Voltage Stabilization in Microgrids
via Quadratic Droop Control
Abstract
We consider the problem of voltage stability and reactive power balancing in islanded smallscale electrical networks outfitted with DC/AC inverters (“microgrids”). A drooplike voltage feedback controller is proposed which is quadratic in the local voltage magnitude, allowing for the application of circuittheoretic analysis techniques to the closedloop system. The operating points of the closedloop microgrid are in exact correspondence with the solutions of a reduced power flow equation, and we provide explicit solutions and smallsignal stability analyses under several static and dynamic load models. Controller optimality is characterized as follows: we show a onetoone correspondence between the highvoltage equilibrium of the microgrid under quadratic droop control, and the solution of an optimization problem which minimizes a tradeoff 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 lowgain 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.
I Introduction
The widespread integration of lowvoltage smallscale 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 lowvoltage 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 selfsufficiency is crucial for reliable power delivery in remote communities, in military outposts, in developing nations lacking largescale 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, microturbines, 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 gridforming 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 higherlevel objectives such as frequency regulation and economic dispatch [6, 7, 8].
Ia Literature Review
The socalled 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 droopcontrolled microgrids has only recently begun to be investigated from a rigorous systemtheoretic point of view [6, 13, 14, 15].
Our focus here is on voltage control, which we now provide some context for. In highvoltage networks, the gridside 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 lowvoltage 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 (socalled 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 IIB.
The widespread deployment of voltage droop has led to several attempts at stability analysis [14, 13]. Both references begin with alltoall Kronreduced 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 constantimpedance or constantcurrent 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 nonconservative 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 wellrecognized 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 interunit 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 nondroopbased 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 droopcontrolled 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.
IB 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 circuittheoretic 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 circuittheoretic foundations. We state and prove a correspondence between the equilibrium points of the closedloop system and the solutions of a reduced power flow equation, before proceeding to analyze the stability of the closedloop 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 IVA 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 tradeoff between total reactive power dissipation and voltage deviations. We comment on possible extensions and generalizations of these results.
Third and finally, in Section IVB 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 lowgain 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 noncollocated 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 III–V. In Section VI we summarize and provide directions for future research.
IC 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 rowstochastic 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 (socalled stability), and componentwise, 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.
Iia Review of Microgrids and AC Circuits
Network Modeling
We adopt the standard model of a quasisynchronous 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 .^{2}^{2}2For 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 offdiagonal 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 nonzero 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 wellknown 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 gridside 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 gridside 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 gridfeeding (i.e., maximum power point tracking) photovoltaic inverters or backtoback converters interfacing wind turbines can be modeled from the grid side as constant power sources, and are therefore included in our framework as loads.
IiB 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 gridconnected 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 nonzero, 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 secondorder 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 highvoltage equilibrium of the closedloop 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
Iiia 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 opencircuit operating point, and does not respect the inherently quadratic nature of reactive power flow. We instead propose a physicallymotivated modification of the conventional voltagedroop 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 closedloop dynamical system is differentialalgebraic, 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.
Circuittheoretic 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].
Indeed, consider the twobus 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 currentvoltage 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 twobus circuits in Figure 2; one for each inverter. If we identify the variablevoltage 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 currentvoltage 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 closedloop 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 controlbyinterconnection with the circuit of Figure 2, the controller (11) represents a more general, densely interconnected circuit with variablevoltage 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).
IiiB 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 differentialalgebraic closedloop 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 IIIC.
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 rowstochastic matrices. Finally we define the opencircuit load voltages via
(16) 
Since is rowstochastic, each component of is a weighted average of inverter set points .
Theorem III.1
(Reduced Power Flow Equation for Quadratic Droop Network) Consider the closedloop system (8) resulting from the quadratic droop controller (6), along with the definitions (13)–(16). The following two statements are equivalent:

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

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 closedloop 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 rowstochastic (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 circuitreduction 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. Leftmultiplying 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 lefthand side of the closedloop system (8) to zero, equilibrium points satisfy
(20)  
Since solves (20), we can leftmultiply 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 fixedpoint 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 closedloop 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 differentialalgebraic 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 differentialalgebraic 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 leftmultiply 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 bottomright block of is positive definite, will be positive definite if and only if the Schur complement with respect to this bottomright 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 offdiagonal 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 IIIC 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 saddlenode and singularityinduced bifurcations. Perhaps surprisingly, we show that even for a star topology, overall network stability is not equivalent to pairwise stability of each inverter with the common load.
IiiC Equilibria and Stability Conditions for ZI, ZIP, and Dynamic Shunt Load Models
In this section we leverage the general results of Section IIIB 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 constantimpedance and constantcurrent loads.
Theorem III.3
(Stability with “ZI” Loads) Consider the reduced power flow equation (17) for constantimpedance/constantcurrent loads
(28) 
where (resp. ) is the vector of constantimpedance loads (resp. constant current loads). Assume that

is an matrix , and

componentwise .
Then the unique solution to (17) is given by
(29) 
and the associated equilibrium point of the closedloop 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 componentwise). Both would be violated only under fault conditions in the microgrid, and are therefore nonconservative. As expected, in the case of opencircuit operation (no loading) when , (29) reduces to , the opencircuit 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.
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 nontrivial stability properties [29]. The following result builds on analysis techniques developed in [30] and provides an approximate characterization of the highvoltage 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 highvoltage solution of (17) for ZI loads as given in Theorem III.3. Furthermore, define the shortcircuit capacity matrix
(33) 
If is sufficiently small, then there exists a unique highvoltage 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 constantimpedance and constantcurrent load components, Theorem III.4 requires the constantpower component of the load model to be sufficiently small. The highvoltage 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 steadystate 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 constantimpedance load model , with the shunt susceptance dynamically adjusted to achieve a constant power consumption in steadystate. This is a common lowfidelity dynamic model for thermostatically controlled loads, induction motors, and loads behind tapchanging 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 droopcontrolled 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 highvoltage solutions can be used for design purposes to (nonuniquely) backcalculate controller gains, or to determine bounds on tolerable loading profiles.
Iv Controller Performance
Having thoroughly investigated the stability properties of the closedloop system (8) in Section III, we now turn to questions of optimality and controller performance. Here we are interested in inverseoptimality 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 systemtheoretic optimality or performance of the controller.
Iva Optimality of Quadratic Droop Control
While in Section III we emphasized the circuittheoretic interpretation of the quadratic droop controller (6), it is also relevant to ask whether the resulting equilibrium point of the system is inverseoptimal with respect to any criteria. For simplicity of exposition, we restrict our attention in this subsection to constantimpedance 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 inverseoptimality 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 tradeoff 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 closedloop microgrid system (8) with constantimpedance 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 closedloop system (8) is equal to the unique minimizer of the optimization problem (39), and both are given by
(40) 
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 closedloop 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 leftmultiplying 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 bottomright 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 tradeoffs are studied). The result can also be interpreted as an application of Maxwell’s minimum heat theorem to controller design [40, Proposition 3.6].