1. EMS Smart Grid Complementarity Constraints in Storage-Concerned Economic Dispatch and a New Exact Relaxation Method Zhengshuo Li PhD candidate, Tsinghua University May, 2015

2. EMS Smart Grid Contents IntroductionProposed MethodNumerical TestsConclusions

3. EMS Smart Grid I. Introduction

4. EMS Smart Grid 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

5. EMS Smart Grid Backgrounds : About Storage Storage has been largely integrated into power grids Battery storage systems Lead-acid batteries (most mature) LI-Ion batteries Sodium–Sulfur batteries (NaS) Most cost-efficient technology on the market, mainly applied to PV systems Super-Capacitor Storage Systems Electrochemical double-layer capacitors (EDLCs) Relatively low internal resistance, higher energy density, long cycle life and can store and deliver energy at higher power rating Mainly employed for power quality applications

6. EMS Smart Grid Flywheel Storage Systems Storing energy in the form of kinetic energy of a rotational mass Their power density is almost ten times greater than the batteries Employed for power quality applications. Due to their fast response time, they are used generally for frequency regulation Superconducting Magnetic Energy Storage (SMES) The energy is stored in the magnetic field generated by the dc flow through the superconductive coil The most important applications are in the field of transmission and distribution grids, in order to improve the voltage stability within a utility network

7. EMS Smart Grid Pumped Hydrostorage When there is an energy surplus, the electricity can be bought at low prices and further used to pump the water from a lower aquifer to a higher aquifer This technology helps make profit, and it can be employed when the electrical energy generation has an intermittent character Some other storage-like devices E.g. Electric Vehicles (EVs) Compressed Air Energy Storage (CAES) The CAES uses the grid surplus to compress air in an aboveground or underground reservoir, and, when the energy demand increases, the compressed air is heated and expanded through a gas turbogenerator in order to produce electricity It can be used for bulk energy storage

8. EMS Smart Grid TransmissionTransmission DistributionDistribution 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

9. EMS Smart Grid Motivation : Solution issue arises with storage integration into ED problems complementarity constraints MPEC problemHence, 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 ） P ch P dc P ch P dc Kuhn-Tucker conditions are valid Kuhn-Tucker conditions are INVALID Charging and discharging rate limits Complementa- rity constraint Physically, no storage can be charged and discharged simultaneously

10. EMS Smart Grid Motivation : Most common methods for MPEC problems LONG 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

11. EMS Smart Grid 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 μ k, which may result in long solution time For instance… Original form

12. EMS Smart Grid 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? P ch P dc non-convex and hard to solve P ch P dc Other constraints Optimal solution A new solution idea arises from empirical observations Origi- nal Form convex and easy to solve Other constraints Relax Form

13. EMS Smart Grid II. Exact Relaxation Method

14. EMS Smart Grid Formulation of a storage-concerned ED problem (general form) Subject to: for each time t, A. Storage device constraints (self-discharging)B. Generator operating constraints C. Network constraints 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

15. EMS Smart Grid Interpretation of the storage cost Typical scenarios Description Scenario 1Both charging and discharging are costs for grid dispatch Scenario 2Storage operational cost is neglected in grid dispatch Scenario 3 Storage pays the grid for charging energy and the grid pays the storage for discharging energy default assumption used in general cases with different signs 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 f i ’ and g i ’

16. EMS Smart Grid Sufficient conditions for exact relaxation Condition 1 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?

17. EMS Smart Grid 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: How to prove that?

18. EMS Smart Grid Mathematical Proof (1) (2)

19. EMS Smart Grid Mathematical Proof By adding (1) and (2) Because of α i,2 (t) ≥ 0, and Cond. 2, it follows from (1) that There exists contradiction ! Hence, no storage can charge and discharge simultaneously !

20. EMS Smart Grid Discussion-1 : Satisfaction of condition 1 in real life

21. EMS Smart Grid 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.

22. EMS Smart Grid 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. MAPEfrom 0.9% to 1.5% 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

23. EMS Smart Grid 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..

24. EMS Smart Grid 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

25. EMS Smart Grid 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. Condition 1 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

26. EMS Smart Grid III. Numerical Tests

27. EMS Smart Grid 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..

28. EMS Smart Grid 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 The loads and maximum wind output in the dispatch horizon Test bed information Simulation scenarios S1 S2 S3

29. EMS Smart Grid Result 1 Exact Lowest LMP ($/MWh) Time of MIP method by CPLEX (s) Time of solving relaxed model by CPLEX (s) S1: (-5, 20)yes12.38< 10 -2 kW 2 4.7441.821 S2: ( 0, 0 )yes13.32< 10 -2 kW 2 6.2481.893 S3: (10, 15)yes15.68< 10 -2 kW 2 5.5811.986 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

30. EMS Smart Grid Result 2 134 2 1 3 4 2 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)

31. EMS Smart Grid Result 3 We 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

32. EMS Smart Grid Result 4 Numerical examples that Cond. 1 is violated Numerical examples that Cond. 2 is violated MUTUALLYConds. 1 and 2 MUTUALLY guarantee the exactness of the exactness of the relaxation Observations

33. EMS Smart Grid IV. Conclusions

34. EMS Smart Grid 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

35. EMS Smart Grid