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Project Presentation E9501: Electrical Power Network Instructor: Javad Lavaei Xiangying Qian PowerPoint Presentation
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Project Presentation E9501: Electrical Power Network Instructor: Javad Lavaei Xiangying Qian xq2120@columbia.edu. Background Knowledge Optimal Power Flow (OPF) Loads are given Each Generator has a cost function (random and nonlinear -> piecewise linear in practice)

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slide1

Project Presentation

E9501: Electrical Power Network

Instructor: JavadLavaei

XiangyingQian

xq2120@columbia.edu

slide2

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)
slide3

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 …

slide4

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:

slide5

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.
slide6

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.
slide7

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.
slide8

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.
slide9

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.
slide10

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.
slide11

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.
slide12

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

slide13

The end!

Thank you !

XiangyingQian

xq2120@columbia.edu