1. Nonsmooth Optimization for Optimal Power Flow over Transmission Networks GlobalSIP 2015 Authors: Y. Shi, H. D. Tuan, S. W. Su and H. H. M. Tam

2. Outline 2 Introduction Mathematical formulation of the OPF Simulation results Literature review Nonsmooth optimization algorithm for OPF GlobalSIP

3. 3 1. Introduction The optimal power flow(OPF) problem was introduced by Carpentier since 1962. (OPF) problem is to locate a steady state operating point in an AC power network such that the cost of electric power generation is minimized subject to operating constraints. This problem is complex economically, electrically and computationally.

4. 4 GlobalSIP 1.1 Potential Savings gross production (GWh) assuming price ($/GWh) cost ($billion) savings ($billion) AU196,987300,000593 World23,131,200300,0006939347 The potential cost savings are based on 5% increase of power dispatch efficiency.

5. 5 GlobalSIP 1.2 OPF overview Optimization problem Classical objective function – Minimize the cost of generation Equality constraints – Power balance at each node Inequality constraints – Network operating limits – Control variables limits

6. 6 GlobalSIP 1.3 Difficulty of OPF The multiple quadratic equality and indefinite quadratic inequality constraints on the voltages variables. The nonlinear constraints are so difficult that most of the state-of- the-art nonlinear optimization solvers often converge to infeasible solutions.

7. 7 GlobalSIP 2. Literature review Gradient Methods Quadratic Programming Interior Point Conventional Methods Artificial Neural Networks Evolutionary Programming Ant Colony Intelligent Methods Semi-definite Relaxation (SDR) Nonsmooth Optimization Algorithm (NOA) Semi-definite Programming

8. 8 GlobalSIP Merits and Demerits Demerits of conventional and intelligent methods: – Local solution – Convergence speed – Initial point decide the quality of solution

9. 9 GlobalSIP Merits and Demerits Merit of SDR If the solution of SDR is rank-one, then the solution is global optimal to the original OPF. Demerit of SDR If the solution of SDR is not rank-one, then the solution is not feasible to the original OPF.

10. 10 GlobalSIP Merits and Demerits Merit of NOA – If the solution of SDR is not rank-one, NOA can iteratively generate a sequence of improved solutions that converge to an optimal rank-one solution. – NOA is applicable for a wide range scale of power networks. (2, 5, 9, 14, 30, 39, 57, 118 nodes)

11. 11 GlobalSIP 3. Mathematical formulation of the OPF

12. 12 GlobalSIP 3. Mathematical formulation of the OPF Objective function: – Minimize total generating cost:

13. 13 GlobalSIP 3. Mathematical formulation of the OPF Equality constraints: – Power balance at each node:

14. 14 GlobalSIP 3. Mathematical formulation of the OPF Inequality constraints: – Limits on active and reactive power at each generator:

15. 15 GlobalSIP 3. Mathematical formulation of the OPF Inequality constraints: – Limits on voltage at each node:

16. 16 GlobalSIP 3. Mathematical formulation of the OPF

17. 17 GlobalSIP 3. Mathematical formulation of the OPF Define the Hermitian symmetric matrix of outer product ： which must satisfy,

18. 18 GlobalSIP 3. Mathematical formulation of the OPF

19. 19 GlobalSIP 4. Nonsmooth optimization algorithm for OPF

20. 20 GlobalSIP 4. Nonsmooth optimization algorithm for OPF

21. 21 GlobalSIP 4. Nonsmooth optimization algorithm for OPF

22. 22 GlobalSIP 4. Nonsmooth optimization algorithm for OPF

23. 23 GlobalSIP 4. Nonsmooth optimization algorithm for OPF

24. 24 GlobalSIP 4. Nonsmooth optimization algorithm for OPF

25. 25 GlobalSIP 5. Simulation results The hardware and software facilities : Processor: Intel(R) Core(TM) i5-3470 CPU @3.20GHz; Software: Matlab version R2013b; Matlab toolbox: Matpower version 5.1 to compute the admittance matrix Y from the power system data; Yalmip with SeDumi 1.3 solver for SDP.

26. 26 GlobalSIP 5.1 WB2mod Network WB2mod is a power network with 1 generator, 1 load bus and 1 transmission line.

27. 27 GlobalSIP 5.1 WB2mod Network

28. 28 GlobalSIP 5.2 WB5 and WB5mod Network WB5 is a power network with 5 buses, 2 generators and 6 transmission lines, which lead to 3 nonlinear equality constraints.

29. 29 GlobalSIP 5.2 WB5 and WB5mod Network

30. 30 GlobalSIP 5.3 Case39mod1 and Case118mod Network

31. 31 GlobalSIP 5.4 Case9, 14, 30, 57 Network

32. 32 GlobalSIP 5.5 Conclusion The proposed nonsmooth optimization algorithm (NOA) is able to overcome the shortcomings of the existing methods to compute its optimal solution efficiency and practically even for networks with reasonably large numbers of buses.

33. 33 GlobalSIP Thank you for your attendance! Any questions are welcome!