# Emulating Rotational Inertia of Synchronous Machines by a New Control Technique in Grid-Interactive Converters

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Inertial Response of Synchronous Machines

_{r}

^{2}/2VA

_{base}) is represented by [24]:

_{AGC}

^{pu}) is neglected. To analyze the effect of rotational inertia on the system frequency response, the transfer function of frequency deviation (Δω) over the load disturbance (ΔP

_{e}) is derived as (4).

_{e}= 5%) is depicted in Figure 2 for different values of H. It is clear that increasing H results in a lower RoCoF level and frequency nadir and, thereby, better primary frequency regulation during the transient time.

## 3. The Proposed Converter Control Scheme

_{c}, L

_{c}), and a balanced power grid (u

_{g}) with the impedance of R

_{g}and L

_{g}. Generally, supercapacitors or battery energy storages can be applied as the DC-side energy buffer for frequency support. Herein, the DC-link capacitor acts as the energy buffer that does not impose any extra cost on the VSC hardware. This is because the DC-link capacitors are essential for voltage support and harmonics filtering in power conversion systems. The variables u

_{p}, e

_{c}, and i in Figure 3 stand for point of common coupling (PCC) voltage, converter terminal voltage, and injected current to the PCC in the natural abc reference frame, respectively.

#### 3.1. Analysis of the Interfaced Converter Under Dynamic Operating Condition

_{c}+ L

_{g}, and r = R

_{c}+ R

_{g}. Transforming (5) into the grid synchronous reference frame (dq

^{g}) using Park Transformation (T

_{abc}

_{/dq}) results in:

_{plant}(s) is expressed as:

_{r}Li

_{d}and ω

_{r}Li

_{q}, which gives rise to the control complexity. Nonetheless, the cross-coupling terms can be eliminated by applying appropriate current feedforward control. The inner current loops (7) are controlled via two separate proportional integral (PI) PI controllers (G

_{i}(s) = k

_{cp}+ k

_{ci}/s) since the dq variables are constant in the synchronous reference frame.

_{dc}) by:

_{dc}reaches its reference by controlling the d-axis current through a PI controller (G

_{u}(s) = −k

_{up}− k

_{ui}/s), which forms the outer control loop. The time-delay introduced by the pulse-width modulation (PWM) PWM updates and reference computations can be approximately modeled using a first-order function G

_{d}(s) in which f

_{s}denotes the sampling frequency [25].

#### 3.2. Dynamic Analysis of the Phase-Locked Loop

^{g}frame corresponding to the grid and dq

^{c}frame corresponding to the VSC controller. In steady state operating conditions, the controller dq frame is aligned with the grid dq frame. When a small contingency occurs in the system, the two frames are no longer aligned due to the dynamics of the PLL. It means that the dq

^{c}frame lags Δθ as demonstrated in Figure 4a.

^{g}frame is rotated to the dq

^{c}frame using matrix T

_{Δθ}for feedback control purpose [26]. During steady state operation, we can conclude (12):

_{p}

^{g}and u

_{p}

^{c}denote the PCC voltage in the grid frame and the controller frame, respectively. Equation (12) points out that the angle difference (Δθ) between PCC voltage vectors in the two frames is zero. Using matrix T

_{Δθ}, (12) can be rewritten as:

^{c}and dq

^{g}frames can be expressed as:

^{g}

_{pq}as:

^{g}

_{pq}= 0 and assuming the VSC is operating in the unity power factor, (15) can be further simplified to (18) as follows:

^{g}

_{pd}·Δθ and + I

_{d}·Δθ terms in the q-axis control loop, as depicted in Figure 5. On the other hand, due to the coupling between two axes, any variations in Δθ leads to the changes in i

_{d}(this correlation is elaborated in Section 4).

#### 3.3. Fast Grid Frequency Support

_{j}. The time delay T

_{j}in discharging the DC-link capacitor is analogous to gradually releasing the kinetic energy of rotational SGs. The controller is added to the outer voltage loop during the initial support period. The proposed technique is described mathematically as:

_{p}and D

_{p}emulate the inertia and damping characteristics of the real SGs, respectively. A disturbance in the power system results in a small deviation in the variables around their steady state values. Thus, the corresponding small-signal model of the synthetic inertia loop can be derived by canceling the steady state values of (19) as:

_{i}= 0 during steady state operation of the system. Indeed, the frequency support level is limited by the maximum acceptable DC-link voltage deviations in the foregoing method. This is because u

_{dc}must be kept in an acceptable range for linear modulation of the converter and limiting current flow through components of the conversion system. Thereby, the d-q control loops augmented with the frequency support controller are formed as Figure 5.

## 4. Dynamics Assessment of the Proposed Control Technique

_{d}, which produces an effect on the outer voltage controller by the synthetic inertia loop. This relationship is visible in Figure 5. Hence, the transfer function between Δθ and i

_{d}is derived as (23).

_{i}and i

_{d}is derived as:

_{i}and i

_{d}can be considered in parallel with G

_{1}. This is because the two corresponding paths start with i

_{d}and reach the identical summer. It means that the dynamics caused by the PLL in the q-axis are reflected in the d-axis by means of the virtual inertia loop. Then, the function that appeared in the feedback path is formed as follows:

_{2}(s) (Figure 5). Thus, the proposed controller has better performance in comparison to [21] regarding the system stability margin.

## 5. Simulation Results and Discussions

_{a}= 0.86. Two scenarios are considered for simulations, step-up and step-down changes in the demand. Then, in each scenario, the proposed controller is compared with the method in [21] and the conventional VSC system in terms of frequency stability metrics (i.e., frequency nadir and RoCoF).

#### 5.1. Scenario 1

_{a}) must increase in order to (1) increase the difference angle between converter voltage and PCC voltage (Δ) and (2) prevent reactive power exchange with the grid (i.e., the equation e

_{d}= m

_{a}× u

_{dc}× cosΔ = u

_{pd}holds true). This can be observed in Figure 11. The modulation index peaks at around 0.93 in the time interval at which the capacitor is discharged. Then, it rebounds to the value of pre-perturbation. The supportive power supplied by the VSC is shown in Figure 12. It is clear that the power reaches approximately 15 kW and then drops to the steady state value when the frequency returns to 50 Hz.

#### 5.2. Scenario 2

_{d}= m

_{a}× u

_{dc}× cosΔ = u

_{pd}holds true). Then, the modulation index decreases as the DC-link capacitor voltage increases. The modulation signal in this scenario is observed in Figure 15. The power absorbed by the VSC with the aim of frequency support is also brought in Figure 16. The power reaches approximately −15 kW after the loss of demand and becomes zero when the grid frequency returns to 50 Hz.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Abbreviations: | |

dq^{g}, dq^{c} | Synchronous reference frames corresponding to the grid and the controller |

ESS | Energy storage system |

PLL | Phase-locked loop |

PCC | Point of common coupling |

RESs | Renewable energy sources |

RoCoF | Rate of change of frequency |

SGs | Synchronous generators |

VSC | Voltage source converter |

VSG | Virtual synchronous generator |

Variables: | |

i | Current flowing into the grid |

m_{a} | Modulation Index |

P_{m} | Input mechanical power of SG |

P_{e} | Electromagnetic power of SG |

u_{dc}, e_{c}, u_{p}, u_{g} | DC-link capacitor voltage, converter voltage, PCC voltage, grid voltage |

u_{p}^{g}, u_{p}^{c} | PCC voltage vectors in the grid frame and the controller frame |

ω | Angular velocity/frequency of the rotor/grid |

Δ | The difference angle between the converter voltage and the PCC voltage |

Parameters: | |

C | DC-link capacitance |

D, D_{p} | Damping coefficient, virtual damping coefficient |

F_{HP} | Turbine coefficient |

f_{s} | Sampling frequency |

H, H_{p} | Inertia constant, virtual inertia constant |

I_{d} | Rated value of the current i in the d-axis |

J | Moment of inertia of the turbine and the generator |

k_{cp}, k_{ci} | Proportional and integral gains of the current controller |

k_{up}, k_{ui} | Proportional and integral gains of the voltage controller |

k_{ppll}, k_{ipll} | Proportional and integral gains of the PLL |

L_{c}, L_{g} | Inductance of the filter and the grid |

R_{c}, R_{g} | Resistance of the filter and the grid |

R | Droop coefficient |

T_{G} | Governor time constant |

T_{CH} | Inlet volume time constant |

T_{RH} | Re-heater time constant |

U_{dc,ref} | DC-link voltage reference |

U_{pd} | Nominal PCC voltage in the d-axis |

ω_{r} | Reference angular velocity/frequency of the rotor/grid |

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**Figure 8.**Bode diagram of the method in [21].

**Figure 9.**Frequency nadir and rate of change of frequency (RoCoF) metrics under a 5% step-up change in the demand.

Parameter | Value |
---|---|

Governor time constant (T_{G}) | 0.1 s |

Inlet volume time constant (T_{CH}) | 0.2 s |

Re-heater time constant (T_{RH}) | 7 s |

Turbine coefficient (F_{HP}) | 0.3 s |

Droop coefficient (R) | 0.05 |

Inertia constant (H) | 5–20 s |

Damping coefficient (D) | 1 |

System Parameters | Values | Controllers Parameters | Values |
---|---|---|---|

SG nominal power | 100 kW | H, D | 3, 1 |

VSC nominal power | 15 kW | H_{p}, D_{p} | 50, 100 |

L_{g} | 0.003 H | k_{cp} | 0.1 |

L_{c} | 0.002 H | k_{ci} | 10 |

r | 0.001 Ω | k_{up} | 0.006 |

U_{pd} | 326.59 V | k_{ui} | 0.001 |

U_{dc,ref} | 750 V | k_{ppll} | 180 |

C | 0.1 F | k_{ipll} | 3200 |

ω_{r} | 100π rad/s | T_{j} | 0.2 |

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**MDPI and ACS Style**

Saeedian, M.; Pournazarian, B.; Seyedalipour, S.S.; Eskandari, B.; Pouresmaeil, E.
Emulating Rotational Inertia of Synchronous Machines by a New Control Technique in Grid-Interactive Converters. *Sustainability* **2020**, *12*, 5346.
https://doi.org/10.3390/su12135346

**AMA Style**

Saeedian M, Pournazarian B, Seyedalipour SS, Eskandari B, Pouresmaeil E.
Emulating Rotational Inertia of Synchronous Machines by a New Control Technique in Grid-Interactive Converters. *Sustainability*. 2020; 12(13):5346.
https://doi.org/10.3390/su12135346

**Chicago/Turabian Style**

Saeedian, Meysam, Bahram Pournazarian, S. Sajjad Seyedalipour, Bahman Eskandari, and Edris Pouresmaeil.
2020. "Emulating Rotational Inertia of Synchronous Machines by a New Control Technique in Grid-Interactive Converters" *Sustainability* 12, no. 13: 5346.
https://doi.org/10.3390/su12135346