bilevel programming-based unit commitment for locational...

9-11 September 2018 - 50th North American Power Symposium

Bilevel Programming-Based UnitCommitment for Locational Marginal Price

ComputationPresentation at 50th North American Power Symposium

Abdullah AlassafDepartment of Electrical

EngineeringUniversity of South Florida

University of Hail

Dr. Lingling FanDepartment of Electrical

EngineeringUniversity of South Florida

Septemper 10, 2018


Outline

Introduction

Model FormulationThe Conventional Unit Commitment ModelThe Bilevel Unit Commitment Model

The Upper-Level ProblemThe Lower-Level ProblemThe Single-Level Equivalent Problem

Case Studies5-Bus PJM System Case StudyIEEE 30-Bus System Case Study

Conclusion


Introduction

I In the US, locational marginal pricing system is applied in allISOs, independent system operators [B. Eldridge, 2017].

I Locational marginal price (LMP) is the cost of the nextincreased demand unit at a bus.

I Generation company bidding is based on the LMP.I The LMP is the dual variable of the power balance equation.I In the day-ahead market, generation companies determine

on/off status of each unit for the next 24 hours.I Unit status causes nonconvexity in formulating unit

commitment participating in the day-ahead market.


Introduction

I Issues– Commercial solvers cannot reach the problem dual variables.– Traditionally, the LMP is obtained by solving two problems.

The first one is unit commitment. The second problem isDCOPF, where units’ statuses are known.

I Solution– We propose a bilevel model that solves the unit commitment

problem and attains the LMP and the other dual variables.– We present solving the bilevel problem using primal-dual

relationship.

I The solution of the bilevel model and the conventional modelexactly match.


Outline

Introduction




Conclusion


Model FormulationThe comparison diagram of the two models

The Mixed-Integer Unit

Commitment Problem

DCOPF Problem

The Locational Marginal Price

ON/OFF Unit

Variables

The Bilevel Unit Commitment Problem

The Locational Marginal Price

The Conventional Method The Bilevel Method


The Conventional Unit Commitment Model

minpf

l,pg

j ,δn

∑j∈J

Kgj p

gj +

∑j∈J

cuj (1a)

subject to cuj ≥ Kuj (vj − vj0);∀j ∈ J (1b)

cuj ≥ 0;∀j ∈ J (1c)vj ∈ {0, 1}; ∀j ∈ J (1d)∑

j∈Jpgj +

∑l|d(l)=n

pfl −∑

l|o(l)=npfl = P dn ; ∀n ∈ N (1e)

pfl = 1xl

(δo(l) − δd(l));∀l ∈ L (1f)

− P fl ≤ pfl ≤ P

fl ; ∀l ∈ L (1g)

P gjvj ≤ pgj ≤ P

gjvj ; ∀j ∈ J (1h)


The Bilevel Unit Commitment Model

I The Upper-Level Problem

maxvj

∑n∈N

P dnλ∗n −

∑j∈J

cuj (2a)

subject to cuj ≥ Kuj (vj − vj0);∀j ∈ J (2b)

cuj ≥ 0;∀j ∈ J (2c)vj ∈ {0, 1}; ∀j ∈ J (2d)



I The Lower-Level Problem

minpf

l,pg

j ,δn

∑j∈J

Kgj p

gj (3a)

subject to pfl = 1xl

(δo(l) − δd(l)); (νl);∀l ∈ L (3b)

− P fl ≤ pfl ≤ P

fl ; (φminl , φmaxl );∀l ∈ L (3c)

P gjv∗j ≤ p

gj ≤ P

gjv∗j ; (θminj , θmaxj );∀j ∈ J (3d)∑

j∈Jpgj +

∑l|d(l)=n

pfl −∑

l|o(l)=npfl = P dn ; (λn);∀n ∈ N (3e)


The Bilevel Unit Commitment ModelThe structure of the bilevel model

Upper-Level Problem (2)

Maximize The Production Profit.

Determine: The Optimal Unit Status.

𝑣𝑗

Lower-Level Problem (3)

Minimize The Operation Cost.

Determine: The Optimal Dispatch.

𝜆𝑛



The lower-level dual problem

maxλ,ν,φ,θ

ZD =∑n∈N P

dnλn −

∑l∈L P

fl (φmaxl + φminl ) +

∑j∈J v

∗j (θminj P gj − θmaxj P

gj ) (4)

subject to

Kgj − λn + θmaxj − θminj = 0;∀j ∈ J (5a)

λo(l)=n − λd(l)=n − νl + φmaxl − φminl = 0;∀l ∈ L (5b)∑l|o(l)=n

1xlνl −

∑l|d(l)=n

1xlνl = 0;∀l ∈ L (5c)

θmaxj , θminj ≥ 0;∀j ∈ J (5d)φmaxl , φminl ≥ 0;∀l ∈ L (5e)



The single-level equivalent

maxpf

l,pg

j ,δn,λn,νl,φl,θj

∑n∈N

P dnλn −∑j∈J

cuj (6a)

subject to Constraints (2b)− (2d) (6b)Constraints (3b)− (3e) (6c)Constraints (5a)− (5e) (6d)

∑j∈J

P dnλn +∑j∈J

(P gjθminj vj − P

gjθmaxj vj)−

∑l∈L

Pfl (φmaxl + φminl ) =

∑n∈N

Kgj p

gj (6e)



Constraint (6e) is linearized

∑j∈J

P dnλn +∑j∈J

(P gjbj − Pgjaj)−

∑l∈L

Pfl (φmaxl + φminl ) =

∑n∈N

Kgj p

gj (7a)

0 ≤ aj ≤ θmaxj vj (7b)0 ≤ θmaxj − aj ≤ θmaxj (1− vj) (7c)

0 ≤ bj ≤ θminj vj (7d)

0 ≤ θminj − bj ≤ θminj (1− vj) (7e)



The linearized single-level equivalent

maxpf

l,pg

j ,δn,λn,νl,φl,θj

∑n∈N

P dnλn −∑j∈J

cuj (8a)

subject to Constraints (2b)− (2d) (8b)Constraints (3b)− (3e) (8c)Constraints (5a)− (5e) (8d)Constraints (7a)− (7e) (8e)


Outline

Introduction




Conclusion


Case Studies

I The model (8) has been tested on different scale systems.I The tested cases are nesta_case5_pjm and

nesta_case30_ieee, taken from [C. Coffrin, 2014].I The model has been implemented in Matlab with CVX using

solver Gurobi 7.51.


5-Bus PJM System Case Study

10 $/MW

5 0 ≤ 𝑝5𝑔≤ 600 MW Limit = 240 MW

3

30 $/MW

0 ≤ 𝑝3𝑔≤ 520 MW

𝑝4𝑑 = 400 MW

𝑝3𝑑 = 300 MW

2

𝑝2𝑑 = 300 MW

Limit = 400 MW

1

14 $/MW

0 ≤ 𝑝11𝑔≤ 40 MW

15 $/MW

0 ≤ 𝑝12𝑔≤ 170 MW

40 $/MW 0 ≤ 𝑝4

𝑔≤ 200 MW


The System Normal Operation

Bus 1 2 3 4 5LMP ($/MWh) 16.9774 26.3845 30 39.9427 10

Total Revenue = $32,892

Table: 5-Bus System LMPs of The Buses and The Revenue

ON/OFF Output (MW) θmax θmin

G11 1 40 2.9774 0G12 1 170 1.9774 0G3 1 323.4948 0 0G4 0 0 0 0G5 1 466.5052 0 0

Total Generation Cost = $17,480

Table: 5-Bus System Generator Primal and Dual Components


The System Normal Operation

pf12 pf14 pf15 pf23 pf34 pf45Flow (MW) 249.7 186.8 -226.5 -50.3 -26.8 -240

φmax ($/MWh) 0 0 0 0 0 62.32φmin ($/MWh) 0 0 0 0 0 0

Table: 5-Bus System Primal and Dual Components of The Lines


IEEE 30-Bus System Case Study

I The system contains 6 generator units and 41 transmissionlines.

I The system is tested under normal operation.I The system is examined under congestion operation. The

maximum capacity limit of line 1-3 is set 72.5 MW.


IEEE 30-Bus System Case StudyThe LMPs of the system buses in both normal and congestedoperations.


IEEE 30-Bus System Case Study


G1 1 215.7540 0 0G2 1 67.6460 0 0

Total Generation Cost = $189.2789

Table: Generator Primal and Dual Components


G1 1 184.3122 0 0G2 1 99.0878 0 0

Total Generation Cost = $208.5774

Table: Generator Primal and Dual Components (Congested)


Outline

Introduction




Conclusion


Conclusion

I The nonconvexity of unit commitment problem causesdifficulty of achieving the problem dual variables that play animportant rule in market participation.

I We develop a bilevel model that consists of two levelproblems, namely, the upper-level and the lower-level.

– The upper-level has binary decision variables.– The lower-level has only continuous decision variables.

I The bilevel problems are transformed to a single-level problemand solved efficiently using Gurobi.

bilevel programming-based unit commitment for locational...

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