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Renewable energy sources, such as photovoltaic wind turbines, and wave power converters, use power converters to connect to the grid which causes a loss in rotational inertia. The attempt to meet the increasing energy demand means that the interest for the integration of renewable energy sources in the existing power system is growing, but such integration poses challenges to the operating stability. Power converters play a major role in the evolution of power system towards SmartGrids, by regulating as virtual synchronous generators. The concept of virtual synchronous generators requires an energy storage system with power converters to emulate virtual inertia similar to the dynamics of traditional synchronous generators. In this paper, a dynamic droop control for the estimation of fundamental reference sources is implemented in the control loop of the converter, including active and reactive power components acting as a mechanical input to the virtual synchronous generator and the virtual excitation controller. An inertia coefficient and a droop coefficient are implemented in the control loop. The proposed controller uses a current synchronous detection scheme to emulate a virtual inertia from the virtual synchronous generators. In this study, a wave energy converter as the power source is used and a power management of virtual synchronous generators to control the frequency deviation and the terminal voltage is implemented. The dynamic control scheme based on a current synchronous detection scheme is presented in detail with a power management control. Finally, we carried out numerical simulations and verified the scheme through the experimental results in a microgrid structure.

The concern for climate change and the demand for energy have increased the demand for renewable energy considerably. For technical, economic and environmental reasons, the share of electricity generation by renewable energy resources (RES), such as solar power, wind power, tidal and wave power etc., is growing rapidly [

Carrsaco et al. [

The electricity generation in a traditional power system relies on fully dispatchable power generation units including rotating SGs enabled with stored kinetic energy. The crucial property of the frequency and stability dynamics is the rotational inertia (rotating mass) which is added by the conventional SGs due to the electro-mechanical coupling [

Many control algorithms have been reported for the voltage regulation using various schemes such as static var-compensation (SVC) [

The increasing shares of inverter connected RES are causing a drop in the rotational inertia of the power system. The kinetic energy delivered/absorbed to/from the grid by a generator’s rotating mass depends to the rate of change of the frequency. Therefore, the RES are coupled with an ESS to mimic the external behavior of the SG. The DC-link represents not only the power taken from the imaginary prime mover but also from the inertia of the rotating part of the imaginary SG. Rotating inertia, i.e. the inertia constant, reduces the rate of change of frequency in a transient process. In a multi-area power system the inertia constant varies in individual grid areas depending on the inverter connected RES.

Motivation and Contribution of the WorkIn the literature, various methods to emulate a virtual inertia of the VSG have been presented with a load mutation and a step change in the wind speed, in a wind energy harvesting case. Shi et al. [

In this paper, we considered a microgrid as a small single-area power system where the inertia constant (H) is highly time variant and fluctuates between 3 and 6 s due to the deployment of Wind-PV-Wave power generation. The idea of using ultra-capacitor (UC) at the DC-link to emulate a virtual inertia of the VSG is implemented while the droop control is emulated by the battery as a virtual speed governor. A hypothetical inertia coefficient is implemented in the control loop which defines a time constant of the capacitor to respond during the transient processes. Due to the inertia coefficient, the UC responds quickly to the transients and protects battery for a fast discharge. This combination of the ESS connected with the UC at the DC-link forms a DC-grid structure. The proposed scheme controls the three phase inverter connected with the ESS with a DC-DC controller in parallel and this structure operates as VSG as shown in

A WEC feeds the rectified power, intermittent by nature, into the DC-link. Hence, a power smoothening and a frequency regulation is desired. The capacitor at the DC-link absorbs the extra power along with the battery storage. The DC-control regulates the DC-link voltage to a stable point and supports the frequency regulation caused by the varying WEC power. The main electrical and mechanical parameters of the linear generator used as a WEC are shown in

The resulting current error from the comparator is used as an input for the PI-controller, designed with proportional gain only, to amplify the error signal. The amplified signal is compared with a fixed frequency (10 kHz) triangular wave to generate pulses to control the DC-DC controller. In this way, a stable DC-link voltage is achieved according to the frequency deviation. Due to the scope of the paper, the control on the WEC side is not discussed here. However, a brief overview of the control on the WEC side is shown in

The dynamic behavior of a synchronous machine can be described in Formula (1) [

T m − T e = J d ω d t (1)

where T m is the input mechanical torque, T e is the output electromagnetic torque, J is the rotational inertia of the SG, ω is the mechanical angular speed and can also be expressed in terms of the inertia constant H and power in (2),

Parameters | Electrical characteristics | Mechanical characteristics | ||
---|---|---|---|---|

Synchronous inductance | 21.2 mH | Nominal speed | 0.7 m/s | |

Winding resistance | 0.36 Ω | Stator length | 1.96 m | |

Rated armature current | 48 A | Stator width | 0.4 m | |

Rated power | 40 kW | Translator length | 2.0 m | |

Rated voltage | 450 V | Translator weight | 2700 kg |

P m − P e = 2 H d ω d t (2)

Assuming T m is constant and J increases in (1) entails that the change in the magnitude of the speed will be smaller caused by the load perturbation. According to the expression of the rotor kinetic energy, E R , and the inertia constant are defined in (3) [^{2} and system rated power, p r a t e d = 40 kW ,

E R = ∫ ( P m − P e ) d t = 1 2 J ω 2 H = E R / p r a t e d } (3)

The variation of the rotor kinetic energy, Δ E R , can be defined as in (4),

Δ E R = 1 2 J [ ( ω + Δ ω ) 2 − ω 2 ] (4)

where P m , and P e are the mechanical and electrical powers, and Δ ω is the change in angular speed. Formula (4) shows that the less kinetic energy is released at the same time if J increases, which results in the low release rate and conversion efficiency of the rotor kinetic energy. This phenomenon brings an unbalanced power in the MG and pulls a heavy burden to the SG or the AC-grid. Moreover, this virtual inertia control requires additional speed or de-loading control techniques, which makes it a complex control structure.

The VSG implementation investigated in the study is based on a conventional swing equation representing the inertia and damping of a traditional SG. A second-order model is usually applied in VSG control topology. The mathematical model of the synchronous generator is expressed as its swing equations in (5) [

P r * − P e − D ω Δ ω = J ω d ω d t Q r * − Q − D q ( V * − V ) = K d E d t d θ d t = ω } (5)

where P r * is the virtual mechanical input power, P e is the electrical power flowing into the grid, Q r * is the reactive power reference, Q is the filtered reactive power measurement, V * , and V are the referenced and output voltage amplitude respectively, θ is the electrical angle, and D, D q are the active and reactive damping factors, respectively.

The power converters regulated as VSGs present a similar frequency dynamic response as SGs, and are incapable of storing or releasing the kinetic energy. Therefore, they require an additional energy storage unit, called an ESS. To use the ESS with power converters as a combined unit to form a VSG, they must satisfy the condition in (6) [

( Δ p b − 2 H d Δ f d t ) , used for the power balancing in real time,

Δ p R E S + Δ p l = [ Δ p b − 2 H d Δ f d t ] (6)

Δ p l = D Δ f (7)

where Δ f is the measured frequency deviation, and H is the inertia constant. Since the Ultra-capacitor (UC) at the DC-link is used to emulate the virtual inertia of the VSG in a transient process. A transient condition occurs when a load change is observed and results in a sharp deviation in the derivative of the change in frequency. The rotational inertia (J) is inversely proportional to the rate of change in the frequency ( d Δ f / d t ), i.e. a transition process, and effects the frequency regulations much faster. The rotational inertia is defined in terms of inertia constant as in (3). The inertia constant quickly responds to this deviation by emulating a virtual inertia. The UC is modeled to respond quickly by implementing a time constant (inertia coefficient of the capacitor) in the control loop of the converter and defined as inertia coefficient of the capacitor in (11). Since D is inversely proportional to the frequency variations Δ f and does not depend on d Δ f / d t which means the damping power only can support in a steady state instead of the transient process [

Δ p b ≈ Δ p d r o o p = Δ f r d . (8)

where r d is the frequency-droop coefficient acting on Δ f during the RES power variations P w e c . The VSG power contribution by the inertia emulation Δ p c a p is defined in (9) and represents the power sharing of the capacitor. Since the emulated inertia is proportional to the energy stored in the capacitor in (10), there is a trade-off for the selection of the inertia constant and the capacitor size to reduce the frequency deviation. Since Δ v d c is adjusted by K c in relation to Δ f as defined with Δ v d c = K c Δ f as shown in

Δ p c a p = − 2 H d Δ f d t (9)

Δ p c a p = − 2 H c a p d ( Δ v d c ) d t Δ p c a p = − 2 H c a p K c d ( Δ f ) d t } (10)

where H c a p = H c is the inertia coefficient of the capacitor and defined in (11) as a function of the capacitance C d c , reference DC-link voltage v d c r e f , and the system rated power, p r a t e d ,

H c a p = 0.5 C d c ( v d c r e f ) 2 p r a t e d . (11)

Therefore, the inertia coefficient of the capacitor H c a p is inversely related with the gain K c and can be expressed as ( H c a p = H / K c ). The emulated power system inertia is considered similar to the inertia in a conventional power system, which is taken as H = 4 as obtained from (3) and a maximum value of K c can be obtained using (11) which is 13.3. The control parameters and the system parameters used in this study are reported in

The emulation of a rotating inertia and the power-balance synchronization mechanisms of this virtual inertia is the main difference between the investigated VSG control structure and conventional control systems for the inverter. The swing equation used for the implementation is linearized with respect to the speed so the

Electrical characteristics | Mechanical characteristics | ||
---|---|---|---|

DC-link voltage ( v d c r e f ) | 700 V | Frequency-droop coefficient (r_{d}) | 0.05 |

DC-link capacitance (C) | 50 mF | Inertia constant (H) | 4 |

DC filter inductance (L_{B}) | 1.5 mH | Damping coefficient (D) | 1 |

Battery voltage (V_{B}) | 250 V | Cap. intertia coeff. (H_{c}) | 0.3 |

Filter inductance (L_{S}) | 3.9 mF | C voltage gain (K_{c}) | 13.3 |

Filter capacitance (C_{f}) | 20 μF | Q-droop gain (k_{q}) | 186 |

Line impedance (Z_{g}) | 0.8 Ω + 1 mH | LPF filter, ω f | 16 Hz |

Rated phase voltage (U) | 400 V | DDC k_{pf}_{, }k_{if} | 0.16, 1.54 |

Voltage reference (U_{ref}) | 400 V | DDC k_{pv}_{, }k_{iv} | 0.22, 786 |

Frequency reference (f_{g}) | 50 Hz | DC, K_{pv}, | 2.5 |

Max frequency deviation | 0.2 Hz | DC, K_{pi}_{,} | 668 |

inertia is determined by the power-balance formula in (12) as shown in

P r e f * − Δ p l + Δ p b − P e 2 H = d ω V S G d t (12)

The mechanical speed ω V S G , of the VSG is given by the integral of the power-balance formula (13) while the phase angle θ V S G , of the VSG is obtained by the integral of the speed. The damping power Δ p l , presenting the damping effect of the VSG, defined by the product of active-damping factor D and the difference between the VSG frequency- f V S G , and the actual grid frequency f g , as in (7) where Δ f = f V S G − f g . The actual grid frequency is measured by a dual second order generalized integrator-PLL (DSOGI-PLL).

P r e f * 2 H − D Δ f 2 H + 1 r d Δ f 2 H − P e 2 H = d ω V S G d t . (13)

The VSG speed in steady state will become equal to the grid frequency and the frequency deviation will approach zero under a stable grid connected condition. The VSG generates the frequency and the active power in steady state condition which is assumed to be the reference frequency and active power reference. However when the load changes the frequency reference values and the active power reference values are obtained by the estimation of the reference frequency and the reference source active power P r e f * , through the CSD theory. The benefit of using a CSD estimation of the references is to reduce the shock currents during the load changing event and improve the stability of the system.

The active power droop control is realized by a CSD estimation of the three-phase reference active power of the source currents. The reference source active power is estimated by a dynamic droop controller (DDC). The DDC estimates the reference source active power by taking the difference of the output of the PI controller P f and the fundamental active power component of the loads P f l , which is the filtered average power of the loads. The frequency error is processed to the PI controller for the estimation of P f in (14). To obtain P f l , a first order low pass filter is applied with a cut-off frequency ω f as shown in

P f ( n ) = P f ( n − 1 ) + k p f { Δ f ( n ) − Δ f ( n − 1 ) } + k i f Δ f ( n ) P r e f * ( n ) = P f ( n ) − P f l ( n ) d P f l d t = − ω f P f l + ω f P l } (14)

where k p f and k i f are the proportional and the integral constants for the PI controller, Δ f ( n ) , and Δ f ( n − 1 ) are the frequency errors at the nth and (n − 1)^{th} sampling instant and P f ( n ) and P f ( n − 1 ) are the output of the voltage PI controller at the nth and (n − 1)^{th} instant.

According to the mathematical model of the SG the loop-voltage equation is given in (15) shown in

E = U + I R + j I X . (15)

where E is the electromotive force, U is the terminal voltage of the stator, I is stator current, R is the armature resistance of the stator, and X is the synchronous reactance. The bold letters E, U, and I represent their phasor form. The power angle, δ , between E and U is very small and hence cos ( δ ) ≈ 1 , which satisfies the formula in (15). Consider the SG stator voltage equations and the excitation controller design. The structure of the excitation controller is shown in

the actual output voltage amplitude at the terminal, namely U t . In

where k p v and k i v are the proportional and the integral constants for the PI controller, V e ( n ) , and V e ( n − 1 ) are the terminal voltage error at the nth and (n − 1)^{th} sampling instant and Q q v ( n ) and Q q v ( n − 1 ) are the output of the voltage PI controller at the nth and (n − 1)^{th} instant required for the voltage regulation.

The filtered reactive power component Q f l corresponds to the three-phase fundamental reactive power of the loads. The filter applied is a first order low pass filter with a cut-off frequency ω f defined in (18),

U 0 = U r e f − k q ( Q r e f * ) (16)

Q r e f * ( n ) = Q q v ( n ) − Q f l ( n ) Q q v ( n ) = Q q v ( n − 1 ) + k p v { V e ( n ) − V e ( n − 1 ) } + k i v V e ( n ) V e ( n ) = U r e f ( n ) − U 0 ( n ) } (17)

d Q f l d t = − ω f Q f l + ω f Q l (18)

The control of the proposed VSG is realized by computing the reference source currents (RSC) through the CSD theory. The RSC are comprised of two components, 1) the in-phase component, the active power current component, i p ( a , b , c ) r for regulating the frequency, and 2) a quadrature component, the reactive power current component, i q ( a , b , c ) r for controlling the magnitude of the generated voltage (the terminal voltage). The reference source active power P r e f * , is estimated by taking the difference of the output of the frequency PI controller P f , and the filtered average load power P f l . The three-phase reference active-power components of the source currents i p ( a , b , c ) r are estimated using the peak amplitudes of the phase voltages V a m , V b m , V c m , the reference source active power ( P r e f * ) and the phase voltages U a , U b , U c . To regulate the terminal voltage at the PCC the VSG requires a reactive power support. The reference source reactive power component ( Q r e f * ) estimation is explained in the previous section. The obtained component is used to extract the three-phase reactive-power components of the reference source currents ( i q ( a , b , c ) r ) using the peak amplitudes of the phase voltages V a m , V b m , V c m and the quadrature phase voltages (QPV), U a q , U b q , U c q . The algebraic sum of the active-power components of the source currents i p ( a , b , c ) r ,and reactive-power components of the reference source currents ( i q ( a , b , c ) r ) yields the total reference source currents i ( a , b , c ) r . The peaks of the phase voltages are detected using a peak detection algorithm, consisting of a sample and hold (SH) circuit, hit crossing (HC) detector and an estimated QPV. Since the QPVs are leading the phase voltages by 90˚, the HC detects the negative slope zero crossing of the quadrature voltage, i.e. the peak of the phase voltage at the same time instant. The estimated reference source currents are compared with the measured currents at the VSG. The resulting errors are processed to a proportional controller and the amplified signals are used for the pulse width modulator (PWM) to control the switches of the power converter.

Assuming that the amplitude of the active power source currents are balanced and equal in magnitude after the compensation in (19),

I p a = I p b = I p c (19)

where I p a , I p b , I p c are the peak amplitude of the active current in each phase. Then,

I p a , b , c = ( 2 P r e f , a , b , c ) / V a , b , c m (20)

Substituting (20), in (19) yields into (21),

( 2 P r e f , a ) / V a m = ( 2 P r e f , b ) / V b m = ( 2 P r e f , c ) / V c m (21)

The total average power is obtained by taking the sum of the reference source active power components in each phase ( P r e f , a , P r e f , b , P r e f , c ) in (22).

P r e f = P r e f , a + P r e f , b + P r e f , c (22)

By rearrangement of (21) and (22) yields (23),

P r e f , a , b , c = ( V a , b , c , m / V T ) P r e f (23)

where V T = V a m + V b m + V c m , is the sum of the peak voltages. The estimation of the reference source currents is done by solving (23) and (20),

i p ( a , b , c ) ( t ) = [ 2 P r e f / ( U t V T ) ] U a , b , c ( t ) (24)

where U t is the terminal voltage and expressed in (25),

U t ( t ) = 2 { U a 2 ( t ) + U b 2 ( t ) + U c 2 ( t ) } / 3 (25)

Using (29) based on the CSD theory estimation, the reference source active power ( P r e f * ) is converted into the active power component of the three-phase source currents ( i p ( a , b , c ) r ) using the phase voltages and the peak of the phase voltages in (26).

i p ( a , b , c ) * ( t ) = [ 2 P r e f * / ( U t V T ) ] U a , b , c ( t ) (26)

where the total power is the reference source active power ( P r e f * = P r e f , a * + P r e f , b * + P r e f , c * ) which has to be generated by the VSG. P r e f * is treated as the virtual mechanical input power P m to the VSG to build the electrical power P e to meet the active power demand which justifies the formula (2).

An assumption similar to the one in the previous section, can be applied to the fundamental reactive power source currents in that they are balanced and equal in magnitude,

I q a = I q b = I q c (27)

The reference source reactive power ( Q r e f * ) is converted to the reactive power component of the three-phase source currents using quadrature phase voltages ( U a q , U b q , U c q ) and the peak of the phase voltages in (28),

i q ( a , b , c ) * ( t ) = [ 2 Q r e f * / ( U t V T ) ] U a , b , c , q ( t ) (28)

The quadrature phase voltages are estimated by using the amplitude of the PCC (terminal voltage, U t ( t ) ) and the quadrature unit templates, u a q , u b q , u c q ( u a , b , c , q ) and the derived expression is explained in (29),

U a , b , c q ( t ) = u a , b , c q ( t ) U t ( t ) (29)

The quadrature ( u a , b , c q ) and the in-phase ( u a , b , c , p ) unit templates are derived and expressed in (30) and (31) at time instant, t [

u a q ( t ) = { − u a p ( t ) + u c p ( t ) / 3 } u b q ( t ) = { ( 3 u a p ( t ) ) + ( u b p ( t ) − u c p ( t ) ) / 2 3 } u c q ( t ) = { { ( − 3 ) u a p ( t ) } + ( u b p ( t ) − u c p ( t ) ) / 2 3 } } (30)

u a , b , c p ( t ) = U a , b , c p ( t ) / U t ( t ) (31)

The total reference source currents are estimated by taking the sum of the estimated active and reactive reference source currents and defined by the formulae (32),

i a r ( t ) = i p a * ( t ) + i q a * ( t ) i b r ( t ) = i p b * ( t ) + i q b * ( t ) i c r ( t ) = i p c * ( t ) + i q c * ( t ) } (32)

To verify the proposed VSG control scheme, numerical simulations are carried out in a MATLAB/Simulink environment. The system structure shown in

To verify the effectiveness of the proposed control strategy, the simulation results are presented with the loads perturbation and a varying WEC power. The verification of the control is presented with the ESS. It is worth noting that the initial grid synchronization is achieved by using a PLL, and the VSG is smoothly synchronized with the grid. The phase synchronization between the grid and the VSG is shown in

2, which is 5 kW, is connected and a drop in frequency is reported immediately just after the load perturbation occurs. At this moment the proposed control quickly takes action to restore the terminal frequency and the active power demand ( P r e f * ). The proposed control strategy based on the CSD scheme, quickly estimates P r e f * , compensates for the required active power demand by increasing the VSG active power with a minor overshoot. The inverter accesses the energy storage to support the active/reactive power and frequency/voltage compensation. The power output of DC/DC converter feeds a small amount of power into the DC-link and surpluses the power until the load 2 (i.e. 5 kW) is disconnected at t = 1.126 s. During the first load perturbation of 5 kW, the terminal voltage U t drops immediately and therefore a reactive power support is released by the VSG to restore the terminal voltage as shown in

with increased or decreased the active power demand. The control restores the frequency within a small deviation of ±0.05 Hz. Furthermore, to investigate the robustness of the control, a reactive load of 400 var, as load 3, is introduced at t = 1.01 s. The purpose of this investigation is to demonstrate the controller performance in a multi load perturbation. In the event at t = 1.01 s when the reactive power demand was raised, the VSG immediately supports the reactive power compensation to maintain the terminal voltage at the operating point at that event. During this event, the load 2 is disconnected at t = 1.126 s and the terminal voltage deviates from the operating point to the nominal terminal voltage.

As a result, the VSG adds a small amount of reactive power at that event and quickly returns to the previous feeding point, since the VSG voltage is detected running up at the nominal terminal voltage. The individual current components to generate the active and reactive power are shown in

mode The drop in the terminal voltage is noticed immediately and the stored reactive power is used to regulate the terminal voltage to a stable operating point. The stored reactive power is an effect of the load 3 disconnection which supports the terminal voltage during the load 4 perturbation as shown in

is achieved.

The ESS effectively participates during each event of the load perturbation. It can be noted that the ESS not only regulates the output droop but also supports the power balance during excessive and insufficient power generation from the WEC as shown in

The DC-link voltage v d c is well controlled within the allowed DC-ripples caused by the intermittent WEC power during the different wave elevations and the load perturbation. In

The experiments were carried out to verify the practical feasibility of the proposed control based on diagram in

behavior of an intermittent power source with the AC-microgrid and to test the effectiveness of the proposed VSG control algorithm. To this end, the wave data is used to generate a torque for the emulator in the microgrid and tested with the algorithm to control the microgrid voltages and the frequency. A medium speed rotatory emulator, originally developed at UCC [

From

battery ( Δ p b ) are shown in a full experimental time and in zoom for one of the load perturbation events.

The frequency dependent loads are changed at different time events as shown in

Since the rotational inertia (J) is inversely proportional to the rate of change in the frequency ( d Δ f / d t ), i.e. a transition process, and effects the frequency regulations much faster. The rotational inertia is realized as an inertia constant (H) in the VSG swing equations. The inertia constant of the VSG swiftly, handles the frequency regulation in a transition process by emulating a virtual inertia. The battery is modeled to emulate the droop-control mechanism to compensate for the long-term power variations, which reduced the stress of the battery. In

The experimental results obtained from the VSG, show the AC-side three-phase

line to neutral voltages and the current in phase-A, in

In this paper, the concept of a VSG for controlling power converter to replicate the behavior of traditional SG is studied. A CSD-scheme based dynamic control to release the potential advantages of distributed control of power electronics is proposed and verified through simulation and experimental results in the SmartGrid context. The VSG based control is extracted from the swing equation of the traditional SG. The DC-link with the ESS is used to emulate the inertia of the VSG and represented as the inertia coefficient, the droop control and a virtual speed governor in a VSG algorithm. The proposed control scheme is designed to regulate the voltage and the frequency controller during the transient and steady state, which is verified by the simulations and the experimental results. The reference active and reactive power components are extracted using the CSD-scheme and used as input references to the VSG algorithm to regulate the voltage and the frequency.

In this study, the experimental data, recorded during the conducted experiments at Lysekil Wave Energy Park on the west coast of Sweden, are used to emulate a real WEC behavior in the time domain. The proposed control scheme effectively addresses the power fluctuations caused by the WEC, a load perturbation and maintains the DC-link voltage at a steady point. Furthermore, the frequency deviation caused by the WEC’s intermittent power, P w e c and the perturbation of the loads P l are well controlled under the allowed maximum frequency deviation Δ f max . Since the grid had excessive real power, the delivery of the real power was slightly reduced by the VSG and the real power delivery remained stable at the loads. However, the slight variation in the real power of the VSG was not clearly visible in the long run test, which appeared to be a constant power delivery for each load perturbation. Moreover, the effectiveness of the VSG real power control could be appreciated in the zoomed results where the real power had slight fluctuations as followed by the variations in the frequency. This verified that the frequency of the system was regulated well within the allowed range by virtual inertia emulation and operated slightly higher than the nominal value (50 Hz), as shown in the results in

Arvind Parwal thanks Swedish Research Council (VR) STandUP for Energy, MaRINET2 and Erasmus Mundus (EMINTE) Ph.D. Scholarship for the support of the work. Also, we are thankful to Ms. Kristin Bryon for an extensive English proofreading.

The authors declare no conflicts of interest regarding the publication of this paper.

Parwal, A., Fregelius, M., Silva, D.C., Potapenko, T., Hjalmarsson, J., Kelly, J., Temiz, I., de Oliveira, J.G., Boström, C. and Leijon, M. (2019) Virtual Synchronous Generator Based Current Synchronous Detection Scheme for a Virtual Inertia Emulation in SmartGrids. Energy and Power Engineering, 11, 99-131. https://doi.org/10.4236/epe.2019.113007

E Electromotive force in phasor form (V)

C d c Capacitance of the capacitor (F)

C f Filter capacitance (F)

D , D q Active and reactive damping coefficients

E R Rotor kinetic energy (J)

f_{g} Frequency reference (Hz)

f_{VSG} VSG frequency (Hz)

H Inertia constant (s)

H c a p Inertia coefficient of the capacitor (s)

I Stator current in phasor form (A)

i a b c Three-phase current (A)

I B DC-current (A)

i f Excitation current (A)

I p a , I p b , I p c The peak amplitude of the active current in each phase (A)

I q a , I q b , I q c The peak amplitude of the reactive current in each phase (A)

i ( a , b , c ) r Total reference source current (A)

i p ( a , b , c ) r Three-phase reference active power currents (A)

i q ( a , b , c ) r Three-phase reference reactive power currents (A)

i d c r e f DC-reference current (A)

J Rotational inertia (kg-m^{2})

K Inertia coefficient for reactive power loop

k c Voltage gain

k q Reactive power droop gain

k p f , k i f Proportional and Integral constants for active power control loop

k p v , k i v Proportional and Integral constants for reactive power control loop

K p v , K i v roportional and Integral constants for DC control loop

L_{B} DC filter inductance (H)

L_{s} Filter inductance (H)

P e , P m Electrical and mechanical powers (W)

p r a t e d Rated power of the WEC (W)

P r e f * Total reference source active-power (W)

P r e f , a , P r e f , b , P r e f , c Reference source active-power in each phase (W)

P f ( n ) , P f ( n − 1 ) Output of the voltage PI controller at the nth and (n − 1)^{th} instant (W)

Q r * Reactive power reference (var)

Q f l Fundamental reactive power component of the loads (var)

Q q v ( n ) , Q q v ( n − 1 ) PI controller output at the nth and (n-1)th instant (var)

R Armature resistance (Ω)

r d Frequency-droop coefficient

Stator terminal voltage in phasor form (V)

U a b c PCC three phase voltages (V)

U t , U 0 Terminal voltage (V)

U r e f Reference terminal voltage (V)

u a , b , c p In-phase unit templates (V)

u a , b , c q Quadrature unit templates (V)

U a , b , c q Quadrature phase voltages (V)

V a m , V b m , V c m , V T Peak amplitude of the phase voltages (V), Sum of peak amplitudes (V)

V d c r e f DC-reference voltage (V)

Z g Line impedance (H)

ω Mechanical angular speed (rad/s)

ω V S G VSG mechanical speed (rad/s)

ω f Cut-off frequency of the filter

θ Electrical angle (˚)

Δ f Frequency deviation (Hz)

Δ v d c DC-voltage deviation (V)

Δ p b Frequency-droop power for the battery (W)

Δ p R E S RES power variation (W)

Δ p l Damping power (W)

Δ E R Variation of rotor kinetic energy (J)

d Δ f / d t Rate of change of the frequency (Hz/s)