Electric Power Network Response to Random Demand Variations: A large deviations view on voltage collapse phenomena Christopher DeMarco Electrical and Computer Engineering University of Wisconsin-Madison Madison, WI 53706 (608) 262-5546 demarco@engr.wisc.edu
Components of my talk: Evangelism: for an audience likely focused on queuing networks, show what makes electric power networks different, and (hopefully) what makes them interesting. Modeling - construct underlying dynamics. Highlight - policy/control decisions available; - state equations associated with nodes; - role of network structure; - near Hamiltonian structure in dynamics; - physical nature of stochastic disturbance. C. L. DeMarco, University of Wisconsin-Madison; page 2
Components of my talk: Stochastic Exit Problem Formulation - associated optimal control problem - Hamiltonian structure closed form solution to cost of control/potential function - potential function characteristics - saddle exit point characteristics, relation to physical loss of stability mechanisms Application - prediction of onset of voltage collapse Interspersed throughout: New Questions & Challenges Raised by Market Driven Policies and Decentralized Control C. L. DeMarco, University of Wisconsin-Madison; page 3
True Confessions/Acknowledgments Basis of this work dates back to mid 80's thru mid-90's, with contributions from former Ph.D. students Jayant Sarlashkar, Tom Overbye, & Ru-Xiang Qian. Resurrected by new questions introduced by competitive markets, new concerns for fragility of US infrastructure. Current work supported by Electric Power Research Institute and Army Research Office, under "Complex Interactive Networks & Systems Initiative." C. L. DeMarco, University of Wisconsin-Madison; page 4
What makes electric power networks different? Flow of interest is power in a sinusoidal quasi-steady state. Hence each flow is a two parameter quantity ("active" and "reactive" powers, at known frequency). Branches are mostly passive flow determined by boundary conditions at nodes (controllable links do exist, but constitute less than 0.1% of all links). Sinusoidal operation two parameter boundary condition at each node. Essentially no storage at nodes. Hence no policy decisions to be made relating to "service" of commodity at nodes. C. L. DeMarco, University of Wisconsin-Madison; page 5
What makes electric power networks different? At equilibrium, nodes satisfy a power conservation law. Policy/control decisions relate to choice of injection at subset of nodes, to exogenous demand at other nodes. Equilibrium requirement #1: flows on links within capacity limits, nodal variables in limits. Non-equilibrium behavior driven by nodal mismatch between external power injection/demand, absorption by network. C. L. DeMarco, University of Wisconsin-Madison; page 6
What makes electric power networks different? Equilibrium requirement #2: Operating point must be stable w.r.t. dynamics, with suitable robustness margin. Current engineering practice often creates surrogate limits on link flows & node injections to approximate stability limits link capacities function of operating point. C. L. DeMarco, University of Wisconsin-Madison; page 7
What makes electric power networks interesting? Dynamics highly nonlinear; stable equilibrium defining desired operating point never has unbounded domain of attraction. Disturbances can cause state divergence from desired operating point, out of attractive domain (here's our stochastic exit problem). State divergence = blackout. Timescales for dynamics relevant to stability on order of 0.1 sec. to perhaps 0.5 hr. "Deregulation" has brought strong impetus for decentralized, competitive market driven operating policies. C. L. DeMarco, University of Wisconsin-Madison; page 8
What makes electric power networks interesting? Choice of power production (and ultimately, consumption) at each node is to become a decentralized decision, based on a market clearing prices. Current time scales: 5 minute periodic update. Competitive power marketers strongly interested in available link capacity. Strong interest in attributing portion of flow on link to individual market players. Nonlinearities make this questionable (but done anyway...) C. L. DeMarco, University of Wisconsin-Madison; page 9
Nature of Basic Power Systems Model: Two parameter "boundary condition" at nodes is simply a convenient representation of an instantaneous voltage that is sinusoidal: v(t) = V(t) cos ( ω t + δ(t) ) Voltage magnitude relative phase angle 50 or 60 Hz (though we'll use radians/sec) V(t) and δ(t) controlled indirectly. Fundamental dynamics are really just rotational Newton's law - consider nodes with generators attached. C. L. DeMarco, University of Wisconsin-Madison; page 10
Approximate torque as proportional to power acceleration on generator shaft can be computed from net power Electric Power Out, P E Mechanical Power Accelerating Shaft, P G Torque from magnetic field opposing motion P G > P E gen accelerates; P G < P E gen decelerates Control exercised through P G. C. L. DeMarco, University of Wisconsin-Madison; page 11
Other Observations: 1) Physics of Synchronous generators mechanical speed and electric frequency are equal. Can be used interchangeably! ω 0 + ω = normalized frequency/speed Deviation from Synchronous speed 2) ω := d ω dt = dω dt = acceleration or rate of change of elec. freq. C. L. DeMarco, University of Wisconsin-Madison; page 12
3) ω (interpreted as frequency deviation) = δ Key step towards differential equations for nodal dynamics - just rotational acceleration: M ω = (P G P E ) Normalized Rotational Inertia power conservation "mismatch" at the node - P_G control ipput from external source, PE absorbed by network C. L. DeMarco, University of Wisconsin-Madison; page 13
4) As noted previously, electrical power absorbed by network (flow into links) is function of V(t)'s and δ(t)'s; specifically: V 1 V 2 X L sin(δ 1 δ 2 ) where X L is fixed, known parameter. Because only phase differences influence flow,one can set δ 2 0 without loss of generality. C. L. DeMarco, University of Wisconsin-Madison; page 14
5) Simplest possible model has one generator, one link, and assumes V(t)'s constant. State variables associated with node #1 are "boundary" variable δ(t), and "internal" variable ω(t) G Node #1 with generator attached V (t) δ 1 1 (t) Node # 2 V (t) δ 2 2 (t) Demand Resulting dynamic equations δ1 = ω 1 ω 1 = M -1 { D ω 1 + P G,1 V 1 V 2 X L sin(δ 1 )} C. L. DeMarco, University of Wisconsin-Madison; page 15
Observation/Question: State dimension of this two example is 2. Problems of interest have # nodes order of 10 2 to 10 4, state dimension 10 3 to 10 5. How to extract useful qualitative structure for high dimensional models? Answer: Structure above is "nearly Hamiltonian;" remains true for arbitrary dimension, and for (some) added modeling detail. C. L. DeMarco, University of Wisconsin-Madison; page 16
What do we mean by "nearly" Hamiltonian? General structure is: x = A Φ(x) ( ) with A full rank (equilibria at critical points of Φ(x)) symmetric part of A diagonal, negative semidefinite, associated with damping terms (Φ(x(t) non-increasing along trajectories) non-zero diagonal entries of (A+A T ) appear precisely in those components where variations in power demand/injections may reasonably be added to model C. L. DeMarco, University of Wisconsin-Madison; page 17
More on general structure of ( ) Φ(x) available in closed form, structured as sum of products of quadratic terms in V(t)'s with cosine terms in δ(t) differences Sum is taken over every link in network; hence intimate tie to network structure Φ(x) is natural Lyapunov function for deterministic system, and constant contours of Φ(x) approximate boundary of domain of attraction for stable equilibrium. C. L. DeMarco, University of Wisconsin-Madison; page 18
Construction of stochastic differential equation Nature of stochastic disturbance: Variations in aggregate demand at nodes represent filtered aggregate of very large number of small magnitude jump processes (i.e., lots of folks turning their lights on/off). Field measurements confirm that demand variation looks very much like white noise. Typical magnitude on order of 1-2% of total demand/injection at a node. Based on above, motivated to look at asymptotic behavior of expected exit time in small white noise case. C. L. DeMarco, University of Wisconsin-Madison; page 19
Construct stochastic differential eq as dx = A Φ(x)dt + ε[(a + A T ) 1/2 ]dw t ( ) if there's a sleight of hand, it's here, in choice of weighting on noise input Following Wentzell-Freidlin approach, we examine associated optimal control problem: x = A Φ(x) + [(A + A T ) 1/2 ]u(t) ( ) C. L. DeMarco, University of Wisconsin-Madison; page 20
Given stable equilibrium of interest, x s, consider cost of control (integral of norm squared u(t)) steering state from x s, to any point x b in basin of attraction for x s. Key observation: for our ( ), this cost of control is precisely Φ(x b ). Some extra machinery to show relation of level sets Φ(x) to domain of attraction of x s. In ( ), let E[τ ε ] denote expected time to exit (deterministic) domain of attraction of x s. In brief, Wentzell-Freidlin theory then yields that for small noise limit, ε 2 ln{e[τ ε ]} approaches Φ(x u ), where x u is lowest potential saddle. C. L. DeMarco, University of Wisconsin-Madison; page 21
Engineering Application Obvious observation: one measure of robustness or "security" of an operating point is depth of its potential well. But does stochastic exit phenomena match any physical observed problems in the power grid? - We argue yes... In late 70's & early 80's, several significant blackouts occurred in Europe in which no large disturbance "set off" the failure. Rather, system was experiencing gradual, quasi-static increase in average demand. C. L. DeMarco, University of Wisconsin-Madison; page 22
Engineering Application (continued) Existing tools for quantifying stability of operating point (local, linearized analysis) showed stable system throughout. Yet as average demand increased, system showed unusually large random variations in V(t)'s, with V(t)'s ultimately diverging in a "collapse" toward zero (at which point system shuts down). C. L. DeMarco, University of Wisconsin-Madison; page 23
New Questions for Market Driven Environment Old regime was centrally controlled: key control quantities were P G 's - injected power at generating stations. In old regime, central calculations chose P G 's, balancing operating cost minimization against effect of P G 's on relative "security" of operating point. How to capture same trade-off in decentralized decision process? Added effect not discussed here: random failures of links. Can tractable approach to exit problem be formulated with link failures? Probability of link failure increases with its loading. Can suitable pricing signals be developed to offer individual market players globally beneficial incentives? C. L. DeMarco, University of Wisconsin-Madison; page 24