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
9-11 September 2018 - 50th North American Power Symposium
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
9-11 September 2018 - 50th North American Power Symposium
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
9-11 September 2018 - 50th North American Power Symposium
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.
9-11 September 2018 - 50th North American Power Symposium
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.
9-11 September 2018 - 50th North American Power Symposium
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
9-11 September 2018 - 50th North American Power Symposium
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
9-11 September 2018 - 50th North American Power Symposium
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)
9-11 September 2018 - 50th North American Power Symposium
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)
9-11 September 2018 - 50th North American Power Symposium
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)
9-11 September 2018 - 50th North American Power Symposium
The Bilevel Unit Commitment Model
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)
9-11 September 2018 - 50th North American Power Symposium
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
9-11 September 2018 - 50th North American Power Symposium
The Bilevel Unit Commitment Model
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)
9-11 September 2018 - 50th North American Power Symposium
The Bilevel Unit Commitment Model
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)
9-11 September 2018 - 50th North American Power Symposium
The Bilevel Unit Commitment Model
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)
9-11 September 2018 - 50th North American Power Symposium
The Bilevel Unit Commitment Model
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)
9-11 September 2018 - 50th North American Power Symposium
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
9-11 September 2018 - 50th North American Power Symposium
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.
9-11 September 2018 - 50th North American Power Symposium
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 $/MW 0 ≤ 𝑝4
𝑔≤ 200 MW
9-11 September 2018 - 50th North American Power Symposium
The System Normal Operation
Bus 1 2 3 4 5LMP ($/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 = $17,480
Table: 5-Bus System Generator Primal and Dual Components
9-11 September 2018 - 50th North American Power Symposium
The System Normal Operation
pf12 pf14 pf15 pf23 pf34 pf45Flow (MW) 249.7 186.8 -226.5 -50.3 -26.8 -240
φmax ($/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
9-11 September 2018 - 50th North American Power Symposium
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.
9-11 September 2018 - 50th North American Power Symposium
IEEE 30-Bus System Case StudyThe LMPs of the system buses in both normal and congestedoperations.
9-11 September 2018 - 50th North American Power Symposium
IEEE 30-Bus System Case Study
ON/OFF Output (MW) θmax θmin
G1 1 215.7540 0 0G2 1 67.6460 0 0
Total Generation Cost = $189.2789
Table: Generator Primal and Dual Components
ON/OFF Output (MW) θmax θmin
G1 1 184.3122 0 0G2 1 99.0878 0 0
Total Generation Cost = $208.5774
Table: Generator Primal and Dual Components (Congested)
9-11 September 2018 - 50th North American Power Symposium
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
9-11 September 2018 - 50th North American Power Symposium
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.