Transcript Document

Project Presentation
E9501: Electrical Power Network
Instructor: Javad Lavaei
Xiangying Qian
[email protected]
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)
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 …
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:
𝑀𝑖𝑛 𝑇𝑟𝐴0 𝑋
𝑠. 𝑡. 𝑇𝑟𝐴𝑖 𝑋 = 𝑏𝑖 𝑖 = 1,2, … , 𝑚
𝑋≽0
𝑀𝑎𝑥 𝑏𝑇 𝑦
𝑠. 𝑡. 𝐴0 − 𝑚
𝑖=1 𝑦𝑖 𝐴𝑖 ≽ 0 𝑖 = 1,2, … , 𝑚
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”
• 𝑦𝑘 =< 𝑓, 𝜑𝑘 >, 𝑘 ∈ 𝑀 (𝑓 is the information signal ∈ 𝑅𝑛 , 𝜑 is the sensing
waveforms, 𝑦 is the measured signal ∈ 𝑅𝑚 )
• Ideally, we would like to measure all N coefficients of 𝑓, but we only get to
observe a subset of these , where 𝑀 ⊂ {1, … , 𝑁}.
• The process of recovering 𝑓 from 𝑦 is ill-posed/underdetermined, as there
are many candidate signals consistent with the data.
• Can recover the signal by 𝑙1 -norm minimization, that is, among all
candidate signals we pick the one whose coefficient sequence has minimal
𝑙1 norm.
Background Knowledge
- 𝒍𝟏 -norm Regularization
• Sparsity-inducing norm to convex optimization
• Consider the convex optimization problem of the form [4]:
𝑚𝑖𝑛𝑤∈𝑅𝑛 (𝑓 𝑤 + 𝜆Ω(𝑤))
where 𝑓: 𝑅𝑛 → 𝑅 is a convex differentiable function;
Ω: 𝑅𝑛 → 𝑅 is a sparsity-inducing, typically non-smooth, non-Euclidean norm.
• When we know a priori that the solutions 𝑤 ∗ only have a few non-zero
coefficients, Ω is often chosen to be 𝑙1 -norm : Ω 𝑤 = 𝑛𝑗=1 |𝑤𝑗 |.
• This means the recovery via 𝑙1 minimization is provably exact when
information signal is sufficiently sparse.
• Regularizing by 𝑙1 -norm is known to induce sparsity in the sense that, a
number of coefficients of 𝑤 ∗ , depending on the strength of the
regularization (𝜆), will be exactly equal to zero.
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.
Motivation and Goal
- Use SDP to solve OPF for a small scale transmission network;
incorporate 𝒍𝟏 -norm regularization into the objective
function so as to obtain sparse OPF solutions.
- Motivation
• Each generator has a high startup cost (𝑆𝑇𝑖𝑡 ) described in [6] as
𝑆𝑇𝑖𝑡 = 𝑇𝑆𝑖𝑡 + (1 − 𝑒 𝐷𝑖𝑡/𝐴𝑆𝑖𝑡 ) 𝐵𝑆𝑖𝑡 + 𝑀𝑆𝑖𝑡 $/hr
(𝑇𝑆𝑖𝑡 turbine startup cost; 𝐵𝑆𝑖𝑡 boiler startup cost; 𝑀𝑆𝑖𝑡 startup maintenance
cost; 𝐷𝑖𝑡 number of hours down; 𝐴𝑆𝑖𝑡 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.
Motivation and Goal
- Look at the problem from two respects:
1. Assume the generation cost is given, find the effectiveness
of sparsification induced by 𝑙1 -norm regularization on power,
as 𝜆 increases from 0 to infinity.
2. 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 on minimizing the
Total Generation Costs + Total Startup Costs.
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 𝜆.
• From the second perspective, I can observe sparsified solutions 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.
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.
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
The end!
Thank you !
Xiangying Qian
[email protected]