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

Motivation – Paradigm Shift in Power Systems

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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.

Uncertainties!

https://www.ee.washington.edu/research/pstca/pf14/pg_tca14bus.htm

Re-dispatch

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

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?>>

power system

power system

Power Systems Modeling (cont’d)

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Standing assumptions:

– Lumped parameter systems

– Steady state

– Alternating current, single phase

Ingredients:

‒ 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

– PCE provides exact solution via tractable SOCP

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Ongoing work:

– AC case (optimal policies, exactness, …)?

– Dynamic/multi-stage OPF?

– PCE truncation errors?

– When is PCE solution = MC solution?

Thank you.