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Complementarity Constraints in Storage-Concerned Economic

Dispatch and a New Exact Relaxation Method

Zhengshuo Li

PhD candidate, Tsinghua University

May, 2015

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Contents

Introduction

Proposed Method

Numerical Tests

Conclusions

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I. Introduction

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Brief Backgrounds : About Storage Various storage has been largely integrated into power grids :

Battery storage systems

Super-Capacitor Storage Systems

Flywheel Storage Systems

Superconducting Magnetic Energy Storage (SMES)

Pumped Hydro-storage

Compressed Air Energy Storage (CAES)

Other storage-like devices

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Transmission

Distribution

• Different types of storage have different features in terms of P and T• Storage is being widely used in both smart transmission and distribution

systems for different purposes, one of which is Economic Dispatch (ED).

Pumped hydro

CAES

Battery SMES

Battery

EVs

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Motivation : Solution issue arises with storage integration into ED problems

• Hence, complementarity constraints, which prevent simultaneous charging and discharging, should be included in a storage-concerned ED model, making the model strongly non-convex and difficult to solve with regular interior-point-based methods（MPEC problem）

Pch

Pdc

Pch

Pdc

Kuhn-Tucker conditions are valid

Kuhn-Tucker conditions are INVALIDdcdc dc

chch ch

P P P

P P P

0dc chP P

Charging and discharging rate limits

Complementa-rity constraint

• Physically, no storage can be charged and discharged simultaneously

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Motivation : Most common methods for MPEC problems

Mixed-integer programming

Penalty function method

Smoothing method

Regularization relaxation

Result in LONG

solution time due to

additional integer

variables or iteratively solving a series of

optimization problems

Linear constraints with binary variables are used to replace the complementarity constraints, and then an equivalent MIP problem is formed and solved

Penalty function is used to “relax” the complementarity constraints as one objective term, but the penalty factor must be determined by solving a series of problems

Smoothing function is used to approach the complementarity constraints. However, iterations are needed for convergence to the “true” optimal solution

Relax the equality in the complementarity constraints as an inequality so that the relaxed problem can be solved with regular method. However, iterations are still needed

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0 1, , 0,1

x x

y y

x y x y

xI x xIx x x

y y y yI y yI

xy I I I I

0 , 0k k

x x x x x x

y y y y y y

xy xy

MIP Transformation

Regularization Relaxation

Though they are linear constraints, it is not so efficient

to solve linear MIP problems yet

Iterations are needed to solve a series of relaxed problems

regarding various μk, which may result in long solution time

For instance…

0

x x x

y y y

xy

Original form

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• Based on large numbers of numerical tests, we found that in some conditions, even if the complementarity constraints were removed, the optimal solution remained unchanged.

• Hence, from that fact arises a new idea : Could we just remove the complementarity constraints and solve

the relaxed model instead? And under what conditions is that relaxation exact?

0

x x x

y y y

xy

Pch

Pdc

non-convex and hard to solve

Pch

Pdc

x x x

y y y

Other constraints

Optimal solution

Optimal solution

A new solution idea arises from empirical observations

Origi-nal

Form

convex and easy to solve

Other constraints

Relax Form?

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II. Exact Relaxation Method

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Formulation of a storage-concerned ED problem (general form)

min ( ) ( ) ( )dc ch Gi i i i i i

i N t T i N t T

F g P t f P t h P t

0 ( ) ( )ch chi iP t P t

0 ( ) ( )dc dci iP t P t

( ) (1 ) ( 1) ( ) ( ) /ch ch dc dci i i i i i iE t E t P t P t t

min max( ) ( ) ( )i i iE t E t E t

Subject to: for each time t,

0( ) ri i iE T E E

( ) ( ) 0ch dci iP t P t

A. Storage device constraints (self-discharging) B. Generator operating constraints

C. Network constraints

( )G G Gi i iP P t P

( 1) ( )dn G G upi i i iR t P t P t R t

( ) ( ) ( ) ( )G dc chi i i i

i N i N i N

P t P t P t D t

( ) ( ) ( ) ( )Ln G dc ch Lnj j i i i i i j

i N

P GSF P t P t P t D t P

Charging and discharging rate limits

Storage capacity limit

Charging/discharging process equation

Complementarity constraint

L to R: storage:{discharging cost, charging income}; generation cost

Ramp limit

Power balance

Line transmission limit

Output limit

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Interpretation of the storage cost

Typical scenarios

DescriptionSigns of

(, )

Scenario 1 Both charging and discharging are costs for grid dispatch (,)

Scenario 2 Storage operational cost is neglected in grid dispatch (, )

Scenario 3Storage pays the grid for charging energy and the grid pays

the storage for discharging energy(, )

( ) ( )dc chi i i i

i N t T

g P t f P t

The format in the objective is based on the default assumption that storage pays the grid for charging and the grid pays the storage for discharging; however, it can be used in general cases with different signs of fi’ and gi’

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Sufficient conditions for exact relaxation

' ( ) ' ( ) ,dc chi i i ig P t f P t t

' ( ) ( ) ,chi i if P t LMP t t

Condition 1: discharging price should be no less than charging price

Condition 2: charging price should be strictly less than the locational marginal price (LMP)

Go back to our question:

If we relax the complementarity constraints in the above model, then under what conditions is the relaxation exact?

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Mathematical Proof

• Convexity of the relaxed model

• KT conditions are valid for the relaxed model

• Proof by contradiction

Key Points

Lagrangian function of the relaxed model:

,1 ,2 ,3 ,4

0 min max 0,1 ,21

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) (1 ) (1 ) ( ) ( ) / ( ) ( ) ( ) (1 ) (1 )

ch ch ch dc dc dci i i i i i i i i i

t t t t

tt t ch ch dc dc t t chi i i i i i i i i i i i i i i i

t

L F t P t t P t P t t P t t P t P t

t E P P t E t t E t E P

1

1

,1 ,2

( ) ( ) /

(1 ) ( ) ( ) /

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

t ch dc dci i

t

T T t ch ch dc dc ri i i i i i it

G dc chi i i i

t i N i N i N

G dc ch Ln Ln Gj j i i i i i j j j j i i

t i N

P t

P t P t t E

t P t P t P t D t

t GSF P t P t P t D t P t P GSF P t

( ) ( ) ( )dc chi i i

t i N

P t P t D t

How to prove that?

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Mathematical Proof• Assume there exists and for storage i at time t. Let ξi denote .

• Then, αi,1(t) and αi,3(t) regarding the charging and discharging rate limits are both ZERO because of the complementary slackness conditions.

• Since KT conditions hold for the relaxed convex optimization, so the optimal solution must satisfy (1), (2) below:

,2 ,1 ,2

,1 ,2

' ( ) ( ) ( ) ( )( )

( ) ( ) ( ) 0

ch ch t T ti i i i i i i i ich t

i

j i j jj

L f P t t tP t

t GSF t t

,4 ,1 ,2

,1 ,2

' ( ) ( ) ( ) ( ) /( )

( ) ( ) ( ) 0

dc t T t dci i i i i i i i idc t

i

j i j jj

L g P t t tP t

t GSF t t

(1)

(2) ,1 ,2( ) ( ) ( ) ( )i j i j jj

LMP t t GSF t t

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Mathematical Proof

By adding (1) and (2) ,1 ,2

,2 ,4

1/ ( ) ( )

' ' ( ) ( ) 0

dc ch t T ti i i i i i it

i i i i

t

g f t t

Because of αi,2(t), αi,4(t) ≥ 0, and Cond. 1 ,1 ,2( ) ( ) 0t T t

i i i i it

Because of αi,2(t) ≥ 0, and Cond. 2, it follows from (1) that

,1 ,2( ) ( ) 0t T ti i i i it

• There exists contradiction !

• Hence, no storage can charge and discharge simultaneously !

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Discussion-1 : Satisfaction of condition 1 in real life

• Obviously, with charging and discharging prices as inputs, Cond. 1 can be easily checked.

• In order to attract storage owners to participate in ED, it is reasonable to expect that ≥ , namely, the marginal compensation paid for discharging a unit of energy must cover the marginal cost for the owner to charge that amount of energy back in real life.

• Hence, Cond. 1 can be usually satisfied in real life no matter who, the utility or a third-party, owns the storage.

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Discussion-2 : Satisfaction of condition 2 in real life

• Obviously, with the charging prices and predicted LMPs (or its lower bound), Cond. 2 can be easily checked.

• If the storage’s charging price is mainly determined by the policy from the government or the power grid company, since the storage’s flexible charging benefits the grid, the storage would be very likely to be rewarded by charging at a low charging price (e.g., pumped hydro), even lower than the actual LMP, so as to be attracted to participate in the economic dispatch.

• Hence, Cond. 2 would be satisfied most likely in reality.

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Discussion-3 : Prediction accuracy of LMP

Citation of the LMP prediction results with the ANN approach in [1]

[1] M. Shahidehpour, H. Yamin, Z. Li, Market Operations in Electric Power Systems: Forecasting, Scheduling, and Risk Management, Wiley-IEEE Press, 2002.

Although LMPs are difficult to forecast, several effective approaches have been reported, e.g., artificial neural network (ANN) approach.

In [1], the mean absolute percent error (MAPE) of the LMP is from 0.9% to 1.5% with different load patterns.

If the standard deviation of the LMP forecasting accuracy can be known, we can estimate and use the lower bound of the actual LMP as well

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Extension-1 : Extension of the ED model

• From the above proof, it can be seen that any ED model in the same form or with convex additions can also be exactly relaxed. E.g., wind-EV coordination problem [2], where EV storage is

coordinated with thermal-wind generation systems.

• The proof can also be applied in all the three scenarios in the table, e.g., the grid also pays the storage for charging

[2] Z. Li, Q. Guo, H. Sun, Y. Wang, and S. Xin, "Emission-Concerned Wind-EV coordination on the transmission grid side with network constraints: Concept and case study," IEEE Trans. Smart Grid, vol. 4, no. 3, pp. 1692-1704, Sept. 2013..

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Extension-2 : Other groups of sufficient conditions for the relaxation

For the above storage-concerned ED problem, we have recently obtained two more groups of sufficient conditions for exact relaxation [3].

[3] Z, Li, Q. Guo, H. Sun, and J. Wang, "Further Discussions on Sufficient Conditions for Exact Relaxation of Complementarity Constraints for Storage-Concerned Economic Dispatch," arxiv, 2015.

The first of the new groups guarantees the exactness under the condition where charging price = LMP

The second of the new groups guarantees the exactness always holds under the condition where LMP is non-negative

NEW

Contribute to wider application of the exact relaxation method in storage-concerned ED problems

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Extension-3 : Proposed methodology applied in other dispatch patterns

For distributed storage dispatch in load leveling problems on distribution side [4], the complementarity constraints can also be proven to be exactly relaxed under two conditions which are a little different from the ones presented above.

[4] Z, Li, Q. Guo, H. Sun, and J. Wang, "Storage-like devices in load leveling: Complementarity constraints and a new and exact relaxation method," Appl Energy 2015; 151: 13-22.

' ( ) ' ( ) ,dc chi i i ig P t f P t t

0' ( ) ( ) ,chi if P t D t t

Condition 1: discharging price should be no less than charging price

Condition 2: weighted charging price should be less than the regular loads in the distribution grids

Conds. for Load Leveling

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III. Numerical Tests

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Test systems• IEEE 30-bus systems

• 5 units @ buses #1, #13, #22, #23, #27, with total generation capacity of 520 MW

• 1 wind farm @ bus #2, with the output in the range of 0-50 MW

• The 50 storages @ PQ buses (each bus has two storage devices on average)

• Each storage has 400-kW bidirectional power rate and 2-MWh capacity, with charging and discharging of 90% [5].

[5] P. Yang, and A. Nehorai, "Joint optimization of hybrid energy storage and generation capacity with renewable energy," IEEE Trans. Smart Grid, vol. 5, no. 4, pp. 1566-1574, Jul. 2014..

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Test systems

• The test environment is in Matlab on a laptop computer with a CPU @ 2.60 GHz and 8 GB RAM.

• The most commonly used MIP method and the proposed exact relaxation method are compared

• The solver is IBM ILOG CPLEX®12.5

• 3 scenarios are considered:

S1: Grid pays for both charging and discharging S2: Storage operational cost is neglected S3: Grid pays for discharging and storage pays for charging

0

10

20

30

40

50

0

100

200

300

400

500

600

1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91

Win

d (

MW

)

Loa

ds

(MW

)

Dispatch horizon

Loads Wind

The loads and maximum wind output in the dispatch horizon Test bed information

Simulation scenariosSigns of (, )

S1 (, )

S2 (,)

S3 (, )

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Result 1

(, ) ExactLowest LMP

($/MWh)Relax gap

Time of MIP method by CPLEX (s)

Time of solving relaxed model by

CPLEX (s)

S1: (-5, 20) yes 12.38 < 10-2 kW2 4.744 1.821

S2: ( 0, 0 ) yes 13.32 < 10-2 kW2 6.248 1.893

S3: (10, 15) yes 15.68 < 10-2 kW2 5.581 1.986

Relaxation exactness and computational time comparison for the three scenarios

• With Conds. 1 and 2 satisfied, the relaxation is EXACT (the relax gap is very small)

• The objectives of the relaxed model and the MIP solutions are exactly THE SAME

• The solution time of solving RM is much SHORTER, decreased by 65%

Observations

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Result 2

0

10

20

30

40

50

0

100

200

300

400

500

600

1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91

Win

d (M

W)

Loa

ds (M

W)

Dispatch horizon

Loads Wind

1 3 42

0 8 16 24 32 40 48 56 64 72 80 88 96

-20

-10

0

10

20

upper discharging limit = 20 MW

Pow

er o

f S

tora

ge(M

W)

Time Slots

upper charging limit = 20 MW

1

3

42

The loads and maximum wind output in the dispatch horizon

• Storage charges most at the load valley (e.g., area 3) and discharges most at the peak load time (e.g., areas 1 and 2) or scarce wind time (e.g., area 4)

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Result 3We also tested cases with storage of larger energy capacity• IEEE 30-bus systems

• 5 units @ buses #1, #13, #22, #23, #27, with total generation capacity of 2000 MW

• 1 wind farm @ bus #2, with the output in the range of 0-200 MW

• The 24 storages @ PQ buses (each bus has one storage device)

• Each storage has 2.5-MW bidirectional power rate and 12-MWh capacity, with charging and discharging of 90% [1].

• As long as the conditions are satisfied, the relaxation is exact no matter what the specific parameters of the model are

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Result 4Numerical examples that Cond. 1 is violated

Numerical examples that Cond. 2 is violated

• If (, ) = (15, 14), the maximum of is over 105 kW2, so the relaxation results is infeasible for the original model

• If = 25 > = 24 > LMP, the maximum of is over 105 kW2, so the relaxation results is infeasible for the original model

• Conds. 1 and 2 MUTUALLY guarantee the exactness of the exactness of the relaxation

Observations

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IV. Conclusions

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• Regarding a general storage-concerned economic dispatch problem, a new exact relaxation method is proposed to relax the tough complementarity constraints with two sufficient conditions satisfied

The research paper has been published in IEEE trans. Power Systems

• The sufficient conditions can be found usually satisfied in reality

• The relaxed problem can be solved much more efficiently than the current solutions of MPEC, e.g., MIP algorithms

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Thanks for your attention!

Q & A