stochastic optimization for unit commitment a … › 588b › 57eaeeec536ca...based on the paper...

$: Stochastic Optimization for Unit Commitment A … › 588b › 57eaeeec536ca...Based on the paper \Stochastic Optimization for Unit Commitment { A Review" to appear in IEEE Trans on$
Stochastic Optimization for Unit CommitmentA Review

Qipeng P. Zheng+, Jianhui Wang? and Andrew L. Liu†

+ Department of Industrial Engineering & Management SystemsUniversity of Central Florida? Argonne National Laboratory

† School of Industrial Engineering, Purdue University

INFORMS Annual Meeting 2014, San Francisco, CA

Based on the paper “Stochastic Optimization for Unit Commitment – A Review” to

appear in IEEE Trans on Power Systems. DOI: 10.1109/TPWRS.2014.2355204

Zheng(UCF)&Wang(Argonne)&Liu(Purdue) () Stochastic Optimization for UC November 11th, 2014 1 / 20

http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=6912028

Outline

Outline

1 Introduction

2 Uncertainty Modeling for UC

3 UC models under uncertainty and solution algorithms

4 Market Operations

5 Conclusions and Future Research


Introduction

Power Generation Unit Commitment

I One the key applications in power system operations. (ISO andGENCO).

I Optimal commitment status of the generating units.

I It is a NP-hard problem.

I Two waves of revolutions of UC research:

X Mixed integer programming solution algorithm.X Deterministic to stochastic optimization.

I Increasing demands to deal with uncertainties in the new age ofpower generation.

X High renewable energy penetration (e.g., wind, solar, etc.).X Electricity market deregulation.


Introduction

Our focus of this review

Because of a large number of papers related to this subject, we focus on:

I Short-term unit commitment (hours-ahead to day-ahead) rather thanlonger-term unit commitment (weekly, seasonal and yearly);

I SO techniques in the formulation and solution of UC, instead ofdeterministic UC problems with additional constraints incorporatinginputs derived from statistical methods (e.g., reserve requirementscalculated based on probabilistic forecasts);

I Optimization algorithms that have explicit formulations and can leadto exact solutions rather than metaheuristic methods such as geneticalgorithms, simulated annealing, or swarm-based approaches, etc.

Disclaimer: Due to the vastness and complexity of the literature, any omissions or

inaccurate characterization of the works cited is strictly due to the limitation of the

authors’ knowledge.


Uncertainty

Uncertainty Modeling

I Scenarios

X A scenario is a realization of the all uncertainties.X Usually need a large number of them for stochastic programming

UC models.X Monte Carlo simulation is usually used to generate them based

on a given distribution or stochastic process.X Trade off between desired accuracy and computational

performance.

I Probabilistic forecasting

X Upper and lower quantiles (can be used as the inputs foruncertainty sets).

X Quantile regression, Kernel density estimators, qantile-Copulaestimator, etc.


Uncertainty

Uncertainty Modeling

I Uncertainty Sets

X Box intervals [max{0, d̄ + zασ}, d̄ + zβσ].X Polyhedral sets (reduces conservativeness).X Ellipsoidal sets using expectations and covariance matrix.X Discrete sets, e.g., contingencies, wind power outputs, etc.X Generate uncertainty sets by risk measures (e.g., VaR and

CVaR).X Particularly, constraints on CVaR can be transformed to

polyhedral sets for certain distributions.


Models and Algorithms

UC models under uncertainty and solution algorithms

I Stochastic Programming

X Two Stage Stochastic ProgrammingX Multi Stage Stochastic ProgrammingX Risk Consideration

I Robust Optimization

I Stochastic Dynamic Programming



Two Stage Stochastic Programming Models for UC

I Day-ahead unit commitment schedules (first stage)

minu∈U

cTu + Eξ[F (u, ξ)]. (1)

X Commitment technical constraints U.X ξ is the random vector/variables.X Expected real time cost (e.g., fuel cost).

I Real-time dispatch decisions (second stage)

F (u, s) = minps ,fs

f (ps) (2a)

s.t. Asu + Bsps + Hs fs ≥ ds , (2b)

X s denotes a specific scenario/realization of the r.v.X Technical constraints (e.g., load balancing, ramping, etc.)

† (As ,Bs ,Hs) model contingencies and/or rescheduling.† (ds) model demands and/or renewable energy outputs.



Solving the two stage models

I Use discrete scenarios and the models reduce to a large-scale (a greatamount of scenarios) deterministic optimization problem.

I Decomposition algorithms.

X Benders decomposition or L-shaped method use cutting planesapproximate the expected real-time cost function, Eξ[F (u, ξ)].(acceleration techniques)

X Lagrangian relaxation.

† Relax the unit commitment constraints and the scenariosare decoupled.† Relax the demand and reserve constraints and the units are

decoupled.† Bundle or regularization method to accelerate computation.



Multi Stage Stochastic Programming Models

𝑛1

𝑛2

𝑛9

𝑛3

𝑛6

𝑛10

𝑛13

𝑛4

𝑛5

𝑛7

𝑛8

𝑛11

𝑛12

𝑛15

𝑛14

Scenario 1

Scenario 2

Scenario 3

Scenario 8

Scenario 4

Scenario 5

Scenario 6

Scenario 7

𝑡1 𝑡2 𝑡3 𝑡4

I Capture dynamics of the uncertaintyand decisions are made along the pro-cess of unfolding uncertainties.

I Scenario tree formulations

min∑s∈S

Probs f (us ,ps , rs , xs)

s.t. (us ,ps , rs , xs) ∈ Us , ∀s ∈ S

(us,t , ps,t , rs,t , xs,t)

= (un, pn, rn, xn), ∀(s, t) ∈ Sn, n ∈ N,



Solving the multi stage models

I Computationally very challenging because of the number of scenariosgrow exponentially (e.g., two outcomes at each node/hour, then 16million scenarios for 24 hours).

I Scenarios aggregations/reductions. But still need to usedecomposition methods.

X Column Generation and Lagrangian relaxation (ProgressiveHedging).

X Decompose by scenarios through relaxing the non-anticipativityconstraints.

X Decompose by units through relaxing the demand/reserveconstraints.

X Acceleration methods (stabilization, bundle, regularization, etc.)



Risk Consideration in Stochastic Programming Models

I Risk averse models in addition to expectation minimization.

I Various risk measures and challenges:

X Expected Load Not Served (ELNS).X Variance of total profit.X Loss of Load Probability (LOLP).X Value at Risk (VaR) as same as LOLP or chance-constrained

programs.Computationally challenging and can use Sampling Average

Approximation (SAA).X Conditional Value at Risk (CVaR) (convex constraints and only

continuous variables).



Robust Optimization UC models

I Minimize the worst-case minimal cost (v.s. expected cost instochastic programming).

minu∈U

{cTu + max

v∈V[F (u, v)]

}, (3)

where

F (u, v) = minp,f qTp (4a)

s.t. Avu + Bvp + Hvf ≥ dv (4b)

I Conservative solutions but no scenarios enumeration.

I Various types of uncertain sets for demands, renewable energyoutputs, contingencies, demand side management, etc.

I Budget of uncertainty constraints to adjust therobustness/conservativeness.



Solving Robust Optimization UC Models

I Reformulating the minimal cost problem by taking its dual.

maxv,π (dv − Avu)Tπ (5a)

s.t. Hvπ ≤ 0 (5b)

Bvπ ≤ q (5c)

v ∈ V, (5d)

I A bilinear programming problem with bilinear terms only appearing inobjective function. Guarantee extreme-point optimal solutions forboth π and v.

I Benders decomposition type of approach.

z ≥ (dv∗ − Av∗u)Tπ∗,

I Column-and-constraint generation or constraints generation approach.

z ≥ qTpi

Av∗u + Bv∗pi + Hv∗f

i ≥ dv∗ .



Stochastic Dynamic Programming UC models

I The stochastic dynamic programming framework

infπ∈Π

Vπ(s0) := E

[T−1∑t=0

Ct(st , µt(st), ξt) + CT (uT )

], (6)

I Bellman backward reduction approach for deterministic approach.

I It is hard for stochastic UC to take policy enumeration approach.

I Approximate dynamic programming.

X Value Function Approximation.X Policy Function Approximation (e.g., Model Predictive Control).X State-space Approximation.


Market Operations

Market Operations

I UC can be used to aid day-ahead market clearing, and to improvepost-day- ahead and intraday reliability.

I Stochastic market clearing needs to ensure revenue adequacy.

I Cooptimiztion of energy and ancillary service needs furtherinvestigation.

I Revenue adequacy and associated issues such as pricing, settlement,market power, and uplift charges all need to be addressed before anymodel can be put into use in practice.

I Models should be fair, transparent, and comprehensible to allparticipants.

I Definitions of scenarios, uncertainty sets all will be highly contentiousamong participants and be barrier to implement SO UC approaches.


Conclusions and Future Research


1. Uncertain Modeling

I Better renewable energy output forecasting (e.g., wind and solar).

I A delicate balance needs to be achieved between economics andreliability.

I Improvements of existing approaches, such as scenario selection,reduction, and evaluation.

I New uncertainty modeling concepts (e.g., data-driven optimization,distributionally robust optimization etc.).

I Muliscale modeling (e.g., detailed modeling of different time-scaledecisions will improve model fidelity).




2. Computational Challenges

I Efficient convexification method for nonconvex second stage problems.

I Acceleration techniques for different types of decompositionalgorithms.

I Advanced techniques based on reformulation and approximation forrobust models with multiple stages and distributionally robust models.

I Value function approximation using post-state decision variables forSDP.

I Model Predictive Control approach to SDP UC need further study.

3. Market Design

I Fair and Transparent settlement rules and pricing.

I Link forecasting errors with prices, requirements, compensation forreserves.

I Incorporating sustainability and environmental concerns.



Acknowledgements

I We would like to thank the discussion with Dr. Feng Qiu.

I We would like to thank the editors and the reviewers for suggestions.

I Dr. Liu would like to thank NSF for support (CMMI-1234057).

I Dr. Wang would like to thank DOE Office of Electricity Delivery andEnergy Reliability for support.

I Dr. Zheng would like to thank NSF for support (CMMI-1355939).



Thank you!

Thank you!

Questions?


stochastic optimization for unit commitment a … › 588b › 57eaeeec536ca...based on the paper...

Documents