LOCALIZED LINEAR QUADRATIC REGULATOR
5.7 Simulation Results
5.7.2 Power System Model
We begin with a 20×20 mesh topology representing the interconnection between subsystems, and drop each edge with probability 0.2. The resulting interconnected topology is shown in Figure 5.10 — we assume that all edges are undirected. The
0 2 4 6 8 10 12 14 16 18 20
0 2 4 6 8 10 12 14 16 18 20
Figure 5.10: Interconnected topology for the simulation example
dynamics of each subsystem is given by the discretized swing equation for power network. Consider the swing dynamic equation
miθÜi+diθÛi =− Õ
j∈Ni
ki j(θi−θj)+wi+ui, (5.58)
where θi, θÛi, mi, di, wi, ui are the phase angle deviation, frequency deviation, inertia, damping, external disturbance, and control action of the controllable load of bus i. The coefficient ki j is the coupling term between buses i and j. We let
xi := [θi θÛi]>be the state of busi and useeA∆t ≈ I+ A∆t to discretize the swing dynamics. Equation (5.58) can then be expressed in the form of (5.1) with
Aii =
"
1 ∆t
−mki
i∆t 1− mdi
i∆t
#
, Ai j =
"
0 0
ki j mi∆t 0
# ,
and Bii = h 0 1
i>
. The parameters ki j,di,mi−1 are randomly generated and uniformly distributed between 0.2 and 1. In addition, we set ∆t = 0.2 and ki =Í
j∈Niki j.
We consider the LLQR problem with a spatiotemporal SLC (5.15) enforced as follows. The localized region for each subsystem j is specified by its two-hop neighborhood.3 This means that each subsystem can only communicate up to its two-hop neighbors during implementation, and use the plant model up to its two- hop neighbors for controller synthesis. For communication delays, we assume that ui[t] can access xj[τ] at time τ ≤ t − k if subsystem (i,j) are k-hop neighbors.
As the disturbance takes two steps to propagate to its neighboring subsystems, the communication speed is twice faster than the speed of disturbance propagation. We impose the FIR constraint with lengthT =15.
Large-Scale Example
We now allow the size of the mesh network to vary and compare the computation time needed to synthesize a centralized, distributed, and localized LQR optimal controller. The distributed LQR controller is computed using the methods described in [29], in which we assume the same communication delay constraints as LLQR.
The empirical relationship obtained between computation time and problem size for the different control schemes is illustrated in the log-log plot in Figure 5.11.
For the LLQR controller, we plot both the total computation time and the average computation time per subsystem. As can be seen in Figure 5.11, the computation time needed for the distributed controller grows rapidly when the size of problem increases. For the centralized controller, the slope of the log-log plot in Fig. 5.11 is 3, which matches the theoretical complexity ofO(n3). The slope for the LLQR (total time) is approximately 1.27, which is larger than the theoretical value 1. This overhead is likely caused by other computational issues such as memory management and data copy. For the largest example that we computed, we are able to synthesize
3For the mesh network, the number of subsystems contained within the two-hop neighborhood region of a subsystem is up to 13. Since each subsystem has 2 scalar states, the number of states for each localized region is up to 26.
a LLQR optimal controller for a system with 51200 states in about 23 minutes using a personal computer. If the computation were to be parallelized across the 25600 subsystems in the large-scale network (as would be done in a practical situation), the synthesis procedure can be performed in under 0.1 second. In contrast, the theoretical computation time for the centralized LQR using the same computer is more than 200 days, and the distributed LQR is intractable.
102 103 104
Number of States 10-2
100 102 104
Computation time (second)
LLQR (total) LLQR (parallel) Centralized LQR Distributed LQR
Figure 5.11: Computation time for the centralized, distributed, and localized LQR Large-Scale Example with Sub-optimality Guarantee
In order to quantify the sub-optimality of the LLQR controller for this large-scale example, we use Algorithm 2 to adaptively update the LLQR spatiotemporal con- straint according to the following rule — if the ratio of the lower bound to the upper bound is less than 98% for a particular subsystem, then the subsystem adaptively increases the FIR lengthT to 20 for itself.
We then compute the LLQR controller and the lower bound for the 51200-state system described above. Among the 25600 subsystems, 156 subsystems increase the FIR length from 15 to 20. The ratio of the overall lower bound to the overall upper bound is 99.01%. This means that the LLQR controller, despite the additional communication delay constraint, localized region constraint, and FIR constraint, is guaranteed to be at least 99.01% optimal compared to the unconstrained centralized LQR optimal controller. The computation of the LLQR controller and its lower bound is finished in 38 minutes. If the computation is parallelized into 25600
subsystems and each subsystem is capable of solving its subproblem using a personal computer, then the LLQR controller with performance guarantee can be computed within 0.1 seconds. In Table 5.2, we summarize the comparison between centralized, distributed and localized LQR for this 51200-state example. It is clear that LLQR has superior performance over centralized and distributed LQR in terms of the scalability of controller synthesis and implementation.
Table 5.2: Comparison Between Centralized, Distributed and Localized LQR on a 51200-State Randomized Example
LQR Distributed LLQR
Affected region Global Global 2-hop
Closed Loop Affected time Long Long 20 steps
Quadratic Cost 1 1.01 1.01
Comp. complexity O(n3) >O(n3) O(n) Parallel complexity O(n3) >O(n3) O(1)
Synthesis Comp. time 200 days Inf 38 mins
Parallel time 200 days Inf 0.1 second
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
C h a p t e r 6