Project Presentation E9501: Electrical Power Network Instructor: Javad Lavaei Xiangying Qian

1. Project Presentation E9501: Electrical Power Network Instructor: JavadLavaei XiangyingQian xq2120@columbia.edu

2. Background Knowledge • Optimal Power Flow (OPF) • Loads are given • Each Generator has a cost function (random and nonlinear -> piecewise linear in practice) • Minimize total generation cost to meet physical, operational and network constraints: Active power balance equations Reactive power balance equations Active and reactive power generation limits Power flow limit of lines From and to side bus voltage limits … … • The optimal prices in a transmission network are resulting from an OPF performed by a centralized dispatcher, e.g. independent system operator (ISO)

3. Background Knowledge • Unit Commitment (UC) , Security-Constrained UC (SCUC ) • Find the least-cost dispatch of available generation resources (units) to meet the electrical load [1]. • Apart from the cost of running these units, we have additional costs and constraints: start-up cost/shut-down cost/ spinning reserve/ ramp-up time ... • So we CANNOT just flip the switch of certain units and use them arbitrarily! • We need to think ahead (say, 24hr before), based on the forecasted load and unit constraints, determine which units to turn on (commit) and which ones to keep down. • Intuitive solution: go over all combinations of choice from hour to hour, for each combination at a given hour, solve the economic dispatch and pick the combination with lowest cost. -> Impossible for large-scale power network! • Deal with discrete decision variables -> possible approaches: Lagrangian relaxation, interior point method …

4. Background Knowledge • Semi-definite Programming (SDP) • Minimize a linear function subject to the constraint that an affine combination of symmetric matrices is positive semi-definite [2]. • SDP unifies several standard problems (e.g. linear and quadratic programming) and finds many applications in engineering and combinatorial optimization. • Standard form of SDP: Primal: Dual:

5. Background Knowledge • Compressed Sensing (CS) • CS theory asserts that we can recover certain signals from far fewer samples or measurements than traditional Nyquist rate requires [3]. • Relies on “Sparsity” and “Incoherency” • ( is the information signal , is the sensing waveforms, is the measured signal ) • Ideally, we would like to measure all N coefficients of , but we only get to observe a subset of these , where . • The process of recovering from is ill-posed/underdetermined, as there are many candidate signals consistent with the data. • Can recover the signal by -norm minimization, that is, among all candidate signals we pick the one whose coefficient sequence has minimal norm.

6. Background Knowledge • -norm Regularization • Sparsity-inducing norm to convex optimization • Consider the convex optimization problem of the form [4]: where : is a convex differentiable function; : is a sparsity-inducing, typically non-smooth, non-Euclidean norm. • When we know a priori that the solutions only have a few non-zero coefficients, is often chosen to be -norm : . • This means the recovery via minimization is provably exact when information signal is sufficiently sparse. • Regularizing by -norm is known to induce sparsity in the sense that, a number of coefficients of , depending on the strength of the regularization (), will be exactly equal to zero.

7. Related Work • Semi-definite programming-based method for SCUC with operational and optimal power flow constraints [5] (by X. Bai, H.Wei, IET Generation, Transmission & Distribution) • Apply SDP model to the SCUC problem and solve it efficiently using interior-point method. • When the solution contains minor mismatches in the integer variables, a simple rounding strategy is used to correct non-integers into integers. • The simulations on 6 to 118 bus networks over a 24hr period show the proposed method is capable of obtaining optimal UC schedules without breaking any constraints and minimizing operation cost at the same time. • Indicate SDP sparsity technique and new round strategy for non-interger variables as the future work. • However, not penalize the relaxation of discrete variables in the objective.

8. Motivation and Goal • Use SDP to solve OPF for a small scale transmission network; incorporate -norm regularization into the objective function so as to obtain sparse OPF solutions. • Motivation • Each generator has a high startup cost () described in [6] as $/hr ( turbine startup cost; boiler startup cost; startup maintenance cost; number of hours down; boiler cool down coefficient ) • Because the startup cost function has a jump and discontinuous characteristic, OPF just ignores it for simplicity. • It is better to operate the grid with as few generators as possible when taking the market into account. • Having sparse solutions of OPF is helpful to decide unit commitment.

9. Motivation and Goal • Look at the problem from two respects: • Assume the generation cost is given, find the effectiveness of sparsification induced by -norm regularization on power, as increases from 0 to infinity. • Assume the generation cost is not important, find a minimum number of generation resources that satisfy all operational and network constraints. • Set up an appropriate startup cost for each generator. • The comparison and evaluation is based onminimizing the Total Generation Costs + Total Startup Costs.

10. Current Progress • Source data: • IEEE 14-bus • Method: • Formulate SCOPF as primal problem of SDP, then solve it using existing SDP solvers such as “Sedumi”. • Simulation Tools: • Matlab R2011a, MATPOWER, CVX, Yalmip • Preliminary Results: • From the first perspective, I use the given quadratic cost functions, but cannot observe the benefit of sparsification by changing . • From the second perspective, I can observe sparsifiedsolutions to the power by setting random cost vector. So I can seek for a minimum number of generation units that satisfy all operational and network constraints.

11. Problem Solving and Future Work • Possible solutions to the unexpected results: • Add more generators into the network. • Use all linear cost functions instead. • Tune the bus voltage limits. • Increase the upper bounds on generators. • Since randomly-picked linear cost vector of the generators results in a sparse solution for current 14-bus network, I will extend it to the IEEE 30-bus network and do some comparison. • Another way to solve SDP is from the dual perspective. I will learn more about the dual OPF.

12. References [1] http://en.wikipedia.org/wiki/Power_system_simulation [2] Vandenberghe L., Boyd S., Semidefinite Programming, SIAM REVIEW, 1996, 38, pp. 49-95 [3] Emmanuel J. Candes, Michael B. Wakin, An Introduction to Compressive Sampling, IEEE SIGNAL PROCESSING MAGAZINE, Mar. 2008 [4] Bach F., Jenatton R., Mairal J., Obozinski G., Convex Optimization with Sparsity-Inducing Norms, available at: http://www.di.ens.fr/~fbach/opt_book.pdf [5]X.Bai, H.Wei, Semi-definite programming-based method for SCUC with operational and optimal power flow constraints, IET Generation, Transmission & Distribution, Apr. 2008 [6] Padhy N.P., Unit Commitment – A Bibliographical Survey, IEEE TRANSACTIONS ON POWER SYSTEMS, 2004, 19, pp. 1196-1205

13. The end! Thank you ! XiangyingQian xq2120@columbia.edu