Sparse Control Synthesis for Uncertain Responsive Loads with Stochastic Stability Guarantees

Significant technological advances in recent years have paved the way for flexible electrical loads to participate in grid ancillary services, including improvement of transient performance via primary and secondary frequency response (see, for example, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19] and the references therein). This type of demand-side control offers relatively faster, cleaner and cost-effective alternatives to more traditional stabilizing control measures (e.g. spinning reserves) deployed in the power grid, and can help mitigate the challenges associated with increasing penetration of renewable generation. In comparison to traditional generator-based reserves, however, load-based reserves can be highly uncertain and hence unreliable in providing the required ancillary services [1]. Engaging flexible and responsive loads for critical grid support, therefore, requires demand-side control synthesis methods that can structurally account for those uncertainties and minimize their impact on the grid transient performance. In the often-used hierarchical load control architectures [4, 6, 9, 8, 7], mitigation of uncertainties may happen at multiple layers, namely, the load aggregation and control layer, as well as the system-level grid operation layer. Several recent efforts explored methods to characterize and mitigate uncertainties in the controllable loads at the aggregator level, including probabilistic assessment of flexibility in providing grid ancillary services at multiple time-scales [7, 20, 21], priority-based coordination schemes to ensure reliability and mitigate uncertainty of the grid services [8, 16], joint parameter-state estimation approaches [17], as well as various Markov-based control schemes and randomized switching strategies for thermostatic loads [12, 13, 14, 15]. In yet other efforts, the grid-operator perspective of managing load uncertainties at the system-level have been considered primarily via various formulations of chance-constrained optimization techniques for capacity reserve determination and optimal dispatch of flexible loads [2, 1, 3].

In comparison, the problem of quantifying and mitigating the impact of uncertain controllable loads on the system-wide transient performance of the bulk power grid has received relatively less attention. Fragility of the closed-loop power systems with load-frequency control to stochastic parameter uncertainties (such as renewable generation) was investigated in [19]. The work [18] studies the effect of stochastic measurement errors on the demand-side frequency control. Robustness of the demand-side frequency control performance to bounded disturbances in the closed-loop power system dynamics, arising due to unknown (model) parameters and unmodeled nonlinearity, have been considered in [6, 11]. The existing methods have mostly focused on designing robust demand-side control strategies that can guarantee acceptable transient performance in presence of model uncertainties which are typically modeled as bounded additive disturbances. A key missing piece, however, is the treatment of stochastic uncertainties that arise in the demand-side feedback control action itself, and appear as multiplicative to the system states. This work addresses this issue by designing a sparse demand-side controller that can guarantee closed-loop stochastic stability performance in presence of multiplicative (and additive) uncertainties.

In particular, [5, 6, 7, 8, 9, 10] considers the distributed and hierarchical demand-side primary frequency control methods. In these methods, the participating thermostatic loads are coordinated in a distributed fashion to extract stabilizing response to disturbances in the grid, often in the form of proportional feedback control such as the power-frequency droop response. Uncertainties in load behavior, however, show up as errors in the controller gains, e.g. discrepancies in the actual vs. desired slope of the droop response [7, 8]. These uncertainties can be modeled as multiplicative to the system states (e.g. frequencies) having a nonlinear impact on the system dynamics, unlike the additive uncertainties. Mean square stability analysis of a stochastic power network where the uncertainty in system inertia appears multiplicative to the system dynamics is studied in [22]. Recent efforts have looked into the problem of control synthesis with closed-loop stochastic stability guarantees in dynamical systems with both multiplicative and additive uncertainties [23, 24, 25, 26]. Specifically, the work in [23] provides sufficient and necessary conditions for mean square exponentially stability (MSES) for stochastic dynamical systems with multiplicative uncertainties. In addition, a static full-state feedback control synthesis was proposed in [23] for the special case of single-input channel uncertainty, with a quantification on the maximum allowable (critical) uncertainty level. The approach in [23], however, has limited applicability in realistic power systems because of the assumptions placed on the nature of the uncertainty (single-input) and the availability of the measurements (full-state); and the non-consideration of design concerns such as sparsity, control efforts and closed-loop transient performance.

The main contributions can be summarized as follows:

1. a)

   modeling of multi-variate uncertainties (both multiplicative and additive) in the controllable and uncontrollable loads resulting in a stochastic power network;

2. b)

   an LMI-based control synthesis problem for the stochastic system to guarantee closed-loop stochastic stability under multi-variate uncertainties along with a quantification of the allowable uncertainty levels;

3. c)

   modification to the LMI control synthesis problem to address practical considerations in real power networks such as partial observation, desired transient performance, and sparsity promoting control; and

4. d)

   numerical studies performed on IEEE 39-bus system to demonstrate the fundamental trade-offs between the allowable uncertainties and the optimal control efforts, and to illustrate the advantages of the proposed sparse controller over a conventional linear-quadratic-regulator (LQR) controller during contingency scenarios (e.g., severe under- and over-frequency events).

The rest of the paper is organized as follows. Sec. II describes the stochastic power system model with controllable and uncontrollable loads. Sec. III presents the stochastic stability definitions and formally introduces the stochastic stabilizing control design problem. The convex optimization formulations for the sparse control synthesis with stochastic stability guarantees is presented in Sec. IV. Sec. V illustrates the closed-loop controller performance via numerical studies on the IEEE 39-bus system; before we end the paper with concluding remarks in Sec. VI.

We consider a power systems network with controllable loads which are coordinated to provide distributed frequency response for enhanced transient stability performance [5, 6, 7, 8, 9, 10]. Close to the nominal operating point, the power system dynamics can be modeled in the following control-affine form:

where $x\!\in\!\mathbb{R}^{n}$ is an $n$-dimensional state vector; $P_{in}\!\in\!\mathbb{R}^{m}$ is the $m$-dimensional vector of power injection inputs at various buses across the network; $y\!\in\!\mathbb{R}^{s}$ is the $s$-dimensional sensor measurements vector; $A_{0}\!\in\!\mathbb{R}^{n\times n},B_{0}\!\in\!\mathbb{R}^{n\times m}$ and $C_{0}\!\in\!\mathbb{R}^{s\times n}$ are, respectively, the system matrix, input matrix and output matrix. While the generic form of the system model shown in (1) is sufficient for the development of the key technical arguments in this paper, a specific example of it is provided in Sec. V for illustration purposes. We assume the matrices $A_{0},B_{0}$ and $C_{0}$ to be deterministic, while the injected power inputs ($P_{in}$) are stochastic in nature. The following, typically non-restrictive, assumptions are made to allow the development of the technical results of this paper:

The power injection at each bus is composed of three major components: the generated power $P_{g}\!\in\!\mathbb{R}_{\geq 0}^{m}$, and the controllable load demand $P_{cd}\!\in\!\mathbb{R}_{\geq 0}^{m}$, and the non-controllable load demand $P_{ncd}\!\in\!\mathbb{R}_{\geq 0}^{m}$. Following the convention that a positive (respectively, negative) value of injection refers to net-generation (respectively, net-load), we have

The controllable demand ($P_{cd}$) consists of various flexible loads, such as residential air-conditioners and electric water-heaters, which can be coordinated in hierarchical and distributed manner to provide grid stabilizing control support, as illustrated in [5, 6, 7, 8, 9, 10]. Generally, such control responsive loads can be modeled as:

where $P_{cd,0}\!\in\!\mathbb{R}_{\geq 0}^{m}$ are the nominal (or, the baseline) demand associated with the responsive loads, while $Ky$ represents the control-responsive part which responds to the grid measurements ($y$) as per the output feedback control gain matrix $K$. Unlike the generators, where the controller parameters can often be tuned accurately, the controller gain ($K$) in responsive loads is uncertain. For example, let us consider the case of distributed and hierarchical frequency control by a large number of residential thermostatic loads as in [5, 6, 7, 8, 9, 10]. Each thermostatic load receives a frequency threshold, and switches ‘on’ (respectively, ‘off’) if the measured frequency increases (respectively, decreases) beyond the associated threshold. Weather-related and end-use behavioral factors, however, drive uncertainties in the load consumption and results in uncertainties in the load response to those control thresholds [7, 8], which reflect in an uncertain aggregate controller gain matrix ($K$). We model these demand-side control uncertainties via

where $K_{0}\!\in\!\mathbb{R}^{m\times s}$ is the nominal control gain matrix obtained from design; $\Xi$ and $\Sigma$ are $m\!\times\!m$ diagonal matrices with $\{\xi_{i}\}_{i=1}^{m}$ and $\{\sigma_{i}\}_{i=1}^{m}$, respectively, as the diagonal entries; for each $i\in\{1,2,\dots,m\}$, $\xi_{i}=\frac{d\Delta_{i}}{dt}$ with $\{\Delta\}_{i=1}^{m}$ being the standard independent Wiener process [23] and each $\xi_{i}$ is a standard Gaussian white noise, while $\sigma_{i}\!\geq\!0$ is a scaling factor for $\xi_{i}$ representing the associated standard deviation of the Gaussian white noise. Notice that the uncertainty modeled in (4) appears as multiplicative to the system output ($y$) and, thereby, to the system states ($x$). Additionally, we allow uncertainties in the non-controllable load demand ($P_{ncd}$) as

where $P_{ncd,0}\!\in\!\mathbb{R}_{\geq 0}^{m}$ are the nominal values of the non-controllable load demand, and for each $i\in\{1,2,\dots,m\}$, $\eta_{i}=\frac{d\zeta_{i}}{dt}$ with $\{\zeta\}_{i=1}^{m}$ being the standard independent Wiener process [23] and each $\eta_{i}$ is a zero-mean standard Gaussian white noise. Note here uncertainty models similar to (5) can also be applicable to renewable generation (if any), as was shown in [27]. However, for the sake of simplicity in this paper, we consider the generated power ($P_{g}$) to be deterministic, while the uncertainties (additive and multiplicative) are modeled in the controllable and non-controllable loads, i.e.

The resulting closed-loop system model (shown in Fig. 1) with stochastic uncertainties in load demand - both additive and multiplicative to the system states ($x$) - is given by

The term $BP_{in,0}$ acts as a constant forcing to the system. Under Assumption 1, when $A$ is Hurwitz, we can choose a new coordinate $z\!=\!x\!+\!A^{-1}BP_{in,0}$, and rewrite (7) as:

Notice that the uncertainty appears as both multiplicative to the system states (the second term on the right hand side), as well as additive (the third and the fourth terms).

This section presents the necessary definitions of stochastic stability and formally describes the problem statement on design of a sparse controller for the linear stochastic power network in the presence of stochastic uncertainties.

We start by applying results from [23] into deriving the following MSES stability conditions for the system in (10).

In view of Remark 1, the controller synthesis problem becomes well posed, once we enforce certain closed-loop transient performance goals for the nominal deterministic system (with $\sigma_{i}\!=\!0$ in (10)). In particular, consider a Lyapunov function candidate $\Psi\!=\!\widetilde{z}^{\top}\mathcal{P}\widetilde{z}$ , for some $\mathcal{P}\!\succ\!0$ [29]. Then the closed-loop transient performance goal of the nominal deterministic system $\dot{\widetilde{z}}\!=\!A\,\widetilde{z}$ can be reformulated as identifying an acceptably large positive scalar $\gamma$ such that

where $I_{n}$ denotes an identity matrix of size $n$ .

The inequalities in (11) form a nonconvex feasibility problem with optimization variables $\mathcal{Q},M,\{\alpha_{i}\}_{i=1}^{m}$ and $\{\sigma_{i}^{2}\}_{i=1}^{m}$ , due to the bilinear constraint (14b). In the following, we present convex reformulations of (11) for synthesizing stochastic stabilizing optimal controller, while maximizing the allowable uncertainties at the controllable loads and incorporating the transient performance constraints.

MATLAB-based Power System Toolbox [38] is used to illustrate the proposed stochastic control framework on the IEEE 39-bus test system [39] consisting of 10 generators and 19 loads. Here we briefly describe the network preserving model with linear frequency dependent real power loads [40], which is used for the numerical studies. Swing equations are used to model the synchronous generator dynamics, while loads (primarily driven by induction motors) are modeled as affinely dependent on the frequencies. In the augmented network model including the internal buses for the generators [40], the (internal) generator buses are denoted by $\mathcal{G}$ and the load buses as $\mathcal{L}$ . Moreover, due to the internal bus representation, the generators and loads in the augmented network are not connected to the same bus, i.e. $\mathcal{G}\!\cap\!\mathcal{L}\!=\!\emptyset$ . Compactly, the power system dynamic model is given by:

where $E_{i}$ and $\delta_{i}$ are, respectively, the magnitude and phase angle of the $i^{\text{th}}$ bus voltages; $\delta$ is the vector of all bus angles ($\delta_{i}$); $Y_{ij}$ and $\theta_{ij}$ are, respectively, the modulus and the phase angle of the transfer admittance between buses $i$ and $j$; while $P_{e_{i}}(\delta)$ is the electrical power injected into the network at bus-$i$ . For each generator at bus $i\!\in\!\mathcal{G}$ , $M_{i}\!>\!0$ is the inertia constant, $D_{i}\!>\!0$ is the damping coefficient, and $P_{g_{i}}\!>\!0$ is the mechanical power input. For each load at bus $i\!\in\!\mathcal{L}$ , $D_{i}\!>\!0$ is the load-frequency coefficient and $P_{d_{i}}\!>\!0$ is the load demand at equilibrium. As described in Sec. II, the loads consist of controllable and non-controllable parts, i.e. $P_{d_{i}}\!=\!P_{cd_{i}}\!+\!P_{ncd_{i}}$ .

After the adjustment for the reference bus (indexed by ‘R’) phase angle [40], we can express (27) more compactly as:

where $x\!\in\!\mathbb{R}^{n}$, $P_{in}\!\in\!\mathbb{R}^{m}$, $F\!:\!\mathbb{R}^{n}\!\mapsto\!\mathbb{R}^{n}$ is locally Lipschitz, and $B\!\in\!\mathbb{R}^{n\times m}$. We obtain (1) by linearizing (28) around the nominal operating point $x_{e}$ satisfying $F(x_{e})\!+\!BP_{in,0}\!=\!0$ , where $P_{in,0}$ is the nominal power introduced in (6b). Generator power inputs are considered deterministic (Sec. II), while loads (controllable and non-controllable) display both additive and multiplicative stochastic uncertainties as in (7).

This work considers the problem of achieving frequency response through demand-side control in the presence of uncertainty in the controllable as well as uncontrollable loads. An LMI-based demand-side frequency control synthesis problem is formulated for the power networks, that ensures stochastic stability and, in a multi-objective setting, promotes sparsity of the controller, maximizes allowable uncertainties, and enhances closed-loop transient performance. Extensive numerical studies on the IEEE 39-bus system show the effect of penalizing for sparsity and different types of sparsity norms, impact of critical uncertainty on the system states which results in a heavy tailed distribution for frequencies. The Pareto front of optimal solutions between the optimal control efforts and critical uncertainties indicate that allowing more uncertainty in the controllable loads results in decreased closed-loop transient performance. Closed-loop stochastic stability is demonstrated even with only the generator states as measurements. Future work will address the extension of this control synthesis approach to uncertain generating resources (e.g. solar and wind); as well as explore the co-design problem of placement of sensing and control nodes, via enforcing both column- and row-sparsity on the control gain matrices.