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

Outline

2

Introduction

Mathematical formulation of the OPF

Simulation results

Literature review

Nonsmooth optimization algorithm for OPF

GlobalS

IP

3

GlobalS

IP

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

GlobalS

IP

1.1 Potential Savings

gross production

(GWh)

assuming price

($/GWh)

cost ($billion)

savings ($billion)

AU 196,987 300,000 59 3

World 23,131,200 300,000 6939 347

The potential cost savings are based on 5% increase of power dispatch efficiency.

5

GlobalS

IP

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

GlobalS

IP

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

GlobalS

IP

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

GlobalS

IP

Merits and Demerits

• Demerits of conventional and intelligent methods:– Local solution– Convergence speed– Initial point decide the quality of

solution

9

GlobalS

IP

Merits and Demerits

• Merit of SDRIf the solution of SDR is rank-one, then the solution is global optimal to the original OPF.

• Demerit of SDRIf the solution of SDR is not rank-one, then the solution is not feasible to the original OPF.

10

GlobalS

IP

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

GlobalS

IP

3. Mathematical formulation of the OPF •

12

GlobalS

IP

3. Mathematical formulation of the OPF

• Objective function:– Minimize total generating cost:

13

GlobalS

IP

3. Mathematical formulation of the OPF

• Equality constraints:– Power balance at each node:

14

GlobalS

IP

3. Mathematical formulation of the OPF

• Inequality constraints:– Limits on active and reactive power at

each generator:

15

GlobalS

IP

3. Mathematical formulation of the OPF

• Inequality constraints:– Limits on voltage at each node:

16

GlobalS

IP

3. Mathematical formulation of the OPF

17

GlobalS

IP

3. Mathematical formulation of the OPF

Define the Hermitian symmetric matrix of outer product：

which must satisfy,

18

GlobalS

IP

3. Mathematical formulation of the OPF

19

GlobalS

IP

4. Nonsmooth optimization algorithm for OPF

• NOA Motivation： It is obvious that a positive semi-definite matrix is of rank-one if and only if it has only one nonzero positive eigenvalue. Under the positive semi-definiteness condition , the matrix rank-one constraint is thus equivalent to the spectral constraint

20

GlobalS

IP

4. Nonsmooth optimization algorithm for OPF

• NOA Motivation： Instead of handling nonconvex constraint, we incorporate it into the objective, resulting in the following alternative formulation

21

GlobalS

IP

4. Nonsmooth optimization algorithm for OPF

• NOA Motivation： The following convex optimization problem provides an upper bound for the nonconvex optimization problem

22

GlobalS

IP

4. Nonsmooth optimization algorithm for OPF•

23

GlobalS

IP

4. Nonsmooth optimization algorithm for OPF•

24

GlobalS

IP

4. Nonsmooth optimization algorithm for OPF

Simulation Experience: is determined to make that F() and are of similar magnitude; 2. If decreasestoo slowly reset ;If decreasestoo quickly reset ;

25

GlobalS

IP

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

GlobalS

IP

5.1 WB2mod Network

• WB2mod is a power network with 1 generator, 1 load bus and 1 transmission line.

27

GlobalS

IP

5.1 WB2mod Network

28

GlobalS

IP

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

GlobalS

IP

5.2 WB5 and WB5mod Network

30

GlobalS

IP

5.3 Case39mod1 and Case118mod Network

31

GlobalS

IP

5.4 Case9, 14, 30, 57 Network

32

GlobalS

IP

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

GlobalS

IP

Thank you for your attendance!

Any questions are welcome!