Efficient Two-Layer MPC

Chapter V: Efficient Distributed Model Predictive Control

5.6 Efficient Two-Layer MPC

Figure 5.4: Topology of example system. We will plot the time trajectories of states, disturbances, and input for the red square subsystem.

distributions — grid size𝑁from[4,11](corresponding to system sizes of up to 121 subsystems), actuation density from [0.2,1.0], spectral radius from [0.5,2.5] and horizon length from [3,20]. In these 200 simulations, we encountered 4 instances where a locality constraint was feasible but resulted in insufficient rank; in the vast majority of cases, if a locality constraint was feasible, the rank condition was satisfied as well.

[𝑥]1

[𝑢]1

[𝑤]₁

[𝑥]2

[𝑢]2

[𝑤]₂

[𝑥]3

[𝑢]3

[𝑤]₃

[𝑥]4

[𝑢]4

[𝑤]₄ [𝑥]1∗

Optimal power flow solver [𝑥]2∗

[𝑥]3∗

[𝑥]4∗

load profile

sub-controller subsystem

communication (constant) sensing and actuation physical interaction communication (periodic)

Figure 5.5: Architecture of example system with optimal power flow solver, sub- controllers, and subsystems. For ease of visualization, we depict a simple 4- subsystem topology instead of the 25-subsystem mesh we’ll be using.

zero and draw inertia𝑚⁻¹

𝑖 from a random uniform distribution over[0,10]. These two changes both make this system more challenging to control compared to the system of the previous section; the decreased damping and inertia render the system unstable, and in our example the spectral radius of the 𝐴matrix is 1.5. We define the state at subsystem𝑖:

[𝑥]𝑖:=

"

𝜃(𝑡+1) 𝜔(𝑡+1)

#

𝑖

(5.33) We will incorporate optimal power flow into the system as follows: periodically, we will generate a random load profile, and solve a direct-current optimal power flow problem to obtain the optimal state setpoint𝑥^∗. We will then send each sub- controller𝑖its individual optimal setpoint[𝑥]^∗

𝑖, and expect each subsystem to reach this setpoint. We add local communication constraints: each sub-controller may only communicate with their immediate neighbors (i.e. subsystems with whom they share an edge) and neighbors of those neighbors, i.e. the local region size𝑑is set to 2. Controllers must reject frequent, random disturbances which have much smaller magnitudes than that of the setpoint changes. Additionally, we enforce actuator saturation constraints of the form ∥𝑢(𝑡) ∥_∞ ≤ 𝑢_{𝑚 𝑎𝑥}. Overall, this simple system captures some common challenges from power grid control — namely, instability, setpoint changes, actuator saturation, and disturbances. A diagram of the system setup is shown in Figure 5.5.

We propose a distributed two-layer MPC scheme for this system, as shown in Figure

[𝑢]_𝑖

[𝑥]_𝑖 MPC Trajectory

Generator

Offline Controller desired

trajectory

[𝑥]_𝑖^∗

[𝑥]_𝑖

Figure 5.6: Layered sub-controller for the 𝑖th subsystem. Horizontal dotted lines indicate communication with neighboring sub-controllers.

5.6. We decompose the main problem into two subproblems: reacting to setpoint changes and rejecting disturbances. We use MPC to deal with setpoint changes — specifically, we use MPC in the top layer to generate a desired trajectory that will accomplish the indicated setpoint change while respecting actuator saturation. In the bottom layer, we use a linear offline controller to track the trajectory generated by the top layer, while rejecting disturbances.

To facilitate scalability, we use the distributed and localized MPC technique pre- viously described [27] in the top layer, and a standard SLS controller [3] in the bottom layer. The resulting two-layer controller is fully distributed; each subsystem synthesizes and implements both layers of its own sub-controller. Synthesis of the two layers is independent of one another, although some synthesis parameters may be shared. Additionally, all cross-layer communication is local, i.e. a top layer sub-controller will never communicate with a bottom layer sub-controller from a different subsystem. Distributed implementation and synthesis allows our con- troller to enjoy a runtime that is independent of total system size, like its component SLS-based controllers.

This two-layer controller allows us to obtain MPC-like behavior, but at a fraction of the computational cost. For any MPC method — including distributed MPC — we must solve an optimization problem at every timestep. However, in this two-layer controller, we only need to solve an optimization problem when the setpoint changes.

If the setpoint changes every𝑇_{𝑂 𝑃 𝐹} timesteps, then this two-layer controller offers a 𝑇_{𝑂 𝑃 𝐹}-fold reduction in computational cost compared to a single MPC controller.

The reader may be tempted to ask whether the bottom layer is required at all: could

Table 5.1: LQR costs corresponding to Figure 5.7

Controller LQR Cost

Centralized LQR 2.09e13 Centralized MPC 1.00 Distributed two-layer 1.02

we simply run MPC periodically without need of an offline controller? Unfortu- nately, the answer is no; in the case of an unstable system, disturbances will persist between MPC runs and severely compromise performance. The reverse question can also be asked: could we avoid MPC altogether and simply use an offline con- troller? Again, the answer is no; the linear offline controller alone can react poorly to setpoint changes and actuator saturation, as we will see in simulation.

We now present results from simulations for𝑇_{𝑂 𝑃 𝐹} =20. We compared the perfor- mance of the distributed two-layer controller with the performance of centralized MPC and centralized LQR (i.e. offline controller). The centralized controllers are free of communication constraints while the layered controller is limited to local communication. The resulting LQR costs, normalized by the centralized MPC cost, are shown in Table 5.1. We plot trajectories from the red square subsystem from Figure 5.4 in Figure 5.7, focusing on a window of time during which only one setpoint change occurs.

In this example, the setpoint change induces unstable behavior from the centralized LQR. This is to be expected due to saturation-induced windup effects. The cen- tralized MPC controller and distributed two-layer controller perform very similarly.

From the previous results in this chapter, it should not be surprising that a reduction in communication does not substantially affect performance. However, this exam- ple additionally shows that we need not solve the MPC problem at every timestep to obtain MPC-like performance — here, the two-layer controller solve an MPC problem every 20 timesteps.

To check general behavior, we re-run the simulation 30 times with different randomly generated grids, plant parameters, disturbances, and load profiles. The resulting LQR costs, normalized by the cost of centralized MPC, are shown in Table 5.2.

The centralized LQR controller loses stability in 4 out of 30 simulations. In these cases, as with the previous example, the two-layer controller maintains stability and performs highly similarly to centralized MPC. For the 26 simulations for which the

0 2 4 6 8 10 12 -6

-4 -2 0

Centralized LQR Centralized MPC Distributed two-layer

0 2 4 6 8 10 12

-10 -5 0 5

0 2 4 6 8 10 12

-10 0 10

0 2 4 6 8 10 12

Time (s) 0

1 2

Figure 5.7: Performance of centralized LQR, centralized MPC, and distributed two- layer controller. The phase setpoint is shown by the dotted line; a setpoint change occurs at 𝑡 = 2. The LQR controller loses stability, resulting in large oscillations in phase and frequency, which are omitted from the plot after around 𝑡 = 4; the associated actuation engages in oscillations as well, which are shown on the plot.

The MPC and two-layer controllers perform similarly.

Table 5.2: LQR costs averaged over 30 simulations

Controller LQR cost

All sims LQR controller stable

Centralized LQR 1.14e7 1.05

Centralized MPC 1.00 1.00

Distributed two-layer 1.01 1.01

LQR controller maintains stability, we show the costs in a separate column. In this case, the LQR controller performs similarly to the MPC and two-layer controllers.

Overall, these examples show that the distributed two-layer controller indeed achieves near-optimal performance, at a fraction of the communication and computational cost required by centralized MPC, which is the ideal (but computationally expensive) controller for this problem.