Voltage Stabilization in Microgridsvia Quadratic Droop Control

Voltage Stabilization in Microgrids
via Quadratic Droop Control

John W. Simpson-Porco,  Florian Dörfler,  and Francesco Bullo,  J. W. Simpson-Porco is with the Department of Electrical and Computer Engineering, University of Waterloo. Email: jwsimpson@uwaterloo.ca. F. Bullo is with the Department of Mechanical Engineering and the Center for Control, Dynamical System and Computation, University of California at Santa Barbara. Email: bullo@engineering.ucsb.edu. F. Dörfler is with the Automatic Control Laboratory, ETH Zürich. Email: dorfler@ethz.ch.This work was supported by NSF grants CNS-1135819 and by the National Science and Engineering Research Council of Canada.
Abstract

We consider the problem of voltage stability and reactive power balancing in islanded small-scale electrical networks outfitted with DC/AC inverters (“microgrids”). A droop-like voltage feedback controller is proposed which is quadratic in the local voltage magnitude, allowing for the application of circuit-theoretic analysis techniques to the closed-loop system. The operating points of the closed-loop microgrid are in exact correspondence with the solutions of a reduced power flow equation, and we provide explicit solutions and small-signal stability analyses under several static and dynamic load models. Controller optimality is characterized as follows: we show a one-to-one correspondence between the high-voltage equilibrium of the microgrid under quadratic droop control, and the solution of an optimization problem which minimizes a trade-off between reactive power dissipation and voltage deviations. Power sharing performance of the controller is characterized as a function of the controller gains, network topology, and parameters. Perhaps surprisingly, proportional sharing of the total load between inverters is achieved in the low-gain limit, independent of the circuit topology or reactances. All results hold for arbitrary grid topologies, with arbitrary numbers of inverters and loads. Numerical results confirm the robustness of the controller to unmodeled dynamics.

Inverter control, microgrids, voltage control, Kron reduction, nonlinear circuits

I Introduction

The wide-spread integration of low-voltage small-scale renewable energy sources requires that the present centralized electric power transmission paradigm to evolve towards a more distributed future. As a flexible bridge between distributed generators and larger distribution grids, microgrids continue to attract attention [1, 2, 3]. Microgrids are low-voltage electrical distribution networks, heterogeneously composed of distributed generation, storage, load, and managed autonomously from the larger primary grid. While often connected to the larger grid through a “point of common coupling”, microgrids are also able to “island” themselves and operate independently [2, 4]. This independent self-sufficiency is crucial for reliable power delivery in remote communities, in military outposts, in developing nations lacking large-scale infrastructure, and in backup systems for critical loads such as hospitals and campuses. Energy generation within a microgrid can be quite heterogeneous, including photovoltaic, wind, micro-turbines, etc. Many of these sources generate either variable frequency AC power or DC power, and are interfaced with a synchronous AC grid via power electronic DC/AC inverters. In islanded operation, at least some of these inverters must operate as grid-forming devices. That is, control actions must be taken through them to ensure synchronization, voltage stability, and load sharing in the network [1, 5, 2, 4], and to establish higher-level objectives such as frequency regulation and economic dispatch [6, 7, 8].

Fig. 1: (a) Schematic of a “parallel” microgrid, in which several inverters supply power to a distribution bus (effectively a single load) (b) A simple non-parallel microgrid consisting of five loads and three inverters .

I-a Literature Review

The so-called droop controllers (and their many derivatives) have been used with some success to achieve primary control goals such as stability and load sharing, see [9, 2, 10, 1, 11, 12, 4]. Despite being the foundational technique for networked operation of islanded microgrids (Figure 1), the stability and fundamental limitations of droop-controlled microgrids has only recently begun to be investigated from a rigorous system-theoretic point of view [6, 13, 14, 15].

Our focus here is on voltage control, which we now provide some context for. In high-voltage networks, the grid-side voltages of transformers which interface synchronous generators are regulated to nominal set points via automatic voltage regulator systems. However, if this strategy was applied to inverters in low-voltage networks, the small mismatches in voltage set points would combine with the large impedances presented by distribution lines, and result in large circulating reactive currents between inverters. Moreover, due to the close electrical proximity of devices in small microgrids, and the small power capacities (ratings) of the sources, it is desirable that controllers establish set percentages of the total load to be supplied by each inverter (so-called power sharing). These technical obstacles motivated the use of voltage droop control [16] as a heuristic proportional controller to establish power sharing between units while maintaining voltages within a reasonable range around their set point values; a technical review of the voltage droop controller is presented in Section II-B.

The widespread deployment of voltage droop has led to several attempts at stability analysis [14, 13]. Both references begin with all-to-all Kron-reduced network models which do not explicitly contain loads. They then assume the existence of a system equilibrium, and derive sufficient stability conditions which depend on the uncharacterized equilibrium. Results apply only for constant-impedance or constant-current load models, and no characterizations are given of reactive power sharing in steady state. Thus, the available literature offers no guidance on the foundational issue of operating point feasibility, i.e., the existence and locations of steady states for the microgrid which satisfy operational constraints. This gap of knowledge means that precise reactive loading limits and security margins are unknown, making system monitoring and non-conservative operation difficult. Moreover, stable network operation is often limited by stiff constant power loads, which do not reduce their current consumption when voltages fall.

Regarding control performance, a well-recognized drawback of voltage droop control is that — while achieving better power sharing performance than mere voltage regulation — the power sharing properties can still be quite poor. This has led some authors to investigate alternative strategies to implement reactive power sharing [11, 17, 18, 7] that are based on directly load measurements or inter-unit communication. Aside from these additional requirements, these approaches do not assist in characterizing the power sharing properties of the standard, decentralized droop control. A general discussion of fundamental limits for voltage control and reactive power sharing can be found in [7, Section III], and some recent non-droop-based control (but nonetheless related) approaches to reactive power compensation can be found in [19, 20, 21, 22, 23, 24, 25, 26, 27, 28].

In summary, two fundamental outstanding problems regarding the voltage stability of droop-controlled microgrids are (i) to establish conditions under which a stable network equilibrium exists for various load models, and (ii) to characterize the resulting power sharing properties in terms of grid topology, branch admittances, loads, and controller gains.

I-B Contributions

In the preliminary version of this work [29], we introduced the quadratic droop controller, a slight modification of the conventional linear droop curve to a quadratic one. Quadratic droop leads to the same steady states as conventional droop control (for particular gains) but allows for circuit-theoretic techniques to be leveraged for analysis purposes. In [29] we studied system stability for the case of an islanded parallel microgrid, where several inverters feed power to a single common load or a common distribution bus (Remark 4). In this paper we depart from the case of a parallel microgrid and study system stability for arbitrary interconnections of inverters and loads, under various load models. In addition, we investigate the optimality and performance characteristics of the controller. Linearization ideas presented in [30] are also relevant to the current presentation at a technical level, and will be used in proofs of several results.

There are three main technical contributions in this paper. First, in Section III we present the quadratic droop controller, highlighting its circuit-theoretic foundations. We state and prove a correspondence between the equilibrium points of the closed-loop system and the solutions of a reduced power flow equation, before proceeding to analyze the stability of the closed-loop system for both static and dynamic load models. In particular, we analyze system stability for the static ZI and ZIP load models, and for a dynamic model describing a variable shunt susceptance. Our stability analyses in Theorem III.1, Theorem III.3, and Theorem III.4 result in succinct and physically intuitive stability conditions, significantly generalizing the results and intuition developed for parallel microgrids in [29].

Second, in Section IV-A we investigate the optimality properties of the quadratic droop controller. We demonstrate that the microgrid dynamics under quadratic droop control can be interpreted as a decentralized primal algorithm for solving a particular optimization problem, leading to an additional interpretation of the quadratic droop controller as an algorithm which attempts to minimize a trade-off between total reactive power dissipation and voltage deviations. We comment on possible extensions and generalizations of these results.

Third and finally, in Section IV-B we investigate the power sharing properties of our controller, and closely examine two asymptotic limits. In the limit of high controller gains, we show that the reactive power is supplied by an inverter to a load in inverse proportion to the electrical distance between the two, independent of relative controller gains. Conversely, in the low-gain limit we find that inverters provide reactive power in proportion to their controller gains, independent of electrical distance. When neither limit is appropriate, we provide a general formula and quantify the power sharing error as a function of the controller gains and network parameters. These predictions along with the robustness of the approach are studied numerically in Section V. Remarks throughout the paper provide details on extensions and interpretations of our results, and can be skipped on a first reading without loss of continuity.

At a more conceptual level, this work has two major features which distinguish it from the primary literature on droop control. The first is our analytical approach, based on circuit theory [31] and approximations of nonlinear circuit equations [30]. While we make specific modeling assumptions, they are standard in the field [4] and allow us to derive strong yet intuitive results regarding stability and optimality of operating points. Our focus is on characterizing the operating points arising from decentralized voltage control, along with characterizing controller performance at the network level. We make use of different load models in different sections of the paper, in part to simplify certain presentations, and in part to illustrate a variety of analysis techniques which should prove useful for other researchers. The second distinguishing feature is our explicit consideration of loads which are not collocated with inverters. Voltages at non-collocated loads are typically lower then the controlled voltages at inverters, and hence it is these load voltages which ultimately limit network stability.

The remainder of this section recalls some basic notation. Section II reviews basic models for microgrids, loads, and inverters, along with the conventional droop controller and the relevant literature. Our main results are housed in Sections IIIV. In Section VI we summarize and provide directions for future research.

I-C Preliminaries and Notation

Sets, vectors and functions: We let (resp. ) denote the set of real (resp. strictly positive real) numbers, and let be the imaginary unit. Given , is the associated diagonal matrix with on the diagonal. Throughout, and are the -dimensional vectors of unit and zero entries, and is a matrix of all zeros of appropriate dimensions. The identity matrix is . A matrix is row-stochastic if its elements are nonnegative and . A matrix is an -matrix if for all and all eigenvalues of have positive real parts. In this case is Hurwitz for any positive definite diagonal (so-called -stability), and component-wise, with strict inequality if is irreducible [32].

Ii Modeling and Problem Setup

We briefly review our microgrid model and recall the conventional voltage droop controller.

Ii-a Review of Microgrids and AC Circuits

Network Modeling

We adopt the standard model of a quasi-synchronous microgrid as a linear circuit represented by a connected and weighted graph , where is the set of vertices (buses) and is the set of edges (branches). There are two types of buses: load buses , and inverter buses , such that . These two sets of buses will receive distinct modeling treatments in what follows.

The edge weights of the graph are the associated branch admittances .222For the short transmission lines of microgrids, line charging and leakage currents are neglected and branches are modeled as series impedances [33]. Shunt capacitors/reactors will be included later as load models. The network is concisely represented by the symmetric bus admittance matrix , where the off-diagonal elements are for each branch (zero if ), and the diagonal elements are given by . To each bus we associate a phasor voltage and a complex power injection . For dominantly inductive lines, transfer conductances may be neglected and is purely imaginary, where is the susceptance matrix. We refer to Remark 1 for a discussion of this assumption. The power flow equations then relate the bus electrical power injections to the bus voltages via

(1a)
(1b)

In this work we focus on dynamics associated with the reactive power flow equation (1b), and refer the reader to [6, 7, 8, 34] and the references therein for analyses of microgrid active power/frequency dynamics. We will work under the standard decoupling approximation, where and hence for each ; see [35, 12]. This can be relaxed to non-zero but constant power angles at the cost of more complicated formulae, but we do not pursue this here; see Remark 1. Under the decoupling assumption, the reactive power injection in (1b) becomes a function of only the voltages , yielding

(2)

or compactly in vector notation as

(3)

where . For later reference, some well-known properties of the susceptance matrix are recorded in Lemma A.1.

Load Modeling

When not specified otherwise, we assume a static load model of the form , where the reactive power injection is a smooth function of the supplied voltage . With our sign convention, corresponds to an inductive load which consumes reactive power. Specific static and dynamic load models will be used throughout the paper, and we introduce these models when needed.

Inverter Modeling

A standard grid-side model for a smart inverter is as a controllable voltage source

(4)

where is a control input to be designed, and is a time constant accounting for sensing, processing, and actuation delays. The model (4) assumes that the control loops which regulate the inverter’s internal voltages and currents are stable, and that these internal loops are fast compared to the grid-side time scales over which loads change. This model is widely adopted among experimentalists in the microgrid field [1, 36, 37], and further explanations can be found in [7] and references therein. We note that grid-feeding (i.e., maximum power point tracking) photovoltaic inverters or back-to-back converters interfacing wind turbines can be modeled from the grid side as constant power sources, and are therefore included in our framework as loads.

Ii-B Review of Conventional Droop Control

The voltage droop controller is a decentralized controller for primary voltage control in islanded microgrids. The controller is a heuristic based on the previously discussed decoupling assumption, and has an extensive history of use; see [16, 2, 9, 1, 11, 17]. For the case of inductive lines, the droop controller specifies the input signal in (4) as the local feedback [4, Chapter 19]***For grid-connected inverters, one sometimes sees this formula augmented with power set points . Here we consider islanded operation where .

(5)

where is the nominal voltage for the inverter, and is the measured reactive power injection; see [38, 4] for details regarding measurement of active/reactive powers. The controller gain is referred to as the droop coefficient. From (5), it is clear that if the inverter reactive power injection is non-zero, the voltage will deviate from .

Remark 1

(Comments on Modeling, Decoupling, and Droop Control) Formally, the decoupling assumption leading to (2) is an assumption about the sensitivity of reactive power injections with respect to phase angles . From (1b), this sensitivity is proportional to , and around normal operating points is therefore roughly zero; phase angles effect reactive power only though second-order effects. The strong relationship between reactive power and voltage magnitudes is why the former is often used as a tool to regulate the latter. The voltage droop controller (5) is designed under the assumption that the susceptance to conductance ratios are large within the microgrid. This assumption is typically typically justified in engineered settings, as the inverter output impedances are controlled to dominate over network impedances giving the network a strongly inductive characteristic [39],[4, Chapter 7]. For dominantly resistive (or capacitive) microgrids, the appropriate droop controllers take different forms [4, Chapter 19], with the following general structure: assuming all lines have uniform reactance/resistance ratios const., the controller is modified by replacing in (5) with , where . A standard calculation [8] yields , where is an admittance matrix with edge weights . This formula is of the same mathematical form as (1b), and for such uniform networks, we can restrict ourselves — without loss of generality — to the decoupled reactive power flow (3) and the conventional droop controller (5). As a special case of the above, we note that all analysis herein also applies to voltage/active power droop control for dominantly resistive microgrids.

Despite its extensive history, the voltage droop controller (5) has so far resisted any rigorous stability analysis. In our opinion, the key obstacle has been — and remains — the difficulty in determining the high-voltage equilibrium of the closed-loop system for arbitrary interconnections of devices. We remove this obstacle in the next section by proposing a controller based on the nonlinear physics of AC power flow.

Iii Quadratic Droop Control

Iii-a Definition and Interpretation

While the droop controller (5) is simple and intuitive, it is based on the linearized behavior of AC power flow around the system’s open-circuit operating point, and does not respect the inherently quadratic nature of reactive power flow. We instead propose a physically-motivated modification of the conventional voltage-droop controller (5). In place of (5), consider instead the quadratic droop controller

(6)

where is the controller gain.While the chosen sign convention is unconventional, it will simplify the resulting formulae and help us interpret the controller physically. Compared to the conventional controller (5), the gain on the regulating term in (6) now scales with the inverter voltage. Combining the power flow (2) with a static load model at each load bus , we must also satisfy the power balance equations

(7)

Combining now the inverter model (4), the quadratic droop controller (6), the load power balance (7), and the power flow equation (3), the closed-loop dynamical system is differential-algebraic, and can be written compactly as

(8)

where , , and and are diagonal matrices with elements and respectively. Since the variables represent voltage magnitudes referenced to ground, they are intrinsically positive, and for physical consistency we restrict our attention to positive voltage magnitudes.

Remark 2

(Interpretations) Before proceeding to our main analysis, we offer two interpretations of the quadratic controller (6), one theoretical and one pragmatic.

Circuit-theoretic interpretation: The design (6) can be interpreted as control by interconnection, where we interconnect the physical electrical network with fictitious “controller circuits” at the inverter buses [40].

Fig. 2: Linear circuit representation of the quadratic droop controller (6).
Fig. 3: Diagram showing network augmentation and reduction. First, each inverter bus of the network in Figure 1 is interconnected with a two-bus controller circuit, consisting of an inverter bus and fictitious bus

at constant voltage . In Theorem III.1, the inverter buses are eliminated via Kron reduction, leaving a reduced network with only fixed voltage buses

and load buses .

Indeed, consider the two-bus circuit of Figure 2, where the blue bus has variable voltage and is connected via a susceptance to the green bus of fixed voltage . The current-voltage relations for this fictitious circuit are

(9)

where (resp. ) is the current injection at the bus with voltage (resp. ). Now, let there be of the two-bus circuits in Figure 2; one for each inverter. If we identify the variable-voltage blue buses of these circuits with the inverter buses of our original network, and impose that the current injected into the former must exit from the latter, we obtain an augmented network with buses, and in vector notation the current-voltage relations in the new network are

(10)

where we have block partitioned all variables according to loads, inverters, and fictitious controller buses. In this augmented circuit, the inverters behave as interior nodes joining the fictitious controller buses to the loads, and do not sink or source power themselves. Left multiplying the first two blocks of equations in (10) by and noting that by the definition of reactive power , we immediately obtain the right hand side of (8). Thus, the closed-loop equilibrium points under quadratic droop control can be interpreted as the solutions of the power balance equations for an expanded linear circuit.

Practical interpretation: Far from being linear, voltage/reactive power capability characteristics of synchronous generators display significant nonlinearities. In the absence of saturation constraints, the characteristics are in fact quadratic [33, Equation 3.105], and thus the quadratic droop controller (6) more accurately mimics the behavior of a synchronous generator with automatic voltage regulation, compared to the classical controller (5). This quadratic dependence means that the marginal voltage drop (voltage drop per unit increase in reactive power) increases with reactive power provided.

Remark 3

(Generalizations) The quadratic droop controller (6) is a special case of the general feedback controller

(11)

where and are gains. One recovers (6) by setting , , and all other parameters to zero. While the decentralized controller (6) can be interpreted as control-by-interconnection with the circuit of Figure 2, the controller (11) represents a more general, densely interconnected circuit with variable-voltage nodes and fixed voltage nodes. Since decentralized control strategies are preferable in microgrids, we focus on the decentralized controller (6) with the understanding that results may be extended to the more general (11).

Iii-B Equilibria and Stability Analysis by Network Reduction

We first pursue the following question: under what conditions on load, network topology, admittances, and controller gains does the differential-algebraic closed-loop system (8) possess a locally exponentially stable equilibrium? By exploiting the structure of the quadratic droop controller (6), we will establish a correspondence between the equilibria of (8) and the solutions of a power flow equation for a reduced network. In this subsection we focus on generic static load models before addressing specific load models in Section III-C.

For notational convenience, we first define a few useful quantities. We block partition the susceptance matrix and nodal voltage variables according to loads and inverters as

(12)

and define the reduced susceptance matrix by

(13)

Moreover, we define the averaging matrices and by

(14)
(15)

It can be shown (Proposition A.2) that is invertible, and that and are both row-stochastic matrices. Finally we define the open-circuit load voltages via

(16)

Since is row-stochastic, each component of is a weighted average of inverter set points .

Theorem III.1

(Reduced Power Flow Equation for Quadratic Droop Network) Consider the closed-loop system (8) resulting from the quadratic droop controller (6), along with the definitions (13)–(16). The following two statements are equivalent:

  1. Original Network: The voltage vector is an equilibrium point of (8);

  2. Reduced Network: The load voltage vector is a solution of the reduced power flow equation

    (17)

    and the inverter voltage vector is recovered via

    (18)

Theorem III.1 states that, to study the existence and uniqueness of equilibria for the closed-loop system (8), we need only study the reduced power flow equation (17). Voltages at inverters may then be recovered uniquely from (18). In fact, since is row-stochastic (Proposition A.2), the inverter voltages (18) are simply weighted averages of load voltages and inverter set point voltages .

The reduced power flow equation (17) is straightforward to interpret in terms of the circuit-reduction of Figure 3. First, as in Remark 2, we augment the original circuit with fictitious controller buses. Eliminating the inverter voltages from the augmented network current balance (10) through Kron reduction [31], one obtains the input/output equivalent circuit

(19)

This reduction process is shown in Figure 3. Left-multiplying the first block in (19) by immediately yields the reduced power flow (17). Hence (17) is exactly the reactive power balance equation in the reduced network of Figure 3 after reduction (cf. [35, Equation 2.10b]). The bottom block of equations in (19) determines the fictitious controller current injections once the top block is solved for .

Proof of Theorem III.1:   (i)(ii): Setting the left-hand side of the closed-loop system (8) to zero, equilibrium points satisfy

(20)

Since solves (20), we can left-multiply the lower block of equations in (20) by to obtain

(21)

and solve for to obtain the inverter voltages (18). Substituting (18) into the first block of equations in (20), we calculate

which is the reduced power flow equation (17).
(ii)(i): Due to (18) and Proposition A.2 (iii), we have that implies that and hence . An easy computation shows that (17) and (18) together imply that satisfy the fixed-point equations (20), and thus is an equilibrium point.

In a similar spirit, the following result states that local exponential stability of an equilibrium point of (8) may be checked by studying a reduced Jacobian matrix.

Theorem III.2

(Stability from Reduced Jacobian) Consider the closed-loop system (8) resulting from the quadratic droop controller (6). If the Jacobian of the reduced power flow equation (17), given in vector notation by

(22)

is a Hurwitz matrix when evaluated at a solution of (17), then the equilibrium point of the differential-algebraic system (8) is locally exponentially stable. Moreover, assuming that for each load bus , a sufficient condition for (22) to be Hurwitz is that

(23)

Proof of Theorem III.2:   We appeal to [41, Theorem 1], which states that local stability of the differential-algebraic system (8) at the equilibrium may be studied by linearizing the differential algebraic system, eliminating the algebraic equations from the system matrix, and checking that the resulting reduced matrix is Hurwitz. Consider the generalized eigenvalue problem (GEP) where , and Jacobian matrix of (8) is given by

(24)

The diagonal matrix has elements for and for . Since is strictly positive, we left-multiply through by and formulate the previous GEP as the symmetric GEP

(25)

where

(26)

Partitioning the eigenvector as , block partitioning , and eliminating the top set of algebraic equations, we arrive at a reduced GEP where . The matrices on both sides are symmetric, and in particular the matrix on the right is diagonal and positive definite. The eigenvalues of this reduced GEP are therefore real, and it holds that for each if and only if is negative definite. We now show indirectly that is positive definite, by combining two standard results on Schur complements. Through some straightforward computations, one may use (13), (16), (18) and (21) to simplify to

where . Since the bottom-right block of is positive definite, will be positive definite if and only if the Schur complement with respect to this bottom-right block is also positive definite. This Schur complement is equal to

(27)

which is exactly times the Jacobian (22) of (17). Since is Hurwitz with nonnegative off-diagonal elements, is an -matrix and is therefore -stable. It follows that the Schur complement (27) is positive definite, and hence that is positive definite, which completes the proof of the main statement. For the moreover statement, proceed along similar arguments and consider the symmetric version of (22), which equals

Solving the reduced power flow equation (17) for and substituting into the third term above, we find that

The first term is diagonal and by assumption negative semidefinite, and hence will be Hurwitz if (23) holds, which completes the proof.

The sufficient condition (23) is intuitive: it states that at equilibrium, the network matrix should be more susceptive than the equivalent load susceptances . The results of Theorem III.2 are implicit, in that checking stability depends on the undetermined solutions of the reduced power flow equation (17). The situation will become clearer in Section III-C when we apply Theorems III.1 and III.2 to specific load models.

Remark 4

(Parallel Microgrids) In [29] we provided an extensive analysis of the closed loop (8) for a “parallel” or “star” topology, where all inverters feed a single common load. In particular, for a constant power load we provided a necessary and sufficient condition for the existence of a stable equilibrium. Depending on the system parameters, the microgrid can have zero, one, or two physically meaningful equilibria, and displays both saddle-node and singularity-induced bifurcations. Perhaps surprisingly, we show that even for a star topology, overall network stability is not equivalent to pair-wise stability of each inverter with the common load.

Iii-C Equilibria and Stability Conditions for ZI, ZIP, and Dynamic Shunt Load Models

In this section we leverage the general results of Section III-B to study the equilibria and stability of the microgrid for some specific, tractable load models. To begin, the reduced power flow equation (17) can be solved exactly for combinations of constant-impedance and constant-current loads.

Theorem III.3

(Stability with “ZI” Loads) Consider the reduced power flow equation (17) for constant-impedance/constant-current loads

(28)

where (resp. ) is the vector of constant-impedance loads (resp. constant current loads). Assume that

  1. is an -matrix , and

  2. component-wise .

Then the unique solution to (17) is given by

(29)

and the associated equilibrium point of the closed-loop system (8) is locally exponentially stable.

The first technical condition in Theorem III.3 restricts the impedance loads from being overly capacitive, while the second restricts the current loads from being overly inductive (since component-wise). Both would be violated only under fault conditions in the microgrid, and are therefore non-conservative. As expected, in the case of open-circuit operation (no loading) when , (29) reduces to , the open-circuit load voltage vector.

Proof of Theorem III.3:    Substituting the ZI load model (28) into the reduced power flow equation (17), we obtain

(30)

where in the second line we have factored out and identified the second term in parentheses with (29). Observe that is a solution to (30). Since is nonsingular and we require that , is the unique solution of (30). It remains only to show that . Since is by assumption an -matrix, the inverse has nonpositive elements [32]. By invertibility, cannot have any zero rows, and therefore the product with the strictly negative vector yields a strictly positive vector as in (29).

To show local stability, we apply Theorem III.2 and check that the Jacobian of (30) evaluated at (29) is Hurwitz. Differentiating (30), we calculate that

and therefore that

(31)

which is Hurwitz since is by assumption an -matrix and -matrices are -stable.

A more general static load model still is the ZIP model which augments the ZI model (28) with an additional constant power demand [35]. The model is depicted in Figure 4.

Fig. 4: Pictorial representation of ZIP load model.

In vector notation, the ZIP model generalizes (28) as

(32)

Unlike the ZI model (28), the reduced power flow equation (17) cannot be solved analytically for ZIP loads, except for the special case of a parallel microgrid. Even in this special case, the network can have multiple equilibrium points with non-trivial stability properties [29]. The following result builds on analysis techniques developed in [30] and provides an approximate characterization of the high-voltage solution to (17) when the constant power term is “small”.

Theorem III.4

(Stability with “ZIP” Loads) Consider the reduced power flow equation (17) with the ZIP load model (32), let the conditions of Theorem III.3 hold, and let be the high-voltage solution of (17) for ZI loads as given in Theorem III.3. Furthermore, define the short-circuit capacity matrix

(33)

If is sufficiently small, then there exists a unique high-voltage solution of the reduced power flow equation (17) with ZIP loads (32), given by

(34)

where , and the corresponding equilibrium point of (8) is locally exponentially stable.

Proof:  See Appendix A.

Similar to the technical conditions (i) and (ii) in Theorem III.3 regarding constant-impedance and constant-current load components, Theorem III.4 requires the constant-power component of the load model to be sufficiently small. The high-voltage solution (34) can be thought of as a regular perturbation of the solution for ZI loads, where is a small error term. As , quadratically, and .

While the ZIP load model can accurately capture the steady-state behavior of most aggregated loads, when considering dynamic stability of a power system it is often important to check results obtained for static load models against those obtained using basic dynamic load models [35]. A common dynamic load model is the dynamic shunt model , or in vector notation

(35)

where is a diagonal matrix of time constants. The model specifies a constant-impedance load model , with the shunt susceptance dynamically adjusted to achieve a constant power consumption in steady-state. This is a common low-fidelity dynamic model for thermostatically controlled loads, induction motors, and loads behind tap-changing transformers [35]. We restrict our attention to inductive loads , as these are the most common in practice, and without loss of generality we assume that for all , since if the unique steady state of (35) is and the equation can be removed.

Theorem III.5

(Stability with Dynamic Shunt Loads) Consider the reduced power flow equation (17) with the dynamic shunt load model (35) and . If is sufficiently small, then there exists a unique solution of (17),(35), given by

(36a)
(36b)

where and , and the corresponding equilibrium point of the system (8),(35) is locally exponentially stable.

Proof:   We sketch the proof in Appendix A.

Remark 5

(From Quadratic to Conventional Droop): While the stability results derived above hold for the quadratic droop-controlled microgrid, under certain selections of controller gains a direct implication can be drawn from the above stability results to stability of the microgrid under conventional, linear voltage droop control. Additional information can be found in [29, Section V].

We conclude this section by commenting on the utility of the preceding analysis and stability results. If hard design limits are imposed on the load voltages , the expressions (29) or (34) for the unique high-voltage solutions can be used for design purposes to (non-uniquely) back-calculate controller gains, or to determine bounds on tolerable loading profiles.

Iv Controller Performance

Having thoroughly investigated the stability properties of the closed-loop system (8) in Section III, we now turn to questions of optimality and controller performance. Here we are interested in inverse-optimality of the resulting equilibrium point of the system in an optimization sense, along with how well the resulting equilibrium achieves the power sharing objective. We do not addresses system-theoretic optimality or performance of the controller.

Iv-a Optimality of Quadratic Droop Control

While in Section III we emphasized the circuit-theoretic interpretation of the quadratic droop controller (6), it is also relevant to ask whether the resulting equilibrium point of the system is inverse-optimal with respect to any criteria. For simplicity of exposition, we restrict our attention in this subsection to constant-impedance load models of the form

(37)

Analogous extensions to ZIP load models are possible using energy function theory [42], at the cost notational complexity and less insight. To begin, note that for nodal voltages , the total reactive power absorbed by the inductive transmission lines of the microgrid is given by

Similarly, the total reactive power consumed by the loads is

Efficient operation of the microgrid stipulates that we minimize reactive power losses and consumption, since transmission and consumption of reactive power contributes to thermal losses.For an inverse-optimality analysis that relates real power generation to frequency droop control, see [8]. Simultaneously however, we have the practical requirement that inverter voltages must remain close to their rated values . We encode this requirement in the quadratic cost

(38)

where are cost coefficients. Consider now the combined optimization problem

(39)

where we attempt to enforce a trade-off between minimizing reactive power dissipation and minimizing voltage deviations. The next result shows that an appropriately designed quadratic droop controller (6) is a decentralized algorithm for solving the optimization problem (39). Conversely, any quadratic droop controller solves an optimization problem of the form (39) for appropriate coefficients .

Proposition IV.1

(Optimality and Quadratic Droop) Consider the closed-loop microgrid system (8) with constant-impedance loads (37) and controller gains , and the optimization problem (39) with cost coefficients . Assume as in Theorem III.3 that is an -matrix. If the parameters are selected such that for each , then the unique locally exponentially stable equilibrium of the closed-loop system (8) is equal to the unique minimizer of the optimization problem (39), and both are given by

(40)

with recovered by inserting (40) into (18).

Proof of Proposition IV.1:    That is the unique locally exponentially stable equilibrium of (8) follows immediately from Theorem III.3 by setting . We now relate the critical points of the optimization problem (39) to the equilibrium equations for the closed-loop system. In vector notation, we have that , , and , where . The total cost may therefore be written as the quadratic form , where and

The first order optimality conditions yield

(41)

Solving the second block of equations in (41) for yields the quadratic droop inverter voltages (18). Substituting this into the first block of equations in (41), simplifying, and left-multiplying by yields

(42)

where and are as in (13)–(16). The result (42) is exactly the reduced power flow equation (30) with , which shows the desired correspondance. It remains only to show that this unique critical point is a minimizer. The (negative of the) Hessian matrix of is given by the matrix of coefficients in (41). Since the bottom-right block of this matrix is negative definite, it follows by Schur complements that the Hessian is positive definite if and only if is positive definite, which by assumption holds.

Note that depending on the heterogeneity of the voltage set points , minimizing may or may not conflict with minimizing . The quadratic droop controller (6) is a primal algorithm for the optimization problem (39), and can therefore be interpreted as striking an optimal balance between maintaining a uniform voltage profile and minimizing total reactive power dissipation (cf. [19, 43], where similar trade-offs are studied). The result can also be interpreted as an application of Maxwell’s minimum heat theorem to controller design [40, Proposition 3.6].

Remark 6

(Generalized Objective Functions) To obtain the generalized feedback controller (11), one may replace given in (38) by