In Part I of this post, I have explained the idea of reverse and forward engineering, applied to TCP congestion control. Here, I will describe how forward engineering can help the design of ubiquitous, continuously-acting, and distributed algorithms for load-side participation in frequency control in power networks. One of the key differences is that, whereas on the Internet, both the TCP dynamics and the router dynamics can be designed to obtain a feedback system that is stable and efficient, a power network has its own physical dynamics with which our active control must interact.

**Frequency control**

Frequency control maintains the frequency of an AC (alternating current) power system tightly around its nominal value (e.g., 60Hz in the US, 50Hz in Europe) when demand or supply fluctuates. The system dynamics are primarily driven by big rotating machines in bulk generators. The frequencies at which these machines rotate determine the frequency in the power network. When demand exceeds supply, these machines slow down, and the system frequency drops; when demand falls short of supply, these machines speed up, and the system frequency rises. Hence, the frequency deviation measures the instantaneous power imbalance.

Frequency control is traditionally implemented on the generation side, i.e., generation is adjusted in response to demand fluctuations. Until recently, demands were not exploited to help balance power, except as a last resort, in the form of voluntary or involuntary load shedding (from utility companies turning off air conditioners of summer program participants to larger-scale brown-outs or black-outs). The generation-side frequency control consists of three mechanisms that work at different timescales in concert. The *primary* frequency control operates at a timescale up to low tens of seconds, and uses a governor to adjust, around a setpoint, the mechanical power input to a power generator based on the local frequency deviation. It is completely decentralized. The primary control can rebalance power and stabilize the frequency, but does not in itself restore the nominal frequency. The *secondary* frequency control (also called automatic generation control) operates at a timescale up to a minute or so, and adjusts the setpoints of governors in a control area in a centralized fashion to drive the frequency back to its nominal value and the inter-area power flows to their scheduled values. The *tertiary* frequency control (also called economic dispatch) operates at a timescale of several minutes or up, and schedules the output levels of generators that are online and the inter-area power flows to optimize economic efficiency.

**Load-side participation **

The idea of ubiquitous continuous fast-acting distributed load participation in frequency control dates back to the late 1970s. Schweppe *et al. *at MIT pioneered this idea in a paper that appeared 35 years ago:

F. C. Schweppe, R. D. Tabors, J. L. Kirtley, H. R. Outhred, F. H. Pickel, and A. J. Cox, Homeostatic utility control, IEEE Trans. Power App. Syst., vol. PAS-99, no. 3, pp. 1151–1163, 1980.

They advocated three ideas in that paper that were ahead of their time, but are now being pursued worldwide. The first is the deployment of controllable loads for frequency regulation to “assist or even replace turbine-governed systems and spinning reserve.” The second is to use spot prices to incentivize the users to adapt their consumption to the true cost of generation at the time of consumption. The third is the need of an information technology infrastructure to coordinate load-side frequency regulation systemwide. Remarkably, it was emphasized back then that such frequency-adaptive loads will “allow the system to accept more readily a stochastically fluctuating energy source, such as wind or solar generation!” Just like in information technologies, we are feeding off of research conducted decades ago.

The idea of load-side frequency control was ignored for almost 25 years, until the last decade, when both the need and the technologies for its implementation started to mature. California has set a goal (RPS) that 33% of electricity retail will come from renewable sources by 2020. A study commissioned by the CA regulators has concluded in 2012 that, to deal with the higher supply volatility from the increased renewable penetration, will require 3x the 2011 level of reserve capacity. This will essentially neutralize the benefits from the higher renewable generation. The alternative to higher reserve capacity is *demand response* (DR) that exploits flexibility in load much more actively and extensively to adapt to the fluctuating supply.

Load-side participation in frequency control is such an example. Its benefit can be substantial, as the total DR capacity of grid-friendly appliances in the U.S. has been estimated to be about 18% of the peak demand, comparable to the required operating reserve, currently at 13% of the peak demand. The feasibility of this approach is confirmed in a Montana Tech report that describes an experiment that measured the correlation between the frequency at a 230kV transmission substation and the frequencies at the 120V wall outlets at various places in a city in Montana. They have concluded that local frequency measurements are adequate for loads to participate in primary frequency control as well as in the damping of electromechanical oscillations due to inter-area modes of large interconnected systems. There have also been small-scale field trials, in the 2000s, by the Pacific Northwest National Lab in the US using dryers and water heaters, and by a consortium in the UK using fridges.

How do we design distributed algorithms for millions of grid-friendly appliances so that they can participate in continuous and ubiquitous frequency regulation?

**Forward engineering**

We model a power network by linearized swing dynamics at generator buses, power flow dynamics on the transmission/distribution lines, and a measure of disutility to users when they participate in frequency control. At steady state, the frequencies at different buses are synchronized to a common nominal value, and the mechanical power is balanced with the electric power at each bus. When a disturbance occurs, i.e., changes in generations or loads on an arbitrary subset of the buses, it will cause the bus frequencies to deviate from their nominal value. Instead of adjusting the remaining generators as in the traditional approach, can we adjust controllable loads in the network to rebalance power in a way that minimizes the aggregate disutility of these loads?

The challenge is to design simple *distributed* algorithms that can be implemented at millions of grid-friendly appliances, and that are *provably* *stable and efficient*.

We propose a design approach based on forward engineering where we start with the control goals and derive the controller and the network’s own natural dynamics as a distributed solution of an optimization problem. Specifically, the **FED (forward-engineering design) **approach is:

- Formalize the goal of the control (rebalance power, and restore nominal frequency and inter-area power flows at minimum user disutility) as a constrained optimization problem, called OLC (optimal load control).
- Derive controller as a part of the first-order primal-dual algorithm for solving OLC.

The first-order primal-dual algorithm to solve OLC has two parts. The first part turns out to be identical to the natural swing dynamics of the power network, and is therefore automatically carried out by the network itself. The second part must be implemented as active control at the controllable loads on the network. This means that *active load control together with the network (swing) dynamics serve as a primal-dual algorithm for solving OLC!*

Within this model, one can prove that the closed-loop system is globally asymptotically stable and, starting from any initial condition, the network will converge to a unique steady state where OLC is optimized.

In the FED approach, *the key design decision is the careful formulation of OLC*; the remaining derivations and analysis follow a standard (though nontrivial!) procedure. One of the key design steps in the formulation of OLC is to decide which variables and constraints to include. This directly determines the control goals at equilibrium, and the information requirement to implement the control.

For instance, if the controllable loads participate only in the primary frequency control (where the goal is to balance power and stabilize frequency), they can adjust their consumptions based only on frequency deviations that are locally measurable. This is a completely decentralized algorithm, like the primary frequency control on the generation side. For more details, see

C. Zhao, U. Topcu, N. Li and S. H. Low. Design and stability of load-side primary frequency control in power systems, IEEE Trans. on Automatic Control, 59(5):1177-1189, May 2014

If the controllable loads participate in the secondary frequency regulation as well (balance power, restore nominal frequency and inter-area flows), then auxiliary variables need to be introduced in OLC, and additional constraints must be *carefully* designed, so that the network dynamics continue to be a part of the primal-dual algorithm for OLC. These auxiliary variables and the Lagrange multipliers associated with their constraints must be computed at each bus and communicated with their neighbors. For more details, see

E. Mallada, C. Zhao and S. H. Low. Fair load-side control for frequency regulation in smart grids, July 2014

**Implications**

These preliminary results have four implications.

First, for primary frequency control, the local frequency deviation at each bus conveys exactly the right information about the global power imbalance for the loads themselves to make local decisions that turn out to be globally optimal (i.e., solve OLC in equilibrium). This allows a completely decentralized solution without explicit communication to or among the loads. For secondary frequency control that restores the nominal frequency and inter-area flows, it is an open question whether completely decentralized control is possible. We know, however, that local computation and communication are sufficient. While a centralized solution may be implementable on the generation side (as is currently practiced), a distributed solution may be the only practical approach for ubiquitous continuous load-side participation.

Second, the global asymptotic stability of the primal-dual algorithm for OLC suggests that ubiquitous continuous distributed load participation in primary and secondary frequency control is stable, addressing a question raised in several prior studies.

Third, our approach actively exploits the natural system dynamics for computation and communication. This enhances the simplicity and scalability of our algorithm. Its implication on the transient behavior of the system, however, is not well-understood, although preliminary simulation results suggest that load-side participation improves both the steady-state and the transient behavior. Note that the global asymptotic stability result implies that the system will eventually converge to a global optimal of OLC, but says nothing about the transient behavior.

Finally, even though we have only presented a “forward engineering” perspective, the opposite perspective of “reverse engineering” is useful as well where, given an appropriate controller design, the network dynamics will converge to a unique equilibrium that *inevitably *solves OLC with a certain objective function that depends on the controller design. In this sense, any frequency adaptation implies a certain disutility function of the load that the control implicitly minimizes. For instance, the linear droop controller in the literature implies a quadratic disutility function, and hence a quadratic objective in OLC.

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