Mathematics for energy systems: an engineer’s view

1. Mathematics for energy systems:an engineer’s view Janusz Bialek

2. Outline • Personal view on collaboration between maths and power communities • How we got here: a short history of power systems research since the 70s • Reflections on INI Maths for Energy Systems – what’s missing? • Examples from my own research

3. What’s this talk about? • A biased view – I hope someone will present a biased view from maths side… • Mathematicians: anyone who is better in maths than me (almost everyone in the room): includes control, OR, optimisation, stats, etcetc • How many power/energy engineers in the audience? • I have collaborated with mathematicians since early 2000s • Why: power networks are, at the end of the day, complex graphs with some specific properties – graph theory, complex networks, control, OR etc. • There must be solutions out there for many problems • I know that I don’t know • Main problem: overcoming jargon and superiority complex • Takes about 2 years

4. Actually for me it is inferiority complex • But I got a revenge: power systems for mathematicians • Hard core fanatics are on both sides: caricature view • Engineers: models must be realistic and must account for and simulate everything • Maths: assume away reality to fit my beautiful method/theorems • Mathematical analysis requires simplified models - then testing by simulation using more realistic models • but how realistic to get accepted by a journal? • Reaction– ban engineers from reviewing papers! • Humility needed from both sides • engineers need more maths, mathematicians need understand physical reality better and we need to learn from each other AC or DC?

5. Debate about maths for energy systems in power system community • PowerGlobe“Diatribe against simulation” , Sept 2016 • My main thesis: today’s power system research relies too much on simulation, too little on mathematical analysis: • Get a model of a power system in PST, PSAT, PowerFactory, etc. • Add, say, a wind farm and see what happens • A point solution for given operating conditions • Understanding about the problem structure by running multiple simulations instead of using mathematical analysis (i.e. brains) • Not unique for power engineering - SPICE Monkeys • over 100 replies, a heated discussion, so I must have hit at something important • Unintended consequences: extra funding for a student who wanted to enhance her maths (Ben Hobbs)

6. 2017 Special session at PowerTech Manchester: • Mathematical foundations of power system analysis: J. Bialek (Skoltech), D Hill (Hong Kong Uni), G. Hug (ETH Zurich), J. Lavaei (UC Berkeley), J-Y Le Boudec and M. Paolone(EPFL) • Huge interest, over 100 people, standing room only • Forthcoming 2019 IEEE PES General Meeting: • Thinking outside the “black box” — analytical foundations of power systems: J. Bialek (Skoltech/Newcastle), I. Hiskens (Michigan), D. Hill (HKU), S. Low (Caltech), F. Milano (UCD, PSAT) defending the simulation camp • Certainly interest and enthusiasm from both sides (although not every dinosaur would agree) but how did we get here?

7. Timeline of power system research since the 70s price • 70’s and 80’s – “golden era” of developing analytical toolsfollowing NYC blackouts in 1965 and 1977: • real-time security assessment • Increase use of computers • Big names: Fred Schweppe (state estimation, spot pricing), Bill Tinney (sparse solution methods – died last week), F. Carpentier (OPF 1962), B. Stott (OPF), Felix Wu, etcetc • 90’s – “black hole” for power system research • Market reforms - the invisible hand of the market will solve ALL the problems • Only economics is important, forget engineering • Power systems seen as mature and requiring little funding, industry downsizing, closing down of power courses • Some mathematical advances (OR) – W. Hogan (LMPs), PravinVarayia with F. Wu Consumer’s surplus supply * = MCP demand Producers’ profit q* quantity

8. Post 2000 – discovery of climate change • Explosion of funding – a feast followed famine • A few of us left standing got handsomely rewarded (M₤) • Technical and economic integration of renewable energy, decentralisation, uncertainty analysis due to renewables • Advances in ICT: sensing, WAMS, IoT, Smart Grids • Recent focus:Energy System Integration • All very exciting and begging for new tools: advanced maths, stats, decentralised control, OR • Revival of engagement by researchers from optimisation and control communities – e.g. this INI programme

9. INI Maths for Energy Systems programme • Excellent! • A maths friend: too much economics! • Is it a fair comment? 4 out of 9 tracks • Look-ahead operational planning under uncertainty • Budgeting and scheduling of maintenance and replacement of power system components   • Future Electricity Markets • Data and analytics for short-term operations (forecasting) • Moving energy through time: storage and demand side response • Equilibria and computation in markets with risk • Pricing and optimization of intraday/day-ahead electricity and futures contracts • Planning Low-Carbon Electricity Systems under UncertaintyConsidering Operational Flexibility and Smart Grid Technologies •  Mechanism Design for the Economics of Future Energy Systems  • I have nothing against economics and some of my best friends are economists • The remaining tracks are on the effects of uncertainty of renewables • Is this all that’s needed from maths?

10. What’s missing in INI programme? • More engineers? • Social science • We are ignorant and arrogant about consumers • How to model consumers? • Is economic rationality assumption good enough? • Where are good old hard maths topics like dynamics, stability analysis, system identification etc? • New sensing and ICT tools open new possibilities • Two examples of utilising Phasor Measurement Units (PMUs) for system identification and stability analysis

11. Phasor Measurement Units • Who knows what PMUs are? • Who wants a short tutorial? • PMU: AC waveform sampling with 1ms accuracy: magnitude, phase angle d, frequency w. • frequency 60 Hz • GPS-stamped(synchronized between different PMUs in the network) • Relative phase angles between locations

12. Source: S, Norris, GE Power (Edinburgh)

13. PMUs vs traditional SCADA (Supervisory Control and Data Acquisition) 4 Hz, non-synchronised 60 Hz, synchronised Source: S, Norris, GE Power (Edinburgh)

14. Wide Area Measurement Systems (WAMS) • Connected PMUs at different locations create Wide Area Measurement Systems (WAMS) • Also low-cost distribution level: see FNET/GridEye Angle contour map, FNET/GridEye

15. Applications of PMUs • Now: mostly just simple monitoring of power flows and dynamics, disturbance recorders, experiments with control • But PMUs contain high-resolution dynamic information about the power system 2006 European disturbance • Interesting un-orthodox applications for big data science, signal processing etc. • Gut feeling: it should be possible to extract information about power system model • Could not solve it properly until I started collaboration with Russian mathematicians • Stochastic: use time-domain correlations between state variables - K. Turitsyn(MIT) • Deterministic – use frequency domain (eigenvalues) – T. Dymarski (Skoltech, Kentucky)

16. Why model estimation/validation is important? Power system analysis is normally undertaken using model-based analysis However power system models are often inaccurate and may not reflect the actual dynamics Example: discrepancy between the actual and modelled response during 1996 WSCC disturbance Can we use advances in WAMS to develop/validate power system (dynamic) models? Kosterev, Taylor, Mittelstadt, IEEE Trans. Power Systems, 1999

17. Estimation of the dynamic Jacobian matrix from stochastic properties of PMU measurements • Assume 2nd order generator model • Assuming that system loads are experiencing Gaussian variation around base case loading, load variation reveals itself in the diagonal elements of the reduced admittance matrix as • W – standard Wiener process, si2variance of load variation • The stochastic power system model is now: Independent Gaussian random variables si • X. Wang; J. Bialek; K. Turitsyn: “PMU-Based Estimation of Dynamic State Jacobian Matrix and Dynamic System State Matrix in Ambient Conditions”, IEEE Trans. on Power Systems, 2018

18. Linearizing around the steady-state A B In the steady-state matrix A is stable, the stationary covariance matrix Cxx can be shown to satisfy the Lyapunov equation Substituting for A and B gives:

19. Jacobian matrix J inertias Key insight: a link between measurements of stochastic variation Cdd , Cwwand the generator physical model. Estimation of J from (13): If Mare known, J can be directly estimated from measurements Cdd , Cww. Network topology and generator parameters are not required! Can be used to verify topology/parameters Estimation of D from (14) Need to know M, emf’s (voltages) E, the conductance matrix  G (and therefore network topology and parameters) and the load variance matrix S2

20. Example: identification of undetected topology change • Topology may be wrongly assumed due to comms errors (see 2003 US/Canada blackout) • Dangerous: significant effect on state estimation and therefore on all security assessment • New England 10 generator 39 bus network. • Undetected trip of two lines close to Gen1 and Gen 8

21. Biggest differences between the estimated and model-based matrices for elements 1-1, 1-8, 8-1 and 8-8 indicating topology changes near/between those generators

22. Extracting modes of oscillation (eigenvalues) and mode shapes (eigenvectors) from time-domain measurements From ring-down response – better estimates but discrete events From ambient noise – continuous monitoring Break down the signal into fundamental frequencies (modes) Source: S, Norris, GE Power (Edinburgh) BPA: Roadmap for Addressing Power Oscillation Risks in Power Systems

23. Mode frequency Mode damping Source: S, Norris, GE Power (Edinburgh)

24. Incompleteness of observations • If all the modes (eigenvalues) and mode shapes (eigenvectors) were observed, a full dynamic model could be reconstructed • Incomplete information: • Modal incompleteness – only relatively slow (a few Hz and below) and weakly damped oscillatory modes can be “caught” • Spatial incompleteness: PMUs are not at every node so not all mode shapes (eigenvectors) are available • How to calculate model parameters using limited information?

25. Estimation of parameters from modal measurements • matrix A is a function of unknown parameters θ • Key idea: calculate such values of θthat eigenequationsA(θ)ϕi = iϕiaresatisfied as closely as possible • Minimise • Gorbunov, A. Dymarski, J. Bialek: "Estimation of parameters of a dynamic generator model from modal PMU measurements” IEEE Trans. Power Systems, under review

26. Estimation of inertia M and damping D Assume 2nd order (classical) linearised power system dynamic model Dynamic Jacobian matrix (assumed known) Spatial completeness (PMUs at every node, all mode shapes for a given mode are observed) minimise Linear in D and M – analytical closed-form solution for estimation of M and/or D Spatial incompleteness (some mode shapes are missing due to missing PMUs) The unknown mode shapes form additional auxiliary unknowns zi Additional unknown mode shapes Nonlinear problem, numerical methods needed to solve the minimisation problem

27. Some simulation results 16 generator New England system 6 electromechanical modes (out of 15) identified by PMUs Estimation of M does not depend on D and vice versa

28. Spatial incompleteness

29. Estimation of damping D • Theoretically a similar procedure as estimation of M • However poor estimation results due to high sensitivity to measurement noise min

30. Conclusions • Draw your own! • Hopefully interesting discussion