Simulation Results

LOCALIZED LINEAR QUADRATIC GAUSSIAN

6.4 Simulation Results

Thus using this approach to solving the iterate updates (6.11a) and (6.11b), the LLQG optimization problem (6.8) can be solved nearly as quickly as the state-feedback problem, as the update equations require first solving a least-squares problem defined on the(d+2)-incoming and(d+2)-outgoing sets of the system and then using matrix multiplication.

6.3.3 Convergence and Stopping Criteria

Assume that the optimization problem (6.10) is feasible, and letΨ^∗ be an optimal solution. Further assume that the matrix [C₁ D₁₂] has full column rank, and [B₁;D₂₁]has full row rank. In this case, the objective function is strongly convex with respect toΨ, and hence any optimal solutionΨ^∗is the unique optimal solution.

As the extended-real-value functionsh^(r)(·)andh^(c)(·)specified in (6.9) are closed, proper, and convex, we have that strong duality holds and that optimization problem (6.10) satisfies the convergence conditions state in [5]. From [5], the objective of (6.10) converges to its optimal value. As the objective function is a continuous function of Ψ and the optimal solution Ψ^∗ is unique, it follows that the primal variable iterates converge to Ψ^∗, i.e., Ψ^k → Ψ^∗ and Φ^k → Ψ^∗. Note that the rank condition on the objective function matrices is only a sufficient condition for primal variable convergence. A less restrictive conditions for the convergence of the ADMM algorithm will be discussed in Appendix 7.B in Chapter 7. The design of the stopping criteria for the ADMM algorithm (6.11) can also be found in Appendix 7.B.

6.4.1 Power System Model

We begin with a randomized spanning tree embedded on a 10×10 mesh network representing the interconnection between subsystems. The resulting interconnected topology is shown in Figure 6.2a — we assume that all edges are undirected. The

0 2 4 6 8 10

0 1 2 3 4 5 6 7 8 9 10 11

(a) Interconnected topol- ogy

✓i

✓˙i ✓˙j

✓j

(b) Interaction between neigh- boring subsystems

Figure 6.2: Simulation example interaction graph.

dynamics of each subsystem is given by the discretized swing equation for power network (5.58). Similar to the model described in Section 5.7, we letxi :=[θi θÛi]^>

be the state of busiand usee^A∆t ≈ I+A∆tto discretize the swing dynamics. Equation (5.58) can then be expressed in the form of (6.1) with

Aii =

"

1 ∆t

−_m^ki

i∆t 1− _m^di

i∆t

#

, Ai j =

"

0 0

k_{i j} m_i∆t 0

# ,Bii =

"

0 1

#

, andCii=

"

1 0 0 1

#

We set∆t = 0.2 andki =Í

j∈N_i ki j. In addition, the parameterski j,di, andm_i⁻¹are randomly generated and uniformly distributed between[0.5,1], [1,1.5], and[0,2], respectively. The instability of the plant is characterized by the spectral radius of the matrix A, which is 1 in the simulated example. The interactions between neighboring subsystems of the discretized model is described by Figure 6.2b. We assume that each subsystem in the power network has a phase measurement unit (PMU), a frequency sensor, and a controllable load that generatesui.

From (5.58), the external disturbancew_ionly directly affects the frequency deviation θÛi. To make the objective functional strongly convex, we introduce small artificial disturbance on the phase deviationθias well. We assume that the process noise on frequency and phase are uncorrelated AWGNs with covariance matrices given byI and 10⁻⁴I, respectively. In addition, we assume that both the phase deviation and the frequency deviation are measured with some sensor noise. The sensor noise of phase and frequency measurements are uncorrelated AWGNs with covariance

matrix given by 10⁻²I. We choose equal penalty on the state deviation and control effort, i.e.,

h

C₁ D₁₂ i = I.

Based on the above setting, we formulate aH₂optimal control (LQG) problem that minimizes the H₂ norm of the transfer matrix from the process and sensor noises to the regulated output. TheH₂norm of the closed loop is given by 13.3169 when a proper centralizedH₂optimal controller is applied, and 16.5441 when a strictly proper centralizedH₂optimal controller is applied. In the rest of this section, we normalized theH₂norm with respect to the proper centralizedH₂optimal controller.

6.4.2 LLQG

The underlying assumption of the centralized optimal control scheme is that the mea- surement can be transmittedinstantaneously withevery subsystemin the network.

To incorporate realistic communication delay constraint and facilitate the scalability of controller design, we impose additional communication delay constraint, local- ized constraint, and FIR constraint on the system response. We introduce these constraints in a sequential order as follows.

For the communication delay constraintC, we assume that each subsystem takes one time step to transmit the information to its neighboring subsystems. Mathematically, the control actionui[t]of subsystemiat timetcan receive(yj[τ], βj[τ])of subsystem jfor timeτ ≤ t−kif the distance between subsystemsiand jisk. The interaction between subsystems illustrated in Figure 6.2b implies that it takes two time steps for a disturbance at subsystem j to propagate to its neighboring subsystems, and hence the communication speed is twice as fast as propagation speed of disturbances through the plant. For the given communication delay constraint C, we use the method described in Section 5.3 to design the sparsest localized constraint L. In this example, we can localize the effect of each process and sensor noise within its two-hop neighbors. This implies that each subsystem j only needs to exchange the information within its two-hop neighbors, and use the restricted plant model within its two-hop neighbors to synthesize its sub-controller.

Once the communication delay constraint C and the localized constraint L are specified, we run some simulation to exploit the tradeoff between the length of the FIR constraint F_T and the transient performance. Figure 6.3 shows the tradeoff curve between the transient performance of the LLQG controller and the lengthTof the FIR constraint. For the given communication delay constraintCand the locality constraintL, the LLQG controller is feasible with the FIR constraintF_Tfor allT ≥ 3.

When the length of the FIR constraint increases, theH₂ norm of the closed loop converges quickly to the unconstrained optimal value. For instance, for FIR length T = 7,10,and 20, the performance degradation compared to the unconstrainedH₂ optimal controller are given by 3.8%, 1.0%, and 0.1%, respectively. This further means that the performance degradation due to the additional communication delay constraint Cand the localized constraintL is less than 0.1%. From Figure 6.3, we show that the LLQG controller, with additional communication delay constraint, localized constraint, and FIR constraint, can achieve similar transient performance to an unconstrained optimalH₂controller.

0 5 10 15 20

1 1.1 1.2 1.3

1.4 Localized

Centralized (p) Centralized (sp)

Figure 6.3: The vertical axis is the normalizedH₂ norm of the closed loop when the LLQG controller is applied. The LLQG controller is subject to the constraint C ∩ L ∩ F_T. The horizontal axis is the horizonT of the FIR constraintF_T, which is also the settling time of the impulse response. We plot the normalizedH₂norm for the centralized unconstrained optimal controller (proper and strictly proper) in the same figure.

To further illustrate the advantages of the LLQG scheme, we chooseT = 20 and com- pare the LLQG controller, distributed LQG optimal controller, and the centralized LQG optimal controller in terms of the closed loop performance, the complexity of controller synthesis, and the complexity of controller implementation in Table 6.1.

The distributed optimal controller is computed using the method described in [28], in which we assume the same communication constraintCas the LLQG controller.

It can be seen that the LLQG controller is vastly preferable in all aspects, except for a slight degradation in the closed-loop performance. In particular, the localized constraintL in this example has almost no effect on the closed loop performance.

Table 6.1: Comparison Between Centralized, Distributed, and Localized LQG Optimal Control

Centralized Distributed Localized

Affected region Global Global 2-hop

Closed Loop Affected time Long Long 20 steps

NormalizedH₂ 1 1.001 1.001

Comp. complexity O(n³) >O(n³) O(n) Synthesis Parallel complexity O(n³) >O(n³) O(1)

Plant model Global Global 2-hop

Redesign Offline Offline Real-time

Implementation Comm. Speed Inf 2 times 2 times

Comm. Range Global Global 2-hop

6.4.3 Large-Scale Example

We now allow the size of the problem to vary and compare the computation time needed to synthesize a centralized, distributed, and localized LQG optimal con- troller. We choose T = 7 for the LLQG controller. The empirical relationship obtained between computation time and problem size for different control schemes is illustrated in Figure 6.4. As can be seen in Figure 6.4, the computation time needed for the distributed controller grows rapidly when the size of problem increases. For the centralized one, the slope in the log-log plot in Figure 6.4 is 3, which matches the theoretical complexityO(n³). The slope for the LLQG controller is about 1.4, which is larger than the theoretical value 1. We believe this overhead is caused by other computational issue such as memory management and data structure. We note that the computational bottleneck that we faced in computing our large-scale example was that we were using a laptop to compute the controller (and hence the localized subproblems were essentially solved in serial) — in practice, if each local subsys- tem is capable of solving its corresponding localized subproblem, our approach

scales to systems of arbitrary size as all computations can be done in parallel. For the largest example we have, we can finish the LLQG synthesis for a system with 12800 states in 22 minutes using a laptop. If the computation is parallelized into all 6400 sub-systems, the synthesis algorithm can be done within 0.2 second. In contrast, the theoretical time to compute the centralized LQG optimal controller for the same example is more than a week, and the distributed LQG optimal controller is intractable.

10² 10³ 10⁴

10^-1 10⁰ 10¹ 10² 10³ 10⁴

Localized Centralized Distributed

Figure 6.4: Computation time for the centralize, distributed, and localized LQG controller. The horizontal axis denotes the number of states of the system, and the vertical axis is the computation time in seconds.

C h a p t e r 7