Viability and analysis of implementing only voltage-power droop for parallel inverter systems

In microgrids that are predominantly resistive, real and reactive power can be controlled by implementation of voltage and frequency droop laws respectively. However, the variable frequency displayed by such a system complicates analysis such that design approaches rely on approximations and linearized models. In this work, we present a modified form of droop control where only the voltage versus real power relationship is upheld and the frequency is held constant. Since the frequency is not explicitly controlled and the reactive power is not measured, the controller can be simplified. In such a setting, the only assumption we make is that all inverters have access to a common time-reference. Because fixed frequency operation is enforced by design, a variety of analytical tools can be leveraged to formulate a comprehensive analytical framework which facilitates a precise design methodology. In particular, closed-form expressions on the output current phase differences are obtained which yield practical selection guidelines on the voltage-power droop gains such that reactive flows between inverters are kept small. As a corollary, it is demonstrated that there are no reactive power flows in the presence of purely resistive loads. For the particular case of a single inverter, an almost exact solution describing the nonlinear dynamics of the inverter output voltage, current, and power are derived. Accompanying simulation results validate the analytical results and demonstrate the feasibility of the proposed control approach.


I. INTRODUCTION
Microgrids are small-scale power systems which utilize local power generation and are generally expected to operate in both grid-connected and islanded modes such that uninterruptible power is delivered to local loads [1], [2], [3]. Because generation is placed in close proximity to loads, a bottom-up approach to system design can be emphasized. Of particular importance is the utilization of modular renewable energy sources, such as wind and photovoltaics, and their accompanying power electronic inverter interfaces. Droop control is a technique which has been applied widely in inverter-based system such that system voltage and frequency stability is maintained while ensuring synchronization [4].
In essence, droop control is a technique of controlling inverters such that they approximate the dynamics of coupled synchronous machines in power systems. However, the variable frequency exhibited by droop controlled systems complicates analysis which often is overcome through linearized models and assumptions on system impedances [5], [6]. In [7], [8], power-flow based phenomenological dynamic models that incorporate droop laws for frequency synchronization and voltage control are analyzed. Sufficient conditions for voltage stability in terms of inequality constraints on various network parameters, the loads, and generation set points are derived in [7]. A quadratic droop law is introduced in [8] which enables analysis of the equilibrium points under the corresponding model such that closed-form load conditions for voltage stability are attained. An improved droop law that achieves more accurate proportional load sharing and is robust to component mismatches, parameter drifts, and disturbances is proposed in [9].
In this work, we focus our investigation on inverterbased microgrids with predominantly resistive networks and propose a simplified droop controller which acts only on the real power output of each inverter. By assuming the existence of a common time-reference among inverters (that is, isochronous operation), the system is designed to operate at a single frequency despite system disturbances.
For networks with small X/R ratios, it is conventional to modulate the inverter voltage and frequency inversely with respect to the real and reactive power output, respectively [10]. However, in contrast to established techniques, we adopt a simplified and isochronous incarnation of droop control where each voltage set-point is altered with respect the real power output and the system frequency is fixed. In such a setting, it is not necessary to measure and compute the reactive power and isochronous operation is facilitated by a common time reference between inverters. We also supplement the controller with a virtual resistance such that network inductances are subsumed. Since isochronous operation is enforced by design, the development of a comprehensive analytical and design framework which describes both steady-state and dynamic parameters is facilitated.
In our analysis, we focus on control architectures that contain a slow droop-based outer loop which derives a sinusoidal voltage reference and a fast inner loop which consists of both a current and voltage controller. This is in contrast to previous analytical efforts which exclude fast inner-loop controllers from analysis and are focused exclusively on the slow droop dynamics. Furthermore, we assume that the design of the inner controller is based on the internal model principle such that the steady-state error at the nominal frequency is zero. Under such conditions, closed-form relationships of the output current phase differences are derived. Despite the fact that the inverter voltage angles and reactive power are not explicitly controlled, we show that the reactive power flows between inverters can be kept small by proper selection of the droop gains and we prove that reactive power flows are zero in the case of a purely resistive load. Bounds on the system voltage in terms of the active power commands and uncertainty in loads are also obtained and the impact of the virtual resistance on system performance is analyzed. Lastly, an almost exact solution describing the nonlinear dynamics of the output voltage, current, and power of a single inverter is derived. Simulation results confirm the validity of the analytical framework and demonstrate the feasibility of the proposed control approach.
II. SYSTEM DESCRIPTION Consider the network of N inverters connected in parallel across a common load, as shown in Fig. 2 To facilitate analysis, high-frequency switching dynamics are disregarded and we model the averaged dynamics [11] of the k th inverter as a controllable voltage source as shown in Fig. 1. By using Kirchoff's laws for the circuit in Fig. 1 wherex(s) represents the Laplace transform of x(t), v invk is the control signal of the k th inverter, and the output voltage v k , inductor current i Lk , and output inverter current i k are measured signals. Therefore for the control law of the form v invk = K refkvrefk − K vkv + K ikîk , the closed-loop system for the k th inverter is given bŷ where v refk is the sinusoidal reference voltage generated by the droop controller (see Fig. 2). Therefore the dynamics of N inverters are described byv Further, if we assume that the load is linear given byî = Y (s)v, where Y (s) is the admittance matrix of the network (see Fig. 2), then the closed-loop expressions forv andî arê Also from the network in Fig. 2 with N branch resistances r k for k = 1, . . . , N, the voltage across the load Z L is given byv where h(s) −1 := r −1 where q(s) := h(s)/Z L (s). The k th inverter current, i k , is given byî A system of N parallel inverters that provide power to a load with impedance Z L (s). The details of the outer-loop droop controllers and network are shown.
The voltage-power droop law, which determines the sinusoidal reference voltage for each inverter, is given by where ω o is the rated system frequency and n k is the k th droop gain. Note that all inverters have access to a common time-reference, t. Rewriting (6) where N = diag(n 1 , . . . , n N ). The active power utilized by each droop controller is derived from the instantaneous power p k (t) := v k (t)i k (t) via a first order filter with time constant τ P . Thus,P B. Inner-control-loop model The inner control loop is composed of cascaded voltage and current controllers. As shown in Fig. 3, the voltage controller acts on the sinusoidal reference generated by the outer droop controller. In turn, the voltage controller outputs a sinusoidal current reference, i ref , which is processed by an innermost current controller. As outlined in [11], the main advantage of this structure is increased robustness of the inverter voltage to load variations. Note that the virtual resistance, R v , is incorporated by subtracting an iR v voltage drop from the voltage reference.
The inner-control loop for voltage regulation with a virtual resistance. The transfer functions of the voltage and current compensators are given by K vol and Kcur, respectively.
Referring to Fig. 3, it follows that where

III. DYNAMICS OF A SINGLE INVERTER DRIVING A LINEAR LOAD
In this section, an approximate solution with quantifiable error bounds for an inverter driving a linear load is obtained. Consider a single inverter driving an impedance load where the measured power, P , is given by (7). For the remaining analysis in this section, indices are dropped since we are studying a single unit. We assume that the time constants of the voltage and current dynamics in (8) are much faster than the dynamics of the active power equation (7). Thus, with respect to the voltage and the current dynamics, the power P is approximated as being constant. In other words, both v and i reach their steady-state values before any appreciable change occurs in P.
For a given P , the product v(t)i(t) at steady-state is given by 1 . Therefore from (7) we obtain where α = 1 ). Note that in the above time-varying nonlinear differential equation, the time constant τ p is designed to be large such that the low-pass filter has a small bandwidth. As a result, the filter rejects the sinusoidal term with frequency 2ω o in (9). Therefore, by design, := 1/(ω o τ p ) 1 and is typically in the range of 10 −3 -10 −4 .
Theorem 1: There exist constants L > 0 and c > 0 such that the solution P (t) to (9) with initial condition P (0) satisfies is the approximate solution of (9),

A. Simulation Results
In this section, the control implementation summarized in Figs. 2-3 is evaluated with a single inverter driving a constant impedance load. The load is designed to consume 3 kW of real power at a power factor of 0.8. In addition, n = 0.2 V/W, r = 0.1 Ω, P * = 1.5 kW, E * = 120 √ 2 V. and τ P = 0.1 s. The time-domain waveforms in Fig. 4(a) compare the evolution of the approximate power response in (11) and the numerical solution to (9). Along similar lines, the approximate and numerically-simulated inverter voltages using are compared in Fig. 4(b). The results confirm that the approximate solution given in Theorem 1 provides a nearly exact solution to active power and voltage signals.

IV. MULTI-INVERTER SYSTEM ANALYSIS
To begin, we assume that the branch resistance between each inverter LC filter and the load is purely resistive. As illustrated in Figs. 2-3, the inner control loop has the objective of tracking the sinusoidal voltage reference given by v r, Theorem 2: Consider the inner control loop shown in Fig. 3. Suppose K volk has a factor 1 s 2 +ω 2 o and the closedloop system is stable. Also, suppose that v refk (t) and i k (t) are sinusoids with frequency ω o . It follows that in steady- where v r,k (t) is the voltage reference processed by the k th inner-loop controller and v r, Thus, if all external inputs to the system in Fig. 3 (that is v refk and i) are sinusoidal with frequency ω o , then the inverter voltage follows the sinusoidal reference with zero steady-state error. Furthermore T (jω o ) = 1 and Q(jω o ) = R v . Proof: The proof is omitted for space reasons.
The following theorem establishes that despite the absence of explicit control over reactive power, voltage phase angle, and frequency, there will be no extraneous reactive power flows if the load is purely resistive. (c) Phase differences in inverter currents for a system with two inverters. We consider the case where P * 2 = 3P * 1 , P * 1 + P * 2 = 6 kW, [L, R, C] = [600 µH, 0 Ω, 100 µF], rated load power is 6 kW, n 1 = n 2 = 0.05 V/W, and examine behavior over a range of load power factors. The phase differences computed using (13) and (14) match results from the numerical simulations.
Theorem 3: Suppose the conditions of Theorem 2 hold and the load is purely resistive such that Z L (s) = R L . In steady-state, the voltage and current output of the k th inverter is given by v k (t) = V k sin ω o t and i k (t) = I k sin ω o t for V k , I k ∈ R + . Proof: The proof is omitted for space reasons.
Next, we characterize the phase difference between inverter currents, i k and i j , in the general setting with complex impedance loads.
Theorem 4: , and suppose the conditions in Theorem 2 hold. Then tan(φ k − φ j ), where φ m denotes the phase of i m , is given by Proof: See Appendix. Remark 1: Note that if the load Z L is real, then the phase differences between inverter currents are zero. This also follows from Theorem 3.
The following corollary provides a more intuitive expression for the phase difference between currents which can provide practical guidelines for design.
Corollary 1: Assume the notation and the assumptions of Theorem 4. Suppose δ k = O( ) and |Z L λ T 1| 1 then Remark 2: If E k is close to E * , which in turn yields small δ k := (E k − E * )/(E * ) = m 1/(r m + R v ), then it is expected that δ k − δ j is small. Typically, the branch resistance, r k , and virtual resistance, R v , are both small, when compared to |Z L |. Consequently, |Z L λ T v 1| 1 should also hold. Thus, the assumptions in Corollary 1 are not restrictive (see Fig. 4

(c) for verification of analytical expressions through simulations).
Remark 3: The relationship in (14) establishes that the inverter current phase differences are small for those designs that achieve values of E k close to E * . In particular, for the current phase differences to be smaller than 1, it follows , it is evident that the deviation of E k from E * is dictated by the outer droop law. Thus, the analysis above translates the concerns of limiting reactive flows exclusively to the design of the outer droop law.
A. Bounds on voltage deviations 1) No branch resistance and no virtual resistance: In geographically small networks, the interconnect resistances, r k , are typically small. Theorem below summarizes the results for case with zero branch and virtual resistances.
Theorem 5: Consider N inverters servicing a complex load Z L (see Fig. 2). Assume there is no branch resistance, no virtual impedance, and suppose the real power consumed by the load is P * L = (E * ) 2 /(2R L ) when v k = E * sin ω o t for all k. We express the summation of commanded power as where ∆ represents the mismatch in actual and commanded power delivery. Furthermore, assume all power generated is consumed by the load. Under such conditions, the steadystate voltage v k = v(t) for all k = 1, . . . , n is given by Proof: See the Appendix.
Remark 5: Consider the case where all inverters have the same droop coefficient (i.e. n i = n) and there are N inverters such that m = N/n. In this circumstance, Thus, the deviation of V g from E * becomes vanishingly small as N increases. This implies that a system with more inverters which delivers power to the same total power provides tighter voltage regulation in comparison to a system with fewer inverters.
In the case where the total commanded power and actual load power are matched (which implies ∆ = 0), we recover V g = E * . Corollary 2: Under the conditions in Theorem 5, if n i and n j are chosen to satisfy n i P * i = n j P * j , then n i P i = n j P j in steady state. Proof: From the droop-laws we have E i = E * −n i (P i −P * i ) and zero branch resistances implies E i = V g . It then follows that n i (P i − P * i ) = n j (P j − P * j ). Thus n i P i = n j P j . Remark 6: Note that Theorem 2 suggests a mechanism for power sharing despite mismatched conditions, that is even when i P i = i P * i .

B. Simulation results
The simulation parameters are R = 0 Ω, L = 600 µH, C = 100 µF for inverter #1 and 1.2R, 1.2L, and 1.2C for inverter #2. As before, the inner-loop voltage and current compensator transfer functions, K vol and K cur , are both assumed to contain poles at ±jω o where ω o = 2πf 0 and f 0 = 60 Hz. The two inverters are connected in parallel across a common load. The inner-loop controllers were designed by following a loop-shaping procedure. The sensitivity transfer functions corresponding to K cur and K vol , as shown in Fig. 5(a), exhibit bandwidths of approximately 1 kHz and 600 Hz, respectively, and the same controller is used for both inverters. The outer droop control is given by The value of n k dictates the deviation of E k from E * which in turn also influences reactive power flows according to Corollary 1. Here two scenarios are studied.
1) Symmetric Outer Droop Laws: Figure 5(b) shows that the difference in the inverter currents is negligibly small when the outer-droop laws are the same for the two inverters even when the load has non-unity power factor and the power commanded does not match the active power rating of the load. Since the droop laws are identical for this particular case, both the inverter output voltages deviate by the same amount and thus δ 1 − δ 2 = E1−E2 E * is small. This results in small phase differences between inverter currents (see Corollary 1). Thus, even under a mismatch between active load power rating and total commanded power, P * k , the phase difference between inverter currents is small. This result also holds for complex loads.
2) Asymmetric Outer Droop Laws: If the load is purely resistive, then the phase difference between inverter currents is zero (see Theorem 3) regardless of mismatches between rated active power of the loads and the total commanded power or mismatched droop slopes among inverters. Simulation results under mismatched droop slopes with a resistive load are presented in Fig. 5(c).
Remark 7: An interesting observation supported both by the analysis and simulations is that if the droop laws are similar then the phase difference in the inverter currents (and therefore the reactive flows) are small. This holds even when the load is complex. In contrast, reactive flow mitigation requires due care under mismatched droop slopes for complex loads. One key insight obtained is that the phase difference between currents can be controlled by ensuring that deviation of all E k from E * are similar. Under the conditions of Theorem 5 and from Corollary 2, we can conclude that E i − E * = n i (P * i − P i ) = n j (P * j − P j ) = E j − E * = 0 if we impose n i P * i = n j P * j on design of the droop laws. Evidently in typical scenarios, the conditions of the theorem are not met since branch and virtual impedances are not zero. However branch resistances r k and virtual resistance R v assume small values and therefore a choice of n k according to n i P * i = n j P * j can still prove an effective design rule. Fig. 6(a) shows that the inverter current phase differences are small when this guideline is used instead of choosing identical droop coefficients. Also smaller values of i n i P * i result in smaller phase differences. Higher virtual resistances and branch resistances also result in lower phase differences but at the expense of power sharing accuracy and power loss (see Fig. 6 (b) and (c)). Thus the analysis and simulations indicate the following measures to limit reactive flows: i) choose n k according to n i P * i = n j P * j (e.g. the reactive phase difference in inverter currents was reduced from -40 deg to -13 deg for a power factor (lagging) of 0.5) ; ii) pick lower i n i P * i ; and iii) increase the values of virtual resistance and or the branch resistance to further limit reactive flows. With these design guidelines, reactive power flows can be mitigated.

APPENDIX
Proof of Theorem 1: Note that (9) can be rewritten in terms of slow time variable τ := ω o t as dPs dτ = where P s (τ ) = P (t). The corresponding averaged equation is given by dPs dτ = ḡ(P s ), that is dP which can be solved to obtain (11). Let D = (P (0) − δ, b + δ) for some δ > 0. Since g(P s , τ ) is in C 1 , therefore for L in (0, P (0)+b+δ 2 sup Ps,τ g(Ps,τ ) ), the solution P s (τ ) ∈ D for 0 ≤ τ ≤ L/ , and we obtain that there exists c > 0 such that (using results from averaging theory such as in [12]) P s (τ ) −P s (τ ) < c for 0 ≤ τ ≤ L . The theorem statement results since P (t) = P s (τ ). Proof of Theorem 4: We will assume steady state for this proof. Now in the isochronous half-droop method assuming that sinusoids with frequency ω o are tracked perfectly we can assume that T Difference in inverter currents is negligibly small when the droop law for both inverters is identical even when the total active power commanded is twice the total rated active power of a complex load (load is rated at 3 kW with a power factor of 0.75 and P * 1 + P * 2 = 6 kW). (c) Two Inverter currents are in phase when the load is purely resistive (rated at 3 kW). The commanded active power P * k is 4.5 kW with P * 1 = 3.365 kW and P * 2 = 1.125 kW. The droop coefficients are n 1 = n 2 = n = 0.02 V/W. n i P * i . Phase differences between inverter currents decrease as the virtual resistances and branch resistances are increased as shown in (b) and (c), respectively. All plots assume asymmetric power sharing P * 1 = 3P * 2 where P * 1 + P * 2 = 3 kW. In all scenarios in (b) and (c), n i P * i = 2.5V for i = 1, 2..
Em rm+Rv ) and tan φ k == sin ∠α cos ∠α − β k |α(jωo)| .The theorem results on expanding tan(φ k − φ j ) using the above expression for tan φ k , where 1 ≤ k, j ≤ N . Proof of Theorem 5: Note that N k=1 P * k = (E * ) 2 2R L + ∆ where ∆ represents the mismatch in power. Assuming all the power generated is consumed by the load and that drooplaws are being followed we have: Define the variable x := E * − V L . Then Substituting V 2 L 2R L from (18) into above equation we have The theorem statement is a direct consequence of solving the above quadratic equation and obtaining the viable solution for V L .