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Voltage Stabilization in Microgrids via Quadratic Droop Control
John W. Simpson-Porco, Florian Dorfler, Francesco Bullo
TL;DR
The paper addresses voltage stability and reactive power sharing in islanded microgrids. It analyzes quadratic droop control, establishing equilibrium and stability results, an optimization interpretation, and gain-dependent sharing laws that range from proportional sharing to electrical-distance-based sharing.
Problem
The paper studies how reactive power injections at inverters change under incremental load-demand changes, alongside voltage stability of islanded microgrids.
Method
The paper analyzes a quadratic droop controller using circuit-theoretic methods, relating closed-loop equilibria to reduced power-flow solutions and studying static and dynamic load models.
Results
The controller interpolates between proportional power sharing in the low-gain regime and electrical-distance-based sharing in the high-gain regime, with an optimization interpretation for the high-voltage equilibrium.
Takeaways & Limitations
Low gains yield reactive power sharing proportional to controller gains and independent of electrical distance, whereas high gains yield sharing inversely proportional to electrical distance.
Takeaways & Limitations
The results apply to microgrids with uniform R/X ratios under a decoupling assumption, and designing a provably stable controller for non-uniform ratios remains open.
Abstract
from arXiv · showhide
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.
I. INTRODUCTION
The paper addresses voltage stability and reactive power sharing in islanded microgrids, where existing droop-control analyses provide limited equilibrium and sharing guarantees. It introduces quadratic droop control and develops circuit-theoretic analyses of stability, optimality, and network-level power sharing.
- Islanded microgrids require inverter controls that maintain voltage stability and load sharing while supporting distributed generation, storage, and autonomous operation.
- Existing droop-control studies offered limited guidance on stable operating points and reactive power sharing across general load models and network structures.
- Quadratic droop control preserves relevant steady-state behavior while enabling circuit-theoretic analysis of equilibria and stability for static and dynamic load models.
- The controller admits an optimization interpretation as a decentralized algorithm minimizing a trade-off between reactive power dissipation and voltage deviations.
- Power sharing interpolates between gain-proportional sharing at low gains and electrical-distance-based sharing at high gains, with intermediate error determined by gains and network parameters.
- The analysis explicitly considers non-collocated loads, whose lower voltages can ultimately limit network stability, and supports arbitrary grid topologies and inverter-load counts.
B. Review of Conventional Droop Control
Conventional voltage droop control is a decentralized heuristic for primary voltage control and load sharing, but its linearized basis complicates rigorous analysis. The quadratic modification preserves circuit-relevant steady-state behavior while enabling circuit-theoretic interpretations and analysis.
- Conventional droop control: Conventional voltage droop control uses local voltage feedback to regulate primary voltage and establish power sharing in islanded microgrids.It is based on a decoupling assumption for inductive lines and has an extensive history of use.
- Limitations of conventional droop control: Determining the high-voltage equilibrium for arbitrary device interconnections is the key obstacle to rigorous stability analysis of conventional droop control.The paper identifies this difficulty as the motivation for the proposed controller.
- Quadratic droop control: Quadratic droop modifies the regulating term so its gain scales with inverter voltage, matching the quadratic dependence of reactive power flow more closely than linear droop.The proposed controller is also motivated by the quadratic voltage/reactive-power characteristics of synchronous generators without saturation constraints.
- Circuit-theoretic interpretation: The quadratic controller can be represented as control by interconnection with fictitious two-bus circuits attached to inverter buses.Each circuit contains a variable-voltage inverter bus connected through susceptance K_i < 0 to a fixed-voltage set-point bus.
- Circuit-theoretic interpretation: Kron reduction of the augmented circuit eliminates inverter buses, yielding a reduced network with fixed-voltage and load buses.The closed-loop equilibrium equations are thereby interpreted as power-balance equations for an expanded linear circuit.
B. Equilibria and Stability Analysis by Network Reduction
The paper reduces equilibrium and stability analysis of the quadratic-droop microgrid to a load-voltage power-flow problem. Inverter voltages are then uniquely recovered as weighted averages, while stability can be tested through a reduced Jacobian.
- Stability analysis: The closed-loop system has a locally exponentially stable equilibrium when the Jacobian of the reduced power-flow equation satisfies the stated stability condition.Under nonpositive load-voltage derivatives, an additional sufficient condition guarantees the relevant Jacobian is Hurwitz.
- Equilibrium correspondence: Theorem 3.1 establishes an equivalence between positive equilibria of the closed-loop system and positive solutions of the reduced power-flow equation.This correspondence lets equilibrium existence and uniqueness be studied using the reduced equation alone.
- Equilibrium correspondence: Inverter voltages are recovered uniquely from load voltages and set points through a row-stochastic averaging matrix.Consequently, each inverter voltage is a weighted average of load voltages and inverter voltage set points.
- Network reduction: Kron reduction converts the augmented network into an input/output equivalent whose top block yields the reduced reactive-power balance equation.The lower block determines fictitious controller current injections after the load-voltage solution is obtained.
- Stability analysis: The stability results remain implicit because stability checks depend on solutions of the undetermined reduced power-flow equation.The paper addresses this dependence later for specific load models.
C. Equilibria and Stability Conditions for ZI, ZIP, and Dynamic Shunt Load Models
For specific load models, the reduced power-flow equation provides tractable equilibrium and stability analyses. For constant-impedance and constant-current loads, explicit conditions yield a unique positive solution.
- ZI loads: The reduced power-flow equation can be solved exactly for combinations of constant-impedance and constant-current loads.This specialization applies the general equilibrium correspondence to tractable static load models.
- ZI loads: Under the stated M-matrix and load-current inequalities, the ZI reduced power-flow equation has a unique positive solution.The assumptions are that −(B_red + [b_shunt]) is an M-matrix and I_shunt > B_red E*_L component-wise.
- ZI loads: The associated positive equilibrium is obtained from the explicit ZI voltage solution and the corresponding inverter-voltage recovery relation.The theorem identifies the resulting equilibrium as the closed-loop operating point for the ZI model.
L , EZI
For ZI loads, the reduced power-flow formulation yields explicit high-voltage equilibria and local exponential stability under matrix-based technical conditions. ZIP loads generalize the model but generally require approximate high-voltage analysis when constant-power demand is small.
- ZI loads: Local exponential stability follows because the relevant Jacobian is Hurwitz when −(Bred + [bshunt]) is an M-matrix.The proof uses the D-stability of M-matrices.
- ZI loads: Open-circuit operation reduces the ZI equilibrium to the open-circuit load-voltage vector.This occurs when Ishunt = bshunt = 0n.
- ZI loads: The ZI load model converts the reduced power-flow equation into an analytically solvable system with a unique high-voltage solution.The solution follows by factoring the reduced network matrix and enforcing positive load voltages.
- ZIP loads: ZIP loads add a constant-power demand to the ZI model and are represented by the ZIP load model.The model combines constant-impedance, constant-current, and constant-power components.
- ZIP loads: For ZIP loads, the reduced power-flow equation is generally not analytically solvable and may have multiple equilibria even in parallel microgrids.When the constant-power term is sufficiently small, the analysis instead approximates the high-voltage solution.
- ZIP loads: If ∥QL∥ is sufficiently small, a unique high-voltage ZIP solution exists under the conditions of Theorem 3.3.The ZIP solution is characterized as a perturbation of the ZI high-voltage solution.
L , EZIP
The dynamic shunt model represents constant-power consumption through dynamically adjusted susceptance. Under inductive loads and sufficiently small constant-power demand, the resulting high-voltage solution is unique and locally exponentially stable.
- Dynamic shunt loads: The dynamic shunt model adjusts susceptance dynamically so that each load achieves constant power consumption in steady state.It is used as a low-fidelity dynamic model for thermostatically controlled loads, induction motors, and loads behind tap-changing transformers.
- Dynamic shunt loads: The analysis restricts dynamic shunt loads to inductive demands with Qi < 0.The Qi = 0 case has a unique steady-state susceptance and is excluded without loss of generality.
- Dynamic shunt loads: If ∥QL∥ is sufficiently small, the dynamic shunt model has a unique solution of the reduced power-flow and load-dynamics equations.The theorem identifies this solution as the relevant high-voltage operating point.
L , EDS
The dynamic shunt operating point is locally exponentially stable under the theorem’s small-load condition. The preceding stability results can also inform gain selection and loading limits.
- Stability: The dynamic shunt equilibrium (EDS_dyn−shunt) is locally exponentially stable when Qi < 0 and ∥QL∥ is sufficiently small.This extends the stability analysis from static ZIP loads to a dynamic shunt model.
- Relation to conventional droop: Under selected controller gains, the quadratic-droop stability results imply stability for conventional linear voltage droop control.The implication applies only under particular gain selections.
- Design utility: Expressions for the unique high-voltage solutions can be used to back-calculate controller gains or bound tolerable loading profiles under voltage limits.The gain back-calculation is generally non-unique.
IV. CONTROLLER PERFORMANCE
Quadratic droop is analyzed as an inverse-optimal decentralized controller for constant-impedance loads. Its equilibrium balances reactive-power dissipation against inverter-voltage deviations, with convexity and gain choices determining the correspondence.
- Optimality formulation: The performance analysis distinguishes inverse-optimality from system-theoretic controller optimality.The optimization study concerns the resulting equilibrium point rather than overall system performance.
- Optimality formulation: The objective combines reactive-power losses, load consumption, and inverter-voltage deviation costs.The formulation targets efficient operation while keeping inverter voltages near rated values.
- Optimality formulation: An appropriately designed quadratic droop controller acts as a decentralized primal algorithm for minimizing the reactive-power and voltage-deviation trade-off.Conversely, any quadratic droop controller corresponds to an optimization problem of this form for suitable coefficients.
- Optimality result: When Ki = κi and −(Bred + [bshunt]) is an M-matrix, the unique locally exponentially stable equilibrium equals the unique optimizer.The equilibrium and optimizer are represented by the same voltage solution.
- Optimality result: The equilibrium equations coincide with the optimization problem’s critical-point equations after eliminating inverter voltages using the quadratic-droop relation.Positive definiteness of the Hessian follows through Schur complements under the stated matrix condition.
- Interpretation and extensions: The controller balances maintaining a uniform voltage profile against minimizing total reactive-power dissipation, although the two goals may conflict depending on voltage-setpoint heterogeneity.General strictly convex voltage costs can generate analogous nonlinear droop controllers.
B. Power Sharing: General Case and Asymptotic Limits
The paper characterizes inverter reactive-power sharing under quadratic droop control, deriving a general formula and two topology-dependent asymptotic limits. High gains favor electrically nearby loads, while low gains yield gain-proportional sharing but can produce large voltage deviations.
- General Case: Power sharing describes how inverter reactive-power injections change when constant-power load demands receive an incremental change.The analysis assumes uniform inverter voltage set points; these assumptions can be relaxed at the cost of more cumbersome formulas.
- General Case: The general result expresses inverter reactive-power injections as a function of network susceptances, constant-power load demands, and inverter controller gains.The derivation uses steady-state inverter injections, load-voltage approximations, and linearization around the open-circuit operating condition.
- Asymptotic Limits: In the high-gain limit Ki → −∞, inverters supply loads according to electrical distance, with nearby sources preferentially supplying power.The limiting behavior corresponds to stiff inverter voltage sources, so power is routed through electrical transfer paths rather than according to relative controller gains.
- Asymptotic Limits: In the low-gain limit Ki → 0−, inverters supply changes in total load reactive power in proportion to their controller gains, independently of network topology.The gains can be selected proportional to unit generation capacity, and the result does not depend on electrical distance or reactances.
- Practical Trade-offs: The proportional-sharing limit is derived under a linear approximation and may produce large voltage deviations that invalidate the approximation or threaten equilibrium and decoupling.Practical implementation is limited by load size, grid stiffness, and the stability bottleneck associated with low load voltages.
- Practical Trade-offs: Large controller gains improve voltage regulation and produce distance-based sharing, whereas small gains reduce reactive-power dissipation and produce proportional sharing with instability risk.This identifies a trade-off between voltage regulation, reactive-power dissipation, sharing structure, and stability.
V. SIMULATIONS
Simulations test quadratic droop control in a five-load, three-inverter islanded microgrid under varying loads and feedback gains. High gains preserve voltage stability but yield poor sharing, whereas low gains enforce sharing while increasing collapse risk.
- Simulation setup: The simulation uses five loads and three inverters with dynamic reactive and active load models, including unmodeled resistive losses and frequency dynamics.The experiments assess robustness beyond the simplified theoretical model.
- Large feedback gains: With stiff tuning, K1 = K2 = 2K3, the controller maintains stability and tracks load demands, but proportional reactive-power sharing is poor.The gains correspond to equally rated inverters 1 and 2 and an inverter 3 rated for half as much power.
- Small feedback gains: Reducing Ki to 5% of its prior value enforces early reactive-power sharing: inverters 1 and 2 share equally and each supplies twice inverter 3.The low-gain traces use Figures 7 and 8 to test the predicted sharing–stability trade-off.
- Small feedback gains: At t = 4s, doubling the load causes voltage collapse under small gains, and the system cannot recover.This supports the predicted increased instability associated with low feedback gains.
VI. CONCLUSIONS
The conclusions present quadratic droop as a tractable controller for analyzing voltage stability, equilibria, optimization, and power sharing in islanded microgrids. They identify low- and high-gain sharing regimes while delimiting assumptions and open practical problems.
- Contributions: Quadratic droop enables circuit-theoretic analysis of closed-loop microgrids, including equilibria and stability under several load models.The controller is presented as a tractable modification of conventional droop control.
- Power sharing: Power sharing interpolates between proportional sharing at low gains and sharing based on electrical distance at high gains.The analysis also provides easily verifiable certificates for system stability.
- Limitations and future work: The results apply to microgrids with uniform R/X ratios under a decoupling assumption, while active/reactive sharing becomes subtler after the associated coordinate transformation mixes power types.The paper identifies non-uniform R/X ratios, inner-loop interactions, and mixed inverter/generator networks as further challenges.
- Limitations and future work: Secondary control for islanded microgrids requires new formulations and greater theoretical attention, including uncertainty about standard voltage-regulation and reactive-sharing objectives.The conclusion specifically questions their importance in sub-distribution-sized microgrids.
APPENDIX A SUPPORTING LEMMAS AND PROOFS
The appendix supplies matrix, reduced-network, electrical-distance, coordinate-transformation, and stability results supporting the paper’s power-flow and dynamic analyses. It establishes local solution behavior and stability for selected load models.
- Matrix properties: The susceptance matrix is structurally constrained: off-diagonal entries are nonnegative, it is negative semidefinite with a simple zero eigenvalue, and principal submatrices are negative definite.These properties underpin the reduced-network analysis.
- Reduced quantities: Reduced quantities satisfy that −Bred is an M-matrix, W1 and W2 are row-stochastic, and E∗L is component-wise positive.The proof uses Schur-complement properties and stochastic-row identities.
- Local power-flow analysis: An invertible coordinate transformation converts the power-flow relation into a form whose Jacobian at the origin is Qsc, enabling the Implicit Function Theorem.For sufficiently small load demand, the resulting expression solves the reduced power-flow equation.
- Local power-flow analysis: Norm bounds on the transformed variable follow from invertibility of Qsc and quadratic inequalities when the load demand is sufficiently small.The appendix then obtains the stated bound through elementary algebraic estimates.
- Stability proofs: Exponential stability follows by continuity from the zero-impedance-load case, while dynamic-shunt equilibria are locally exponentially stable under an extended-dynamics argument.The dynamic-shunt steady state is equivalent to a constant-power model.
- Electrical distances: Differential effective reactance is derived from current-balance equations by relating load voltages to open-circuit voltages and direct line reactances.The result supports the paper’s electrical-distance characterization of power sharing.