complexity of finding an inscribed ellipsoid, whose volume is at least

a factor of e−ε of the MIE to be O(m3.5 ln(mR/ε)) where m is the number of inequalities

defining the polytope and R is the radius of a ball centered at the origin, containing the

83

Chapter 6. Inner Approximations 84

polytope.

6.2 Ellipsoidal approximations of convex sets

We approach this problem by trying to find the MIE [16] of a set. Ellipsoids are defined

by ε = {Gu+ h| ‖u‖2 ≤ 1}; here h is the centre of the ellipse and G is a real, symmetric,

positive definite matrix that scales the ellipse relative to a hypersphere centred at h [16].

To maximize the volume, we maximize the log of the determinant of G as the volume of

the ellipse is proportional to the determinant of G with a multiplicative pre-factor such

as π in R2 and 4π3

in R3. Pre-factors for higher dimensions can be found in [45]. A simple

rendering of the idea may be seen in following figure.

Figure 6.1: Maximizing an inscribed ellipsoid inside a polytope


When given an arbitrary convex set C, we can find the MIE ε in C by finding G and

h that solve the following optimization problem [16]:

maximizeG,h

log det G

subject to sup‖u‖2≤1

IC(Gu+ h) ≤ 0

IC(x) = 0 for x ∈ C, else IC(x) =∞

The resulting constraint is convex.

When the above result is applied to a polytopic set P = {x | aTi x ≤ bi, i =

1, . . .m}, this reduces to the following [16]:

maximizeG,h

log det G

subject to ‖Gai‖2 + aTi h ≤ bi, i = 1, . . .m and

G � 0

We may also apply the above result to a set defined as the intersection of m ellipsoids

defined as εi = {x : xTAix+ 2bTi x+ ci ≤ 0}, A ∈ Sn++, i = 1, . . . ,m (where Sn++ is the set

of symmetric positive definite matrices), obtaining the resulting semidefinite constraints

on the ellipsoidal maximization problem [16].

maximizeG,h,λ

log det G

subject to

−λi − ci + bTi A

−1i bi 0 (h+ A−1bi)

T

0 λiI G

(h+ A−1bi)T G A−1i

� 0, i = 1, . . . ,m,

G � 0


6.3 Ellipsoidal inner bounds of the Minkowski sum

Given the ability to find the MIE of a polytope (or in general a convex set), it is obvious

that an inner bound on the Minkowski sum of polytopes can be found by either finding

an inner bound to the Minkowski sum of the MIEs of the individual polytopes, or by

finding the MIE for the exact Minkowski sum.

As we know from the previous section, obtaining the desired MIE for a polytope re-

quires solution of a semidefinite program. To implement the first approach, MIE’s would

need to be calculated for a large number of polytopes, which would be computation-

ally expensive. Nevertheless, once that step is complete, an inner approximation to the

Minkowski sum of ellipsoids has been derived in [41].

We focus on the second approach, which results in needing to solve only one (large)

semidefinite optimization problem. Let us consider (without loss of generality), the

Minkowski sum of two polytopes, P = {x|ATPx ≤ bP} and Q = {y|ATQy ≤ bQ}, where

both x and y are vectors in RD and the matrices AP and AQ have m and n rows

respectively. From the definition of the Minkowski sum, we can write the H-representation

of the Minkowski sum as follows:

S =

z

∣∣∣∣∣∣∣ AP 0

−AQ AQ

xz

≤bPbQ

Here, we have a matrix of linear constraints (i.e. a polytope) that (implicitly) de-

scribes the Minkowski Sum S. We write y as z − x to eliminate D equalities from the

definition of the sum, which allows us to have (in general) a full-dimensional polytope.

This is essential because the optimizations involved in finding the MIE seek to maximize

the volume of the relevant ellipsoid, which is a highly ill-conditioned problem when the

ellipsoid has no volume.

However, the system of equations has m + n rows and 2D variables. Obviously, this

system would become extremely unwieldy when dealing with potentially millions of load


models, in high dimensions. In a similar way that we found an outer approximation

to the Minkowski sum with a compact representation, we seek to do so with the inner

approximation, by considering the projection of the above polytope to the Minkowski

sum space.

We then determine to find the MIE to this (implicitly defined) Minkowski Sum in

RD, using only the z variables, as an inner approximation to it. All ellipsoids in RD are

described by a symmetric positive definite D by D matrix, G, centred at h that describe

it. However in [67], we see that the problem of explicitly representing the Minkowski sum

is np-hard, and thus the straightforward solution of this problem results in a suboptimal

result. We describe both approaches in the following subsections.

6.4 Finding an MIE over the Minkowski sum from

the MIE of the combined load and sum spaces

We may apply the ellipsoidal transformation described previously to the system described

above, with variables x and z − x, to find G and h for the total space of the loads and

the Minkowski sum. The resulting second-order cone constraints define an ellipsoid over

this entire space (the total ellipsoid). This overall problem is an semidefinite program,

because of the additional constraint that the matrix G be positive definite. The total

ellipsoid is high-dimensional; we will take the subset of results relevant to the Minkowski

sum. Let us consider the following system of linear constraints:

S =

z

∣∣∣∣∣∣∣ AP 0

−AQ AQ

xz

≤bPbQ

(6.1)

which describe the Minkowski sum. We apply the ellipsoidal transformation to find


the matrix G =

Gx Gxz

GTxz Gz

and the vector h =

hxhz

These yield the following second-order cone constraints for the total space MIE opti-

mization problem.

‖G[aiP 0]T‖2 + [aiP 0]h ≤ biP, i = 1, . . .m and

‖G[−aiQ aiQ]T‖2 + [−aiQ aiQ]h ≤ biQ, i = 1, . . . n

We find, as expected, that the best results for the MIE over the Minkowski sum space

are found by maximizing the determinant of the sub-matrix of G associated with the

Minkowski sum, i.e. Gz, rather than maximizing the determinant of G as a whole, as

we wish to maximize the volume associated with the Minkowski sum. Finally, we can

express the optimization problem to find the MIE of the Minkowski sum as finding Gz

and h that solve:

maximizeG,h

log det Gz (6.2)

‖G[aiP 0]T‖2 + [aiP 0]h ≤ biP, i = 1, . . .m and (6.3)

‖G[−aiQ aiQ]T‖2 + [−aiQ aiQ]h ≤ biQ, i = 1, . . . n and (6.4)

G � 0 (6.5)

6.4.1 Example

Let us apply the above procedure to the Minkowski sum of two simple polytopes in R2.

We define P = {x|AP x ≤ bP} and Q = {y|AQ y ≤ bQ} with exact Minkowski sum R.

We seek to find the MIE for the approximate Minkowski sum S of P and Q. The loads

are modified deferrable loads, the first with an additional constraint to encourage early

charging, and the second with an additional constraint to encourage late charging. The


values of AP, bP, AQ, and bQ are given below.

P =

x

∣∣∣∣∣∣∣∣∣∣∣∣∣

I2×2

−I2×2

−1 −1

−2 −1

x ≤

202×1

02×1

−30

−50

and Q =

y

∣∣∣∣∣∣∣∣∣∣∣∣∣

I2×2

−I2×2

−1 −1

−1 −2

y ≤

252×1

02×1

−45

−70

(6.6)

The outer Minkowski approximation to the Minkowski sum (after preconditioning)

may be found as:

S =

z

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

I2×2

−I2×2

−1 −1

−1 −2

−2 −1

y ≤

452×1

−35

−32.5

−75

−115

−110

Because this problem is in R2, the exact Minkowski sum, R is equal to the outer

Minkowski approximation S, by Corollary 1. The volume of the Minkowski sum is 81.25.

Now, let us find the MIE to the exact Minkowski sum of P and Q using the technique

described in this section; where . We set up and solve the semidefinite cone problem for

G and h, according to Equation 6.2, using CVX. We take the rows aP and aQ from the

descriptions of P and Q given in Equation 6.6 and find:

Gz =

3.1328 −0.6109

−0.6109 3.4089

and hz =

41.4761

40.1708

The ellipsoid has the volume 32.38 which is 39.85% of the volume of the Minkowski

sum. We plot the ellipsoid that we find, inside the Minkowski sum in Figure 6.2


Figure 6.2: The MIE (in blue), computed via projection of the total ellipsoid and theactual MIE (in green), both overlaid on the Minkowski sum (in red) .

6.5 Finding an MIE using a Linear Decision Rule

Reference [67] gives a superior approach to finding the maximum inscribed ellipsoid for

the Minkowski sum. The previous approach essentially forced the projection of the total

ellipsoid down to the z-space using a fixed, feasible value for the remaining variable

space. This is however not optimal, as seen in the previous example. In [67], the use of

a linear decision rule is proposed, in place of the previously fixed auxiliary variables in

the problem.

When the linear decision rule formulation is applied to Equation 6.1, we obtain the

following total space MIE optimization problem, with the associated second-order cone

constraints. Here we solve the optimization problem to maximize the log of the determi-


nant of the matrix E.

maximizeE,hE

log det E∣∣∣∣∣∣∣∣∣∣∣∣∣∣EV

[0 aiP]T

∣∣∣∣∣∣∣∣∣∣∣∣∣∣2

+ [0 aiP]

hEhV

≤ biP , i = 1, . . .m and

∣∣∣∣∣∣∣∣∣∣∣∣∣∣EV

[−aiQ aiQ]T

∣∣∣∣∣∣∣∣∣∣∣∣∣∣2

+ [−aiQ aiQ]

hEhV

≤ biQ, i = 1, . . . n

We substitute in the rows aP and aQ from the descriptions of P and Q given in

Equation 6.6. The results show a marked improvement, as can be see in Figure 6.3. We

find

Gz =

3.4218 −0.6777

−0.6777 4.4705

This time, the volume of the inscribed ellipsoid that we calculate is 62.15, which is

76.49% of the volume of the exact Minkowski sum.

We see that the results using this method show a marked improvement relative to a

simple projection of the total ellipsoid.

6.6 Numerical Results for DR Loads

A numerical study was carried out, examining the performance of the inner approximation

for different numbers of loads (2 loads, 4 loads, 8 loads and 16 loads), for loads in

various dimensions. The computations were limited due to the need to calculate the

exact Minkowski sum for comparison, which is intractable.

We see that the number of loads has little to no effect on the performance of the

approximation, but that the accuracy degrades quickly with increase in dimension. This is

to be expected as most of the volume of a high-dimensional object lies near its boundaries,


Figure 6.3: The MIE (in blue), computed via a linear decision rule, overlaid on theMinkowski sum (in red) for the same problem as Figure 6.2.

and an ellipsoid is by nature unable to approach these boundaries except tangentially at

a few points. As a result, the inner approximation is primarily of academic interest.

6.7 Conclusion

This chapter presents an alternative approach to the outer approximations developed in

Section 4.2. The use of an inner approximation is complementary to the use of an outer

approximation as it provides a two-sided bound on the size of the load aggregation.

We consider ellipsoidal approaches to finding inner approximations of load aggrega-

tions. We describe the general approach to finding an maximal inscribed ellipsoid to

polytopic sets. We then seek to extend it to an implicitly defined aggregated set via

finding maximal inscribed ellipsoids on the relevant projection of the overall polytope

(which contains a representation of every load). We compare this approach to finding a

projection using a linear decision rule.


2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Number of Loads

Elli

psoi

d V

olum

e /

Exa

ct M

inko

wsk

i Sum

Vol

ume

Performance of Inner Ellipsoidal Approximation to Minkowski Sum vs. Number of Loads summed

R2

R3

R4

R5

Figure 6.4: Accuracy of MIE approximation vs. number of loads.

We find that the linear decision rule is the superior approach in finding an inner ellip-

soidal approximation, however the result is not maximal. However, the overall approach

is limited in utility as the error in the approximation grows quickly with dimension,

which we show via numerical studies. This is because the overall volume of sets in high

dimensions is concentrated at the set boundaries (particularly at extreme points), which

an ellipsoidal set cannot effectively approximate.

This approach may prove to have higher utility if used on non-polytopic convex sets,

such as those described in the previous chapter. An alternative approach that could be

pursued would be to maximize an inner polytopic set instead of an ellipsoidal set [27].

Chapter 7

Conclusion

7.1 Summary

Demand Response is now an important source of flexibility in the management of the

power grid. It finds use in fast-response activities such as primary or secondary control,

as well as for longer time horizon activities such as load shifting. The most valuable DR

resources are those that are capable of high-resolution control across different time scales,

while still carrying out their primary functions. DR increases the spatial diversity of the

grid, improves its environmental performance and increases reliability.

A significant challenge in working with DR resources is that loads are very numer-

ous (potentially 106 loads in an aggregation), and heterogeneous. Exactly representing

their capabilities in operational programs such as multi-period optimal power flow and

unit commitment would make them intractable. Thus, there is need for ways to find

concise representations of DR aggregations that are compatible with the aforementioned

programs.

In this dissertation, we developed an (approximate) framework for concise DR aggre-

gation based around the representation of resources as convex sets and their aggregation

via the Minkowski sum. When realistic, convex representations are much preferred be-

94

Chapter 7. Conclusion 95

cause they allow the use of a vast array of algorithms developed for solving convex op-

timization problems and provide a unique optimal solution. The Minkowski sum is the

collection of all element-wise sums of members of two (or more) sets. One nice property

of the Minkowski sum is that the sum of two convex sets is also a convex set.

The simplest convex sets are polytopes; sets with flat boundaries derived from hyper-

planes in Rn. Each of these hyperplanes is described by a linear inequality (or constraint).

We describe polytopic representations of deterministic DR resources such as storage-like

loads (with power and energy constraints, dissipation and input / output efficiencies),

deferrable loads (a simplified class of storage-like loads with arrival / departure con-

straints) and thermostatic loads (loads like air-conditioners and heaters), for use with

our framework. We also mention differential power and apparent power constraints as

less common constraint types that may find use in load models.

The primary limitations of the framework are the inability to model non-convex loads,

and that as outer approximation, it will contain some infeasible solutions (i.e. some of

the elements in the aggregated Minkowski sum will not be actually achievable by sums

of elements in the constituent sets). However feasible solutions can be recovered by

minimizing the distance from the result onto to the feasible set.

The polytopic description of a set takes the form Ax ≤ b where we refer to the matrix

A as the A-matrix. This representation is a simple list of all the inequalities that define

the polytope. To approximate the Minkowski sum, for polytopes described by the same

A-matrices, we simply add the right hand-side vectors. Where an inequality does not

exist, we can generate an inequality tangent to the original polytope, by running a linear

program to maximize the inequality over the original set. We can also use this method

to “tighten” existing constraints in the polytopes description; a method we refer to as

preconditioning, and that improves the accuracy of the approximation.

To make use of this framework, the following steps will take place:

• Obtain polytopic load models from loads, or parameters from which load models


can be developed

• Perform load aggregation calculation

• Aggregator communicates result to System Operator

• System operator uses aggregate model in multi-period optimal flow

• System operator communicates dispatch instruction to aggregator

• Aggregator determines controls for individual loads

• Aggregator communicates specific controls to individual loads

Using this approximation, we perform numerical simulations of the Minkowski sums

of thermostatic and storage-like loads and compare them to the exact sums, as well as

to results from other papers. We find that the approximate method shows very good

performance - less than 1% error for adding storage loads together and approx 15% error

for adding thermostatic and storage loads together in the highest dimension computed.

Additionally, we show that the approximation is exact for two special cases: for resources

in two dimensions (two time periods) and for resources with power constraints only.

Next, we extended this framework to loads modelled using semidefinite and second-

order cone constraints. Such constraints can be used to enable us to consider loads

with uncertainty in addition to the deterministic loads modelled previously. We describe

how constraints can be “added together” when they share certain terms for both these

constraint types. Additionally, we make use of the Gershgorin Circle Theorem to create

virtual constraints when needed for sets that do not share the same constraint types. We

carry out simulations to characterize an aggregation of stochastic loads with uncertainty

in efficiency and total energy demand. We see an volume error of about 25 % for an

aggregation of 100 loads, over 5 time periods. This is, as expected, more than the

polytopic case, but still very good. Note that a 25% error in volume in R5 corresponds

to a 4.5 % average error in power.


Finally, we consider the problem of developing an inner bound for the Minkowski sum

of sets, since we have an outer bound. We take the approach of finding an ellipsoidal set

inscribed inside the set describing the Minkowski sum. The exact maximally inscribed

ellipsoid is not a tractable problem, so we use a linear decision rule to find a “large”

volume ellipsoid that gives an inner bound to the Minkowski sum. Unfortunately, the

performance of this method is not as good as that of the outer approximation, because el-

lipsoidal sets are not good at filling space inside polytopes. We confirm this via numerical

simulations.

The methods that we have developed will allow for the aggregation of large num-

bers of deterministic and stochastic loads modeled via polytopic and second-order cone

constraints. The outer approximations for the Minkowski sum developed in this thesis

are novel and have good numerical performance. They may also find use in other con-

texts where the Minkowski sum of convex sets (specified via the intersection of various

inequalities) is desired.

7.2 Challenges for Implementation

In this thesis, we have developed a general framework for load aggregation for the pur-

pose of DR. In order to implement this framework, the following challenges need to be

overcome.

• Today, load models for DR exist in a variety of forms, with no standard rep-

resentations. For our method to see widespread adoption, loads would need to

communicate their capabilities in terms of linear or second-order cone constraints.

• System operators only make limited use of multi-period optimal power flow. In

Ontario, the IESO uses multi-interval power flow only for the real-time market. As

load aggregations for DR, similar to storage are fundamentally dynamic, it only

makes sense to use them in multi-period optimal power flow. Increased usage of


multi-period optimal flow is a prerequisite for effective use of load aggregations. We

note that unit-commitment uses a multi-period algorithm so our load aggregations

may be directly used for that problem.

• While we have shown the approximation to be quite accurate and how to recover

feasible solutions should the result of an optimization give an infeasible solution, no

theoretical bounds on the size of the approximation currently exist. These would

be very helpful in reliably making use of the load aggregations.

• We would like to see these load aggregations made use of in longer term planning

problems for generation and transmission. However operational time-scales for DR

may not align well with planning time scales. Appropriate models for this purpose

would need to be developed.

7.3 Recommendations for Future Work

We now summarize potential extensions to our work. Firstly, theoretical results on

exactness for sets with semidefinite and /or second order cone constraints would be

of interest, in a manner similar to our exactness results for hypercubes and polytopes

in R2. Development of theoretical bounds on the sizes of these outer approximations

relative to exact sets would be useful. As our model only handles PQ loads, an possible

extension could include handle impedance loads. Additional numerical studies for both

the polytopic and second-order cone cases would also be worthwhile, to identify cases

where performance is of acceptable levels for use by utilities.

Extensions of the chance constraint framework to handle additional types of uncer-

tainty would also be useful. A chance constrained formulation of thermostatic loads may

be found for uncertainty in outside temperatures. Additionally, ways to model uncer-

tain arrival / departure times would be extremely useful for problems involving electric

vehicles though they are challenging because of their discrete nature. We have consid-


ered representing uncertainty in the arrival / departure of loads through superpositions

of polytopes, but these do not have the property of being outer or inner approxima-

tions. Other topics include consideration of DR loads with joint chance constraints and

of chance constraints without Gaussian probability distributions.

Non-ellipsoidal inner bounds represent another area of interest as does finding any

type of inner bounds for sets with arbitrary second-order cone and semidefinite con-

straints. Ideally, we would like to see a polytopic formulation for inner approximations

as these are likely to perform better for polytopic sets. Ellipsoidal approximations suf-

fer from the property that they only approach the boundaries of a polytopic set at a

few points and never at its vertices, while most of the volume in high-dimensional sets

is located near the boundary of the set. We are particularly interested in the method

proposed by Firsching [27] which looks at maximal inner polyhedral bounds.

Bibliography

[1] MOSEK optimization software. http://www.mosek.com/.

[2] Pankaj K. Agarwal, Eyal Flato, and Dan Halperin. Polygon decomposition for

efficient construction of Minkowski sums. Computational Geometry, 21(12):39 – 61,

2002. Sixteenth European Workshop on Computational Geometry - EUROCG-2000.

[3] M. Alizadeh, A. Scaglione, A. Applebaum, G. Kesidis, and K. Levitt. Reduced-

order load models for large populations of flexible appliances. IEEE Transactions

on Power Systems, 30(4):1758–1774, July 2015.

[4] Kurt M. Anstreicher. Improved complexity for maximum volume inscribed ellipsoids.

SIAM Journal on Optimization, 13(2):309–320, 2002.

[5] Alper Atamturk and Vishnu Narayanan. Conic mixed-integer rounding cuts. Math-

ematical Programming, 122:1–20, 2010.

[6] D. Avis, K. Fukada, and S. Piccozi. On canonical representations of convex polyhe-

dra. World Scientific, pages 351–360, April 2002.

[7] David Avis and Komei Fukuda. A pivoting algorithm for convex hulls and vertex

enumeration of arrangements and polyhedra. Discrete & Computational Geometry,

8(3):295–313, 1992.

[8] S. F. Barot and J.A. Taylor. Load aggregation for demand response using polytopic

models and the Minkowski sum. In CIGRE Canada Conference, Aug. 2015.

100

http://www.mosek.com/

Bibliography 101

[9] S.F. Barot and J.A. Taylor. An outer approximation of the Minkowski sum of convex

conic sets with application to demand response. 2016 IEEE 55th Annual Conference

on Decision and Control, 2016.

[10] Suhail Barot and Josh A. Taylor. A concise, approximate representation of a collec-

tion of loads described by polytopes. International Journal of Electrical Power &

Energy Systems, 84:55 – 63, 2017.

[11] S. Bashash and H.K. Fathy. Modeling and control of aggregate air conditioning

loads for robust renewable power management. Control Systems Technology, IEEE

Transactions on, 21(4):1318–1327, July 2013.

[12] Mokhtar S. Bazaraa, John J. Jarvis, and Hanif D. Sherali. Linear Programming and

Network Flows. Wiley-Interscience, 2004.

[13] Mark de Berg, Otfried Cheong, Marc van Kreveld, and Mark Overmars. Computa-

tional Geometry: Algorithms and Applications. 2008.

[14] D. Bienstock, M. Chertkov, and S. Harnett. Chance-constrained optimal power flow:

Risk-aware network control under uncertainty. SIAM Review, 56(3):461–495, 2014.

[15] E. Boros, K. Elbassioni, V. Gurvich, and K. Makino. Generating vertices of poly-

hedra and related problems of monotone generation. CRM Proceedings and Lecture

Notes, 48, 2009.

[16] S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press,

New York, NY, USA, 2004.

[17] D.S. Callaway and I. A. Hiskens. Achieving controllability of electric loads. Pro-

ceedings of the IEEE, 99(1):184–199, Jan. 2011.

Bibliography 102

[18] Duncan S. Callaway. Tapping the energy storage potential in electric loads to deliver

load following and regulation, with application to wind energy. Energy Conversion

and Management, 50(5):1389 – 1400, 2009.

[19] Anya Castillo and Dennice F. Gayme. Grid-scale energy storage applications in

renewable energy integration: A survey. Energy Conversion and Management,

87(0):885 – 894, 2014.

[20] Zhi Chen, Lei Wu, and Yong Fu. Real-time price-based demand response manage-

ment for residential appliances via stochastic optimization and robust optimization.

Smart grid, IEEE Transactions on, 3(4):1822–1831, 2012.

[21] G. Constable and B. Somerville. A Century of Innovation: Twenty Engineering

Achievements that Transformed Our Lives. Joseph Henry Press, 2003.

[22] Fabrizio Dabbene, Paolo Gay, and Boris T. Polyak. Recursive algorithms for inner

ellipsoidal approximation of convex polytopes. Automatica, 39(10):1773 – 1781,

2003.

[23] M. E. Dyer and A. M. Frieze. On the complexity of computing the volume of a

polyhedron. SIAM Journal on Computing, 17(5):967–974, 1988.

[24] Martin Dyer, Peter Gritzmann, and Alexander Hufnagel. On the complexity of

computing mixed volumes. SIAM Journal on Computing, 27(2):356–400, 1998.

[25] Ioannis Z. Emiris, Vissarion Fisikopoulos, Christos Konaxis, and Luis Penaranda.

An output-sensitive algorithm for computing projections of resultant polytopes. In

Proceedings of the Twenty-eighth Annual Symposium on Computational Geometry,

SoCG ’12, pages 179–188, New York, NY, USA, 2012. ACM.

Bibliography 103

[26] Federal Energy Regulatory Commission. Reports on demand response and

advanced metering. http://www.ferc.gov/industries/electric/indus-act/

demand-response/dem-res-adv-metering.asp, Dec. 2013.

[27] Moritz Firsching. Computing maximal copies of polyhedra contained in a polyhe-

dron. Experimental Mathematics, 24(1):98–105, 2015.

[28] Komei Fukuda. Polyhedral computation faq. http://www.inf.ethz.ch/personal/

fukudak/polyfaq/polyfaq.html, June 2004.

[29] Joao Gouveia, Pablo A. Parrilo, and Rekha R. Thomas. Approximate cone factor-

izations and lifts of polytopes. Mathematical Programming, 151(2):613–637, 2015.

[30] Michael Grant and Stephen Boyd. CVX: Matlab software for disciplined convex

programming, version 2.1. http://cvxr.com/cvx, March 2014.

[31] Xijuan Guo, Lei Xie, and Yongliang Gao. Optimal Accurate Minkowski Sum Approx-

imation of Polyhedral Models, pages 179–188. Springer Berlin Heidelberg, Berlin,

Heidelberg, 2008.

[32] H. Hao, B. Sanandaji, K. Poolla, and T. Vincent. A generalized battery model of

a collection of thermostatically controlled loads for providing ancillary service. 51th

Annual Allerton Conference on Communication, Control and Computing, August

2013.

[33] Rene Henrion. Chance-constrained programming. http://stoprog.org/

what-stochastic-programming. Stochastic Programming Society.

[34] Roger A Horn and Charles R Johnson. Matrix analysis. Cambridge university press,

2012.

http://www.ferc.gov/industries/electric/indus-act/demand-response/dem-res-adv-metering.asp

http://www.ferc.gov/industries/electric/indus-act/demand-response/dem-res-adv-metering.asp

http://www.inf.ethz.ch/personal/fukudak/polyfaq/polyfaq.html

http://www.inf.ethz.ch/personal/fukudak/polyfaq/polyfaq.html

http://cvxr.com/cvx

http://stoprog.org/what-stochastic-programming

http://stoprog.org/what-stochastic-programming

Bibliography 104

[35] Independent Electricity System Operator (IESO). Introduction to the

demand response auction. http://www.ieso.ca/Documents/training/

DR-Auction-Workbook.pdf, 2016.

[36] Libin Jiang and Steven Low. Multi-period optimal energy procurement and demand

response in smart grid with uncertain supply. In 2011 50th IEEE Conference on

Decision and Control and European Control Conference, pages 4348–4353. IEEE,

2011.

[37] Leonid Khachiyan. Complexity of polytope volume computation. In New trends in

discrete and computational geometry, pages 91–101. Springer, 1993.

[38] Leonid Khachiyan, Endre Boros, Konrad Borys, Khaled Elbassioni, and Vladimir

Gurvich. Generating all vertices of a polyhedron is hard. Discrete and Computational

Geometry, 39(1-3):174–190, 2008.

[39] Leonid G. Khachiyan and Michael J. Todd. On the complexity of approximating the

maximal inscribed ellipsoid for a polytope. Mathematical Programming, 61(1):137–

159, 1993.

[40] A. Khodaei, M. Shahidehpour, and S. Bahramirad. SCUC with hourly demand

response considering intertemporal load characteristics. Smart Grid, IEEE Trans-

actions on, 2(3):564–571, Sept 2011.

[41] A.B. Kurzhanskiı and I. Valyi. Ellipsoidal Calculus for Estimation and Control.

Systems & Control: Foundations & Applications. IIASA, 1997.

[42] M. Kvasnica, P. Grieder, and M. Baotic. Multi-Parametric Toolbox (MPT). http:

//control.ee.ethz.ch/~mpt/, 2004.

[43] J. Kwac and R. Rajagopal. Demand response targeting using big data analytics. In

Big Data, 2013 IEEE International Conference on, pages 683–690, Oct 2013.

http://www.ieso.ca/Documents/training/DR-Auction-Workbook.pdf

http://www.ieso.ca/Documents/training/DR-Auction-Workbook.pdf

http://control.ee.ethz.ch/~mpt/

http://control.ee.ethz.ch/~mpt/

Bibliography 105

[44] Hyung-Geun Kwag and Jin-O Kim. Reliability modeling of demand response con-

sidering uncertainty of customer behavior. Applied Energy, 122:24–33, 2014.

[45] Pavel Loskot and Norman C. Beaulieu. On monotonicity of the hypersphere volume

and area. Journal of Geometry, 87(1):96–98, 2007.

[46] R. Malhame and C.-Y. Chong. Electric load model synthesis by diffusion approxima-

tion of a high-order hybrid-state stochastic system. IEEE Transactions on Automatic

Control, 30(9):854–860, 1985.

[47] Johanna L Mathieu, Marina Gonzalez Vaya, and Goran Andersson. Uncertainty in

the flexibility of aggregations of demand response resources. In Industrial Electronics

Society, IECON 2013-39th Annual Conference of the IEEE, pages 8052–8057. IEEE,

2013.

[48] Johanna L. Mathieu, Maryam Kamgarpour, John Lygeros, Goran Andersson, and

Duncan S. Callaway. Arbitraging intraday wholesale energy market prices with ag-

gregations of thermostatic loads. Power Systems, IEEE Transactions on, 30(2):763–

772, 2015.

[49] Johanna L. Mathieu, Maryam Kamgarpour, John Lygeros, and Duncan S. Call-

away. Energy arbitrage with thermostatically controlled loads. In European Control

Conference, pages 2519–2526, 2013.

[50] A. Molina-Garcia, M. Kessler, J.A. Fuentes, and E. Gomez-Lazaro. Probabilistic

characterization of thermostatically controlled loads to model the impact of demand

response programs. Power Systems, IEEE Transactions on, 26(1):241–251, Feb 2011.

[51] A Nayyar, J. Taylor, A Subramanian, K. Poolla, and P. Varaiya. Aggregate flexibility

of a collection of loads. In Decision and Control (CDC), 2013 IEEE 52nd Annual

Conference on, pages 5600–5607, Dec 2013. Invited.

Bibliography 106

[52] A. Nemirovski and A. Shapiro. Convex approximations of chance constrained pro-

grams. SIAM Journal on Optimization, 17(4):969–996, 2007.

[53] Office of Energy Efficiency and Renewable Energy, U.S. Department of En-

ergy. Buildings energy data book. http://buildingsdatabook.eren.doe.gov/

default.aspx.

[54] P. Palensky and D. Dietrich. Demand side management: Demand response, intel-

ligent energy systems, and smart loads. Industrial Informatics, IEEE Transactions

on, 7(3):381–388, 2011.

[55] A. Papavasiliou and S.S. Oren. Supplying renewable energy to deferrable loads:

Algorithms and economic analysis. In Power and Energy Society General Meeting,

2010 IEEE, pages 1–8, July 2010.

[56] C. Perfumo, E. Kofman, J. Braslavsky, and J. Ward. Load management: Model-

based control of aggregate power for populations of thermostatically controlled loads.

Energy Conversion and Management, 55:36–48, March 2012.

[57] Paul Scott, Sylvie Thibaux, Menkes van den Briel, and Pascal Van Hentenryck. Res-

idential demand response under uncertainty. In Christian Schulte, editor, Principles

and Practice of Constraint Programming, volume 8124 of Lecture Notes in Computer

Science, pages 645–660. Springer Berlin Heidelberg, 2013.

[58] Pierluigi Siano. Demand response and smart grids, a survey. Renewable and Sus-

tainable Energy Reviews, 30:461–478, 2014.

[59] Chua-Liang Su and D. Kirschen. Quantifying the effect of demand response on

electricity markets. Power Systems, IEEE Transactions on, 24(3):1199–1207, Aug.

2009.

http://buildingsdatabook.eren.doe.gov/default.aspx

http://buildingsdatabook.eren.doe.gov/default.aspx

Bibliography 107

[60] J.A. Taylor. Convex Optimization of Power Systems. Cambridge University Press,

2015.

[61] The Mathworks Inc. Matlab r2013a, 2013.

[62] Hans Raj Tiwary. On the hardness of computing intersection, union and Minkowski

sum of polytopes. Discrete & Computational Geometry, 40(3):469–479, 2008.

[63] Andreas Ulbig and Goran Andersson. Analyzing operational flexibility of electric

power systems. International Journal of Electrical Power & Energy Systems, 72:155

– 164, 2015. The Special Issue for 18th Power Systems Computation Conference.

[64] Gokul Varadhan and Dinesh Manocha. Accurate Minkowski sum approximation of

polyhedral models. Graphical Models, 68(4):343 – 355, 2006.

[65] Murat Yilmaz and Philip T Krein. Review of battery charger topologies, charging

power levels, and infrastructure for plug-in electric and hybrid vehicles. Power

Electronics, IEEE Transactions on, 28(5):2151–2169, 2013.

[66] Chaoyue Zhao, Jianhui Wang, Jean-Paul Watson, and Yongpei Guan. Multi-

stage robust unit commitment considering wind and demand response uncertainties.

Power Systems, IEEE Transactions on, 28(3):2708–2717, 2013.

[67] Jianzhe Zhen and Dick Den Hertog. Computing the maximum volume inscribed

ellipsoid of a polytopic projection. CentER Discussion Paper Series No. 2015-004,

2015.

[68] G.M. Ziegler. Lectures on Polytopes. Graduate texts in mathematics. Springer, 1995.

[69] R. D. Zimmerman, C. E. Murillo-Sanchez, and R. J. Thomas. MATPOWER: Steady-

state operations, planning, and analysis tools for power systems research and edu-

cation. Power Systems, IEEE Transactions on, 26(1):12–19, Feb 2011.