12. Elgersburg Workshop, 26. Februar 2018 Optimaler Lastfluss unter nicht-Gauss'schen Unsicherheiten – Ein L2-Problem Tillmann Mühlpfordt, Timm Faulwasser, Veit Hagenmeyer Optimization and Control Group Institut für Automation und angewandte Informatik Karlsruher Institut für Technologie www.iai.kit.edu/control
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Optimaler Lastfluss unter nicht-Gauss'schen Unsicherheiten ... · Objective & Outline Mühlpfordt et al. Optimaler Lastfluss – ein L2-Problem 3 Power systems modeling Simulation
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12. Elgersburg Workshop, 26. Februar 2018
Optimaler Lastfluss unter nicht-Gauss'schen Unsicherheiten –
Images courtesy of https://energypress.eu/wp-content/uploads/2015/05/energy-markets.jpg, http://callmepower.com/images/trading-screen.jpg, visited November 02, 2017Source: Renewables 2014 Global Status Report, REN 21 Steering Committee
RenewablesDe-regulatedelectricity markets
Here: Fast & reliable optimal power flow in the presence of uncertainties.
Tractable formulation & exact solution of OPF in presence of non-Gaussian uncertainties
Power Systems Modeling
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Components (non-exhaustive):
– Generators
– Transformers
– Loads
– Lines
– Inverters
Mathematical model?
Images courtesy of https://feuerzangenbowle-mdg.de.tl/Professor-B.oe.mmel.html, https://www2.ee.washington.edu/research/pstca/pf14/pg_tca14bus.htm, visited February 21, 2018
<<Wo simmer denn dran?Aha, heute krieje merde Dampfmaschin.Also, wat is en Dampfmaschin?>>
‒ Kirchhoff‘s current law (conservation of charge)‒ Kirchhoff‘s voltage law (conservation of energy)
Balance equations:
Constitutive law: ‒ Ohm‘s law (cf. Newton, Fick, Hagen-Poiseuille)
Graph theory:‒ Incidence matrix‒ Graph Laplacian
Power Systems Modeling – AC Optimal Power Flow
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‒ Stochastic uncertainties
‒ Meaningful policies viability
‒ (Dynamics via storages)
How to compute stationary set points that are optimal?
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Power Systems Modeling – Stochastic AC Power Flow
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Assumption:
AC power flow must hold for all realizations
AC power flow holds for all realizations, ifSimplify!
Power Systems Modeling – DC Power Flow
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‒ So-called DC power flow equations—unfortunate name
‒ Linear equations that model AC steady-state operation
One more thing…
Power Systems Modeling – DC Power Flow (cont‘d)
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That means…
Power Systems Modeling – Stochastic DC Power Flow
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Assumption:
DC power flow must hold for all realizations
DC power flow holds for all realizations, ifSimplified!
Recap
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‒ OPF = compute optimal set points
‒ Power flow = nonlinear algebraic constraints
‒ Power flow + uncertainties = equality constraints with random variables
‒ AC hard, DC easier
Stochastic optimal power flow under DC conditions?
Deterministic optimal power flow under DC conditions?
Stochastic Optimal Power Flow – Preliminaries (cont'd)
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Uncertainties modeled as continuous random variables
D. Bienstock, M. Chertkov, and S. Harnett. "Chance-Constrained Optimal Power Flow: Risk-Aware Network Control under Uncertainty,"SIAM Review 2014 56:3, 461-495 .L. Roald, F. Oldewurtel, T. Krause and G. Andersson, "Analytical reformulation of security constrained optimal power flow with probabilistic constraints," 2013 IEEE Grenoble Conference, Grenoble, 2013, pp. 1-6.M. Vrakopoulou, K. Margellos, J. Lygeros and G. Andersson, "A Probabilistic Framework for Reserve Scheduling and N-1 Security Assessment of Systems With High Wind Power Penetration," in IEEE Transactions on Power Systems, vol. 28, no. 4, pp. 3885-3896, Nov. 2013.
Unifying problem formulation using random variables Tractable and exact re-formulation thereof Non-Gaussians considered natively
Here:
Stochastic Optimal Power Flow – L2-Formulation
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Infinite-dimensional
Modeling choice
Random-variable power balance
L. Roald, S. Misra, T. Krause, and G. Andersson. “Corrective Control to Handle Forecast Uncertainty: A Chance Constrained Optimal Power Flow”. In: IEEE Transactions on Power Systems PP.99 (2016), (in press).L. Roald, F. Oldewurtel, B. Van Parys, and G. Andersson. “Security Constrained Optimal Power Flow with Distributionally Robust Chance Constraints”. In: ArXiv e-prints (Aug. 2015). 1508.06061.
Intractability?
Polynomial Chaos Expansion!
Objective & Outline
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Power systems modeling
Simulation example
Conclusions
Polynomial chaos expansion
?!
Tractable formulation & exact solution of OPF in presence of non-Gaussian uncertainties
Polynomial chaos expansion Hilbert space method for random variables
Polynomial Chaos Expansion (PCE)
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L. Fagiano and M. Khammash. “Nonlinear Stochastic Model Predictive Control via Regularized Polynomial Chaos Expansions”. In: Proc. of 51st IEEE Conference on Decision and Control. 2012, pp. 142–147.A. Mesbah, S. Streif, R. Findeisen, and R.D. Braatz. “Stochastic Nonlinear Model Predictive Control with Probabilistic Constraints”. In: American Control Conference. 2014, pp. 2413–2419.J.A. Paulson, A. Mesbah, S. Streif, R. Findeisen, and R.D. Braatz. “Fast Stochastic Model Predictive Control of High-dimensional Systems”. In: 53rd IEEE Conference on Decision and Control. 2014, pp. 2802–2809.
Fourier Series vs. Polynomial Chaos
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Polynomial Chaos Expansion (cont'd)
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T. J. Sullivan. Introduction to Uncertainty Quantification. 1st ed. Vol. 63. Springer International Publishing, 2015.D. Xiu and George E. M. Karniadakis. “The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations”. In: SIAM Journal on Scientific Computing 24.2 (2002), pp. 619–644.
Advantages for stochastic OPF?Accuracy?
Polynomial Chaos Expansion – Advantages
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PCE exact with 2 coefficients for
Need for non-Gaussians:
E. Carpaneto and G. Chicco. “Probabilistic Characterisation of the Aggregated Residential Load Patterns”. In: IET Generation, Transmission Distribution 2 (2008), pp. 373–382.
T. Soubdhan, R. Emilion, and R. Calif. “Classification of Daily Solar Radiation Distributions Using a Mixture of Dirichlet distributions”. In: Solar Energy 83.7 (2009), pp. 1056–1063.
Advantages for stochastic OPF?
– Non-Gaussian random variables
– Moments via PCE coefficients
UniformGaussianGammaBeta
Images taken from Wikipedia entries.
Polynomial Chaos Expansion – Advantages
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Advantages for stochastic OPF?
– Non-Gaussian random variables
– Moments via PCE coefficients
Mean
Variance
Skewness
No sampling required!
– Finite-dimensional
– Standard SOCP
– Optimal coefficients
Correspondence of solutions?
PCE for sOPF – Tractable Formulation
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– Moments
– Galerkin projection
– SOC reformulation
(sOPF)
(SOCP)
Main Result – Correspondence of Solutions
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E.g. Beta, Gamma, Gaussian, uniform,
or combination thereof
Properties of solution?
T. Mühlpfordt, T. Faulwasser, L. Roald and V. Hagenmeyer, "Solving optimal power flow with non-Gaussian uncertainties via polynomial chaos expansion," 2017 IEEE 56th Annual Conference on Decision and Control (CDC), Melbourne, VIC, 2017, pp. 4490-4496.
(SOCP)
(sOPF)
Corollary – Properties of Solution
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Simulation example!
Outline
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Power systems modeling
Simulation example
Conclusions
Polynomial chaos expansion
?!
Simulation Example – IEEE 300-bus
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Compare PCE solution to hypothetical fully-informed case (aka Monte Carlo)
Simulation Example – IEEE 300-bus (cont'd)
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3 min
vs.
40 ms
Results?
Simulation Example – IEEE 300-bus (cont'd)
– Compare statistics per bus
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What happens at bus 48?
– PCE is more conservative
– Histogram for bus 48
– Optimal cost barely affected
– 40 ms (PCE) vs. 3 min (MC)
Simulation Example – IEEE 300-bus (cont'd)
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Consistent results, yet faster computation
Outline
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Power systems modeling
Simulation example
Conclusions
Polynomial chaos expansion
?!
Conclusions
Recap:
– Uncertainties are important
– sOPF is infinite-dimensional in terms of random variables