Virtual Synchronous Machines – Classification of Implementations and Analysis of Equivalence to Droop

Controllers for Microgrids

Salvatore D'Arco*

*SINTEF Energy Research 7465 Trondheim, Norway

Salvatore.D'Arco@sintef.no

Jon Are Suul*†

†Department of Electric Power Engineering Norwegian University of Science and Technology

7495 Trondheim, Norway

Abstract—The concept of Virtual Synchronous Machines (VSM) is emerging as an alternative approach for control of power electronic converters operating in the power system. One main motivation for applying VSM-based control is to achieve a simple approach for emulating the inertia effect of traditional synchronous machines. This paper provides a comprehensive literature review on VSM and a possible classification of the different schemes. In addition, the small-signal response of the inertia emulation characteristics of VSM-based control is proved to be equivalent to conventional droop-based control for standalone and microgrid operation of converters. Thus, the droop gain and the filter time constant of the power feedback in a droop controller can be directly related to the damping factor and the inertia constant of a Virtual Synchronous Machine. The derived results are providing additional physics-based insight into the operation and tuning of both types of controllers.

Index Terms—Distributed Generation, Droop Control, Inertia Emulation, Power Electronic Converters, Virtual Synchronous Machine

I. INTRODUCTION

The traditional power system structure dominated by centralized generation and unidirectional power flow is currently being challenged by the proliferation of Distributed Generation (DG). This trend has initiated significant research and development efforts on control strategies for grid connected power converters in DG systems [1], [2].

Most of the research on control of converters for feeding power into existing power systems has been concerned with maximizing the energy production from DG sources while providing robustness and operability during grid faults and disturbances [2]. It is then assumed that the operation of DG

This work was partly funded by the project "Power Electronics for

Reliable and Energy Efficient Renewable Energy Systems," financed by the Norwegian Research Council, Wärtsilä, Statkraft and GE, and sponsored by Westcode Semiconductors, http://sintef.no/OPE

power converters mainly affects the local conditions in the nearby grid with a negligible impact on the overall power system.

A parallel line of research has focused on the development of droop-based control systems for converters in stand-alone operation and in microgrids dominated by power electronic converters [3]-[9]. In such configurations the converters must be able to establish the system voltage without the presence of dominant synchronous machines and must be explicitly controlled to participate in the frequency control and the power balancing of the system.

In the near future it can be expected that power electronic converters in many cases will have a significant impact on large scale power systems as a non-negligible share of the traditional synchronous generators in operation will be replaced by converter-interfaced generation. A potential problem under such conditions can be the decrease of total system inertia compared to the overall power level and, thus, a reduction of corresponding stabilizing effects [10]. Therefore, it has been proposed to control power electronic converters to emulate the behavior of traditional Synchronous machines (SMs) and by that adding virtual inertia to the power system.

The first proposal of a "Virtual Synchronous Machine" (VSM), was published in English by Beck and Hesse in 2007, and labeled as "VISMA" [11]. Later, several other approaches for adding virtual inertia to power electronic converters by emulating the synchronous machine characteristics have been proposed but with significant differences in the terminology, the targeted applications and the proposed implementations. Thus, the panorama on VSMs from existing literature is rather dispersed and fragmented. This paper will review most of the proposed VSM implementations, and attempt to classify them according to their functional characteristics and their control system structures. The paper will also consider the droop control methods widely applied for microgrids and stand-alone systems, as a conventional approach for achieving parallel operation and load sharing among power electronic

converters. The similarities in the control structures applied for droop controllers and VSM-based control will then be highlighted, and it will be demonstrated how the droop-based approach under certain conditions is equivalent to a VSM. Thus, the parameters of a droop controller can be reinterpreted in the perspective of the inertia and the damping constant of a VSM. This explicit equivalence can bridge the VSM and the droop-based control approaches and can provide useful insights for the design, the tuning and the stability analysis of both these concepts.

II. INERTIA EMULATION EFFECT OF VIRTUAL

SYNCHRONOUS MACHINES

The underlying idea behind the VSM concept is to emulate the essential behavior of a real SM by controlling a power electronic converter. Thus, any VSM implementation contains more or less explicitly a mathematical model of a SM. The specific model of the SM and its parameters is largely an arbitrary design choice as proved by the many different solutions discussed in literature. However, the emulation of the inertial characteristic and damping of the electromechanical oscillations are common features for every VSM implementation. Additional aspects as the transient and sub-transient dynamics can be included or neglected, depending on the desired degree of complexity and accuracy in reproducing the SM dynamics. Furthermore, the parameters selected for VSM implementations are not constrained by the physical design of any real SM. Thus, the VSM parameters can be selected to replicate the behavior of a particular SM design or can be specified during the control system design to achieve a desired behavior.

If the purpose of VSM is to accurately replicate the dynamic behavior of a SM, a full order model of the SM has to be included in the converter control system. This includes a 5th order electrical model with dq-representation of stator windings, damper windings and the field winding, together with a 2nd order mechanical model resulting in a 7th order model [12], [13].

A. Swing Equation for VSM Inertia Emulation

While a full order model faithfully represents the behavior of a real SM, it adds unnecessary complexity if the goal of the VSM is to emulate the inertia and damping properties of the SM. Indeed, these two main aspects can be already captured by the swing equation (1) well known from the literature on power system stability and dynamics [12], [13].

0 el g

dJ T T D

dt

(1)

In (1), J is the rotor inertia, ω the rotating speed of the machine, ωg the angular frequency of the grid, T0 the mechanical torque, Tel the electromagnetic torque, and D a coefficient accounting for the damping torque associated with the damper windings during transient conditions. It should be remarked that the coefficient D in a real SM is not constant but depends on the operating point of the machine. Thus, a reduced order model with a fixed value of D will not be able to match the SM behavior in the entire operating range.

The swing equation can be even more conveniently expressed in terms of power instead of torque by multiplying

all terms of (1) by the frequency ω. For small oscillations around the synchronous conditions, the power balance can be expressed in the Laplace domain by the approximation given by (2) where P0 is the emulated mechanical input power, Pel is the electrical power, and Kd is the damping constant associated with D.

0g el d gJ s P P K (2)

By introducing the mechanical time constant Ta (=2H) the mechanical swing equation can then be expressed in p.u. by:

0 ,a pu el d pu g puT s p p k (3)

The virtual rotor angular position of the VSM is given by the integral of the frequency ω, and this angular position corresponds to the phase angle of the voltage induced by the VSM model. The amplitude of the VSM voltage can then be given by either a reduced order electrical model of the SM or directly by a separate reactive power control loop. Since the active power flow of the VSM is associated with the speed and angular position of the virtual inertia with respect to the grid voltage, in the following the voltage or reactive power control of the VSM will be considered as decoupled from the inertia emulation. A block diagram implementation of the swing equation intended for VSM is shown in Fig. 1.

B. Inertia Effect by Estimating Power Response

From (3) it can also be derived an alternative approach for inertia emulation. Assuming that the electrical power pel can be directly controlled by the converter, and replacing the actual grid frequency ωg,pu by its reference value while the inertia frequency ωpu is replaced by the measured grid frequency, this equation can be reformulated as given by (4).

* *0 , , ,el g pu g pu i g pup p k k s (4)

In (4) the damping constant kd from (3) is replaced by a steady state droop constant kω, while the mechanical time constant Ta is multiplied by −1 and represented by an inertia constant ki. The grid frequency and the derivative of the grid frequency (s·ωg,pu) must then be estimated from the grid voltage, for instance by using a Phase Locked Loop (PLL) [10]. However, this approach is only emulating the inertia effect with respect to the response to changes in the grid frequency, together with a steady state power droop but does not establish an internal model of the machine inertia. Thus, the implementation of virtual inertia by (4) requires the presence of an external voltage with a physical inertia, and is not suitable for islanded operation in contrast with a real SM or a VSM implementation based on (3).

0p

1

aT1

spu pu

,g N

s

,g pudk

p

Electrical system and power converter control

Inertia emulation Fig. 1. Inertia emulation for Virtual Synchronous Machines

III. REVIEW OF VIRTUAL SYNCHRONOUS MACHINE

CONTROL SCHEMES

In the previous section, several alternatives for SM models to be used in VSMs have been discussed. These models need to be interfaced with the power electronic converter through additional controllers which should receive reference signals from the VSM implementation and translate them into gate signals for the converter. The control schemes proposed in literature can be categorized into three main groups based on the nature of the output reference from the SM model.

A. Current References from the SM model

In such schemes, the full order or reduced order model of a SM generates a current reference i*. This allows a quite natural implementation of high order electrical models for the SM since the measured voltage at the converter interface to the grid can feed a simulation model calculating the currents that would result from a real SM. This approach was applied by the VISMA concept which was the first proposal of a VSM implementation [11]. A simplified block diagram illustrating this approach is shown in Fig. 2.

The current controller in Fig. 2 can be realized by hysteresis controllers on phase currents as initially proposed in [11], by PI-controllers as discussed in [15], or by any other conventional current controller in the stationary or synchronous reference frames. In principle, the current regulators can be easily tuned while saturations and limitations can be embedded directly on the iref. However, in practical implementations these schemes can be prone to numerical instability, especially for high order SM models, and this requires specific attention on the practical discrete time implementation of the control system, as discussed in [15]. The origin of these stability problems have not yet been thoroughly analyzed and it is therefore not clear if this is purely related to numerical issues solvable by a proper choice of integration method in the implementation or if such schemes results in intrinsically poor numerical stability conditions.

B. Voltage References from the SM model

Another possible approach for VSM implementation is to configure a SM model to provide a voltage reference output. This is for instance discussed in [16] where implementation of the VISMA concept with current and voltage outputs are compared. However, if a reduced order model of the SM is applied, the power flow is mainly related to the inertia emulation and the phase angle resulting from the swing equation, while the voltage amplitude and reactive power control can be handled separately. This is indicated in Fig. 3 a) and b), where the voltage amplitude is resulting from a reactive power control loop (Q-reg) while the VSM frequency and phase angle is resulting from inertia emulation by the swing equation from Fig. 1. The scheme shown in Fig. 3 a) is arguably the simplest and most direct implementation of the VSM concept, since the voltage amplitude and phase angle are directly used for generation of PWM gate signals for the converter. For instance, this approach is followed by the "Syncronverter" concept discussed in [17], and in several other recent publications [18]-[23]. However, limitations or

controlled saturation of the voltages and currents of the converter cannot be easily and explicitly included in this structure. Protections can be implemented at the hardware level or as parallel loops overriding the references from the VSM, but their interaction with the inertia emulation and reactive power control and the resulting behavior can be difficult to predict.

The controllability can be enhanced by applying a classical cascaded control scheme where the voltage output from the VSM is used as the reference for an external voltage loop cascaded with an internal current loop as shown in Fig. 3 b). This scheme ensures more flexibility for embedding protection strategies since both voltages and currents can be limited by saturating the regulators outputs [24], [25]. Although these schemes are currently not very common for VSM applications, they are becoming widespread in controls for microgrid applications where the voltage and frequency references are provided by droop controllers [6]-[9]. Due to the cascaded loops, the appropriate tuning of the controllers in such structures is critical to ensure stability in all operating conditions, especially in case of low switching frequencies limiting the bandwidth of the inner current control loop [25].

C. Power reference from the SM model

The inertia response can be also emulated by tracking the grid frequency without implementing any full order or reduced order SM model as in (4). The resulting structure is shown in Fig. 4 where the current reference corresponding to a given power reference can be calculated from the measured grid voltage. The inertia emulation can be achieved in a relatively simple way and can be combined with any conventional strategy for current controlled operation of converters. This approach can be easily applied for making converters contribute to the total apparent inertia of large scale power systems, and has therefore been pursued by the European Union VSYNC project [26]-[28]. However, the control will rely on grid synchronization by a PLL and the presence of en external grid with a rotating inertia. Operation in very weak grids can therefore be dubious, and the control system will not have any inherent capability for black-start or islanded operation.

DCC

DCv

1L

,c abcI

PWMg

Modulation

ci

1C

,g totZ

sv

Current

&Control

*ci

gV

SynchronousMachine

SimulationVSMi

svVSM

fvQ-reg0q

0p

PQ-calcq

p

Options for Virtual Synchronous Machine (VSM) Simulation:

- Mechanical model combined with

- Full 5th order electrical model

- Recuced order electrical model

VISMA Concept

Fig. 2. VSM implementation providing current references for based on

the VISMA concept proposed in [11]

IV. DROOP CONTROL FOR MICROGRIDS

A major advantage of the VSM concept is the possibility for load sharing among parallel connected units, as well as the inherent capability for stand-alone and microgrid operation. As mentioned in the introduction, design of control systems for converter dominated microgrids has been a line of research which has received significant attention during the last years. Although a large range of control structures for converter dominated microgrid operation has been proposed, the simplest and most common approach for control system design is based on droop regulators for the active and the reactive power according to (5) and (6) [6], [7].

* *, 0

1,pu g pu p m pum p p

s (5)

*0g q mv v m q q (6)

In such a scheme, the converter frequency is determined by a droop gain mp and the deviation between the measured power pm and the power set-point p0. Similarly, the converter voltage is determined by the droop gain mq and the deviation between the measured reactive power qm and the reactive power set-point q0. However, the active and reactive powers pel and qel at the output of the power electronic converter must be low pass filtered, as given by the first order transfer functions of (7), before they are used as the measurement feedback signals pm and qm in (5) and (6) [6], [7]. This is necessary for stabilizing the control loops and rejecting disturbances and oscillations in the measurements.

,f fm el m el

f f

p p q qs s

(7)

Similarly as for the VSM model, the instantaneous phase angle resulting from the droop controllers are given by the integral of the frequency resulting from (5). The block diagrams representing the implementation of the active and reactive power drops are shown in Fig. 5. Since the output of these blocks will be the voltage amplitude and phase angle, it can easily be seen that they can be combined with either the open loop PWM or the cascaded voltage and current loops shown in Fig. 3 a) and b) respectively.

As the frequency and phase angle is directly associated with the power flow while the voltage is associated with the reactive power flow, the control scheme in Fig. 3 is based on the assumption of a predominantly inductive system impedance, in order to decouple the action of the two droop regulators. If this is not the case, virtual inductance in the converter control system or the application of reference signal transformations based on the impedance angle of the grid can be applied to ensure the stable and satisfactory operation of the droop controllers [7], [9].

DCC

DCv

1L

,c abcI

PWMg

ci

1C

,g totZ

sv

PWM*

gV

Inertia

Emulation

*v̂Q-reg0q

0p

PQ-calcq

p

Open loop control of the converter:

- No explicit overcurrent protection in the control

a) VSM with open loop PWM control

DCC

DCv

1L

,c abcI

PWMg

ci

1C

,g totZ

sv

gV

Inertia

Emulation

*v̂Q-reg0q

0p

PQ-calcq

p

Closed loop control of the converter:

- Cascaded voltage and current loops

- Protection and controller saturation can be explicitly included

- Carefull tuning of controller parameters is necessary

Modulation

Current

&Control

*ciVoltage

Control

b) VSM with cascaded voltage and current loops

Fig. 3. VSM Implementations providing voltage references

DCC

DCv

1L

,c abcI

PWMg

ci

1C

,g totZ

sv

gV

Inertia

Emulation

*qQ-reg0q

0p

PQ-calcq

p

Inertia emulation combined with conventional control:

- Inner loop current controller with limitations

Depends on strong grid and frequency tracking by PLL -

- No inherent capability for island mode operation

Modulation

Current

&Control

*ci

CurrentReference*p Calculation

PLL

sv

Fig. 4. Inertia emulation as outer loop of conventional Current Controlled Voltage Source Converters

0p Pm

0

f

fs

p

Active Power Droop Controller

P-regmp

N

s

*v̂0q

Qm

0v̂

f

fs

q

Reactive Power Droop Controller

Q-regmq

Fig. 5. Active and Reactive Power Droop Controllers for microgrids

V. EQUIVALENCE BETWEEN VIRTUAL SYNCHRONOUS

MACHINES AND MICROGRID DROOP CONTROL

The control system structures for droop controlled converters and the VSM implementations providing voltage reference outputs present strong similarities. Indeed, assuming a reduced order SM model, the only difference is that the frequency and phase angle are provided by the swing equation in (3) for the VSM and by (5) for the droop controller. However, these two equations can be proved to be equivalent for small signal behavior around a steady state operating condition when assuming the grid frequency ωg and the reference for the active power p0 as constant.

Combining (7) and (5) leads to:

*, 0

fpu g pu p el

f

m p ps

(8)

The active power pel can be calculated from (8) as:

*, 0

11el g pu pu

f p

sp p

m

(9)

Expanding the products of (9) results in:

,g pu

el

sp

* *

, 0pu g pu pu

f p p f

s s p

m m

0p (10)

As indicated, the expression in (10) can be simplified by eliminating the derivatives of constant terms, leading to:

*0 ,

Inertia term Damping term

1 1el pu g pu

f p p

s p pm m

(11)

Equation (11) has exactly the same form as (3). Thus, equivalent inertia and damping terms can be clearly identified from the active power droop regulator of (5). The formal equivalence between the VSM model from (3) and the droop regulator from (5) can then be explicitly expressed by:

1 1

,a df p p

T km m

(12)

The relations in (12) provide a further insight in the functional meaning of the terms in (3) and (5). Indeed, the damping gain, kd, in the VSM swing equation is inversely linked to the droop gain mp. Moreover, the first order low pass filter on the active power serve an analogous function of the virtual inertia. Thus, the parameters of the droop regulator can be tuned in order to emulate the small signal behavior of a specific synchronous machine. For example, the low pass filtering of the active power can be designed not only to reduce the effect of high frequency power harmonics in the control, but also to provide an inertia function. As long as the inertial dynamics are dominant, without significant influence of any inner loop dynamics, simplified stability analysis based on the swing equation of synchronous machines and VSMs

can therefore be directly applied to conventional droop controllers. Similarly, stability considerations performed on the droop based controls for microgrids could be extended to the VSM schemes by applying (12).

To verify the equivalence derived in (12), the simple electrical system shown in Fig. 6 has been simulated both with VSM-based control and with a droop controller. The main parameters of the simulated system are given in Table 1, and three different cases have been simulated. Firstly, the VSM swing equation from Fig. 1 combined with cascaded voltage and current loops according to Fig. 3 b) have been compared to the droop controller from Fig. 5 while using the same parameters of the controllers. Then, the results from both these control structures have been compared to the VSM with open loop PWM according to Fig. 3 a), and all results are plotted together in Fig. 7. From this figure, it is clearly seen that the VSM with cascaded loops is identical to the microgrid droop controller with the same inner loops. The simulation of the VSM with open loop PWM verifies that the inertial dynamics corresponding to the swing equation is dominant under these conditions and is common to all three control systems, while the small oscillations shown in the two first curves must be associated with the dynamics of the cascaded voltage and current loops.

VI. CLASSIFICATION OF VSM IMPLEMENTATIONS AND

DROOP CONTROLLERS

Since the concept of Virtual Synchronous Machines is still relatively new, the proposed implementations have been mainly based on individual preferences and objectives resulting in a rather fragmented state-of-the art literature. However, as discussed in section II, it is possible to identify

gR gL

FilterEnergy Storage

1L

1CDCC

VSC

fVgV1R

Grid equivalent

Fig. 6. Schematic of the simulated system

Table 1 System parameters

Parameter Value Parameter Value

Rated voltage Vn 690 V Grid resistance Rg 0.003 pu

Rated power Sn 2.75 MVA Grid inductance Lg 0.08 pu

Rated grid frequency fn 50 Hz Filter capacitance C1 0.074 pu

Moment inertia Rotor J 50 Kg m2 Proportional and integral gains for current regulator kpc kic

1.27, 14.25 Damping coefficient Kd 50 MW s-1

Filter resistance R1 0.003 pu Proportional and integral gain for voltage regulator kpv kiv

0.59, 736.09Filter inductance L1 0.08 pu

Fig. 7. Comparison of results from simulation of various concepts for VSM and droop controllers

several detailing levels for the SM model implementation while section III has shown that the VSM concept can be adapted to several control structures for the power electronic converters. A classification of the literature proposals is suggested in Table 2 with direct references to related publications. The table indicates also configurations that are not relevant and configurations that are technically feasible but not yet proposed in literature.

VII. CONCLUSIONS

The concept of Virtual Synchronous Machines (VSM) has been introduced in the last decade with the aim of providing the controllability and stability features of a classical synchronous machine to a power electronics converter. This paper has reviewed the background for the various implementations and control structures proposed for the VSM concept in the literature, and classified the presented proposals according to the VSM detailing level and the control structure used for implementation. The paper has also proved the equivalence between the VSM concept and droop controllers commonly applied for converter control in microgrids by derivations and time domain simulations. This equivalence provides a new perspective for the VSM concept and a deeper physical insight into the interpretation of the parameters used in conventional power-frequency droop controllers for converter-based microgrids.

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Table 2 Classification of Control Methods for VSM and Droop Control implementation

Model output Voltage vector reference Current vector reference

Power Reference General Comments

VSM Implementation Direct PWM Cascaded Voltage and

Current Controllers

7th order model – Full order SM model.

Possible Possible VISMA concept as in [11], [15], [29], [32], and in [34]. (ref. Fig. 2)

Not relevant

Represents the full dynamics of the SM. May result in unnecessary level of detail.

5th order or 4th order model – Reduced order SM model (2nd or 3rd order) combined with mechanical dynamic.

Voltage output formulation of the VISMA concept [16]

Possible VISMA concept as in [16], [30], [31], [33]. (ref. Fig. 2)

Not relevant

Typically considering only stator windings. Voltage amplitude is provided by a reactive power control loop.

2nd order model –Swing Equation and voltage amplitude given by reactive power controller

"Syncronverter" concept as in [17], [18] and others by same authors. Also in [19], [20], [21]. Similar concept in [35] and others by same authors. (ref. Fig. 3 a)

Analysed in [24]. Ongoing study of control system tuning in [25]. (ref. Fig. 3 b)

A similar concept as in [22], [23] and other publications by the same authors.

Not relevant

Simple implementation that can be combined with virtually any control scheme for the converter.

1st order model – Inertia emulation with power (or current) output calculated from grid voltage.

Not relevant Not relevant Possible, but not explicitly found in literature.

EU VSYNC project [27]. Discussed in [10], [26], [28], [36], [37], [38], [39]. Also in [41], [42], [44], [45]. (ref. Fig. 4)

No inherent capability for islanded operation or black-start. Depends on PLL for tracking grid frequency dynamic.

Microgrid Droop Controllers

Common approach as for instance in [4]

Common approach, as for in [6], [8], [9]

Possible, and partly discussed in [5]

Not relevant Extensive literature available. Only selected examples are given here.

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