New Tools for Analysing Power System Dynamics

Transcription

1 1 New Tools for Analysing Power System Dynamics Ian A. Hiskens Department of Electrical and Computer Engineering University of Wisconsin - Madison (With support from many talented people.) PSerc Research Tele-Seminar March 2, 2004 PSERC

2 Two main themes 2 Dealing with parameter uncertainty How much confidence is there in load model parameters? Generator parameters? Simulation is too time-consuming to repeat studies for multiple parameter sets. Inverse problems What does this mean? Trajectory sensitivities form a common link between these two areas.

3 3 Inverse problems Direct problem: input x process K model output? Inverse problem (system identification): input process output x? model y Inverse problem (causation): input process output? K y model

4 Inverse problems (continued) 4 Power system context Identification problems Load models and parameters Generator parameters What controller parameters ensure appropriate fault-recovery dynamics? What load changes maintain dynamic security? Under what conditions could a fault induce incidental protection operation? Protection operation caused by post-fault transients.

5 Trajectory sensitivities 5 Let φ(t; λ) describe the trajectory (flow), at time t, due to parameters λ. Determined by simulation. The corresponding trajectory sensitivities are Φ(t; λ) φ λ (t; λ). Characteristics of sensitivities Φ: Quantify the change in a trajectory due to a small change in parameters. Can be computed efficiently, as a by-product of simulating the nominal trajectory. Well defined for nonlinear non-smooth systems. Provide gradient information that underlies Newton-type algorithms (shooting methods).

6 Trajectory approximation 6 Taylor series expansion of the flow gives, φ(t; λ + λ) =φ(t; λ)+φ(t; λ) λ Example: Generator field voltage Field voltage, E fd (pu) Actual computed trajectory, t cl =0.23 sec, E fdmax =5.8 0 Approximate perturbed trajectory, t cl =0.21 sec, E fdmax =5.0 Actual perturbed trajectory (for comparison) Time (sec)

7 Error bounds 7 Many system parameters are not known precisely, but are better described by probability distributions. Select parameter sets using a Monte-Carlo process. Form an approximate trajectory for each set Terminal voltage (pu) Nominal trajectory Error bound Time (sec)

8 Parameter estimation 8 Desire a systematic approach to: Determine which parameters are well conditioned (identifiable). Estimate those parameters Algorithm: Minimize the nonlinear least-squares cost J (λ) = 1 m ( ) 2 φ(t k ; λ) ms(t k ) 2 k=1 via the Gauss-Newton iterative scheme Φ(λ j ) T Φ(λ j ) λ j+1 =Φ(λ j ) T ( φ(λ j ) ms ) λ j+1 = λ j α j+1 λ j+1

9 Parameter estimation (continued) 9 Nordel example 450 Bus 12 voltage, V 12 (kv) Measured Initial parameter values Estimated parameter values Time (s)

10 Performance specification (grazing) 10 Trajectory is tangential to a performance boundary. Parameter values λ g λ+ λ Boundary x + x g

11 11 Performance specification (continued) Example: Generator field control. What value of E fd,max ensures voltage does not overshoot abnormally following a fault? V ref E fdmax + V t 1 1+sT R Σ 1+sT C 1+sT B K A 1+sT A E fd + V PSS E fdmin V max 1+sT 1 1+sT 2 V min st w 1+sT w K PSS ω

12 Performance specification (continued) 12 Terminal voltage constrained below 1.2 pu during initial transient Hypersurface b(x,y): V t 1.2= Terminal voltage, V t (pu) Initial parameter value Grazing parameter value Time (sec)

13 Performance specification (continued) 13 Example: Distance protection. How far can load be increased before a fault disturbance induces incidental protection operation? 1.2 Initial trajectory Grazing trajectory 1 b(x,y)= X c 2 c 1 a 2 a 1 0 b b R

14 Performance specification (continued) 14 Example: Distance protection with timing. Replace conditions enforcing tangential contact by conditions specifying time inside mho characteristic. Initial trajectory 1.2 Time constraint trajectory 1 b(x,y)= X a 2 a 1 c 2 0 b 2 c b R

15 Performance specification (continued) 15 Initial, grazing, and time-constraint cases. 3 Initial trajectory 2.5 Grazing trajectory Time constraint trajectory b(x,y) τ=0.3 s Time (s)

16 Dynamic embedded optimization 16 Minimize the cost J (λ) =ϕ ( φ(t f ; λ),λ,t f ) + tf t 0 ψ ( φ(t; λ),λ,t ) dt where φ(t; λ) satisfies the dynamic model. Closely related to optimal control, but optimizing over finite dimensional design parameters. Some technical issues arise for hybrid (switched) systems if event order changes.

17 17 Dynamic embedded optimization (continued) Example: AVR/PSS tuning What values of PSS output limits give best damping? V ref E fdmax + V t 1 1+sT R Σ 1+sT C 1+sT B K A 1+sT A E fd + V PSS E fdmin V max 1+sT 1 1+sT 2 V min st w 1+sT w K PSS ω

18 Dynamic embedded optimization (continued) 18 Optimization adjusted the lower PSS limit from -0.1 to pu. Noticable damping improvement Generator angle, δ (rad) Initial limit values Optimal limit values Time (sec)

19 Conclusions 19 Parameter uncertainty is unavoidable in power systems, and should be considered in decision making. This is computationally feasible using first-order approximations of trajectories. Many analysis and design processes are effectively inverse problems. Such problems can be solved using gradient-based iterative algorithms. In both cases, efficient computation of trajectory sensitivities underlies practical algorithms.

20 20 Extra Material

21 Simulation model 21 Differential Algebraic Impulsive Switched (DAIS) model ẋ = f(x, y)+ r i=1 0=g(x, y) g (0) (x, y)+ where ( ) δ(y r,i ) h i (x, y) x s g (j) (x, y) j=1 ẋ = f(x, y) x + = h i (x,y ), y r,i =0 g (j) (x, y) = g (j ) (x, y) g (j+) (x, y) y s,j < 0 y s,j > 0 j =1,..., s

22 Trajectory sensitivity computation 22 Smooth sections of trajectories evolve according to ẋ = f(x, y) 0=g(x, y) Differentiating with respect to x 0 gives Initial conditions Φ x = f x (t)φ x + f y (t)φ y 0= g x (t)φ x + g y (t)φ y Φ x (t 0 )=I

23 ff Trajectory sensitivities at events 23 At an event (at time τ), sensitivities evolve according to jump conditions, Φ x (τ + )=Φ x (τ ) ( f + f ) τ x 0 Nominal trajectory @ Φ x 0 Ω ΩΩffi@ f τ - ΦΦ* x(τ)+ x Triggering f + τ Φ + x Φ ΦΦΦΦ 0 Φ ΦΦΦΦ ff x(τ + τ)

24 Grazing and time-constraint formulation 24 For simplicity, the ODE form of equations is presented rather than the DAE form. Grazing φ(t g ; λ) x g =0 b(x g )=0 b x (x g )f(x g )=0 Time constraint φ(t 1 ; λ) x 1 =0 φ(t 2 ; λ) x 2 =0 b(x 1 )=0 b(x 2 )=0 t 1 t 2 τ spec =0