# Virtual Inertia Control of Variable Speed Heat Pumps for the Provision of Frequency Support

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Modelling and Control Architecture

#### 2.1. Overview

#### 2.2. Thermal Model of a Residential Building

#### 2.3. Dynamic Model of the Heat Pump

#### 2.4. Enhanced Phase-Locked Loop

#### Small-Signal Model of EPLL

#### 2.5. Virtual Inertia Control

#### 2.6. Induction Machine Power Control

#### 2.7. Overall Small-Signal Model

#### 2.8. Induction Motor Power Controller Design

#### 2.9. Virtual Inertia Control Parameter Selection

#### Selection Based on the Worst Case Scenario

## 3. Hardware Validation and Single Device Characterisation

#### 3.1. Single Device Characteristics

#### 3.2. Modified VSHP System

- Case 4: ${M}_{VI}$ = 300 Ws${}^{2}$${D}_{VI}$ = 1000 Ws.
- Case 5: ${M}_{VI}$ = 100 Ws${}^{2}$${D}_{VI}$ = 1000 Ws.
- Case 6: ${M}_{VI}$ = 0 Ws${}^{2}$${D}_{VI}$ = 1000 Ws.
- Case 7: ${M}_{VI}$ = 100 Ws${}^{2}$${D}_{VI}$ = 250 Ws.
- Case 8: ${M}_{VI}$ = 0 Ws${}^{2}$${D}_{VI}$ = 250 Ws.
- Case 9: ${M}_{VI}$ = 300 Ws${}^{2}$${D}_{VI}$ = 250 Ws.

#### 3.3. Validation of VSHP Small-Signal Model

## 4. Case Study: Aggregated Response from a Population of Heat Pumps

#### 4.1. Simulation Results in the Test Distribution System

#### 4.1.1. Droop vs. Virtual Inertia Controller

#### 4.1.2. Increased Virtual Inertia Controller Installations

#### 4.1.3. Different Outdoor Temperatures

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**A simplified schematic diagram of the VSHP model with VSMcontrol and indoor air temperature control. PFC, power factor correction; ETP, equivalent thermal parameter.

**Figure 3.**Single-phase enhanced phase-locked loop (EPLL) system with a direct estimation of amplitude, phase, frequency, and the rate of change of frequency (RoCoF).

**Figure 6.**Bode plot: (

**a**) open loop transfer function from the change in ${\widehat{\Gamma}}_{e,ref}$ to the resulting change in measured DC link power ${\widehat{P}}_{LPF}^{DC}$; (

**b**) closed loop transfer function from a change in ${\widehat{P}}_{ref}$ to the resulting change in DC power ${\widehat{P}}_{LPF}^{DC}$.

**Figure 10.**Comparison of the small-signal model with the hardware setup. (

**a**) Case 10. (

**b**) Case 11. (

**c**) Case 12. PHIL, power hardware in the loop.

**Figure 13.**Response of the distribution system for different virtual inertia controller installations.

**Table 1.**Building model parameters [5].

C (J/°C) | ${\mathit{R}}_{1}$ (°C/W) |
---|---|

3600 | 0.121 |

Number | Equation | Description |
---|---|---|

(25) | ${\widehat{\Gamma}}_{e,ref}$ = $\left(\frac{{K}_{ppc}s+{K}_{ipc}}{s}\right)({\widehat{P}}_{ref}-{\widehat{P}}_{LPF}^{DC})$ | PI controller for power to torque reference |

(26) | ${\widehat{i}}_{sq,ref}$ = $\frac{2(1+{\sigma}_{r})}{3{L}_{m}{i}_{mr}}{\widehat{\Gamma}}_{e,ref}$ | Relation between reference q-axis component |

of stator current and reference electrical torque | ||

(27) | ${\widehat{i}}_{sq}$ = $\frac{1}{{\tau}_{i}s+1}{\widehat{i}}_{sq,ref}$ | First-order system representing current controller response |

(28) | ${\widehat{\Gamma}}_{e}$ = $\frac{3{L}_{m}{i}_{mr}}{2(1+{\sigma}_{r})}{\widehat{i}}_{sq}$ | Relation between electrical torque |

and the q-axis component of stator current | ||

(31) | ${\widehat{\omega}}_{r}$ = $\frac{1}{sJ+B+\frac{{\widehat{\Gamma}}_{m}}{{\widehat{\omega}}_{r}}}{\widehat{\Gamma}}_{e}$ | Relation between compressor speed |

and electrical torque | ||

(37) | ${\widehat{P}}_{DC}$ = 3$({i}_{sq0}({L}_{a}s+{R}_{a})$ + ${v}_{sq0})$${\widehat{i}}_{sq}$ | Equation for DC bus power variation from |

+$3{L}_{s}{i}_{sq0}{i}_{sd0}{\widehat{\omega}}_{r}$ | ${\widehat{\omega}}_{r}$ and ${\widehat{i}}_{sq}$ | |

(38) | ${\widehat{P}}_{LPF}^{DC}$ = $\frac{1}{{\tau}_{LPF}^{power}s+1}{\widehat{P}}_{DC}$ | Filter used for measuring the DC bus power |

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

${n}_{w0}$ | 0.004 | - | ${n}_{w1}$ | 0.004 | - |

${d}_{w0}$ | 4 | - | ${d}_{w1}$ | 5 | - |

${\Gamma}_{m0}$ | 1 | Nm | ${\omega}_{r0}$ | 125.664 | rads${}^{-1}$ |

${K}_{1}$ | 100 | - | ${K}_{2}$ | 10,000 | - |

${K}_{3}$ | 400 | - | ${\tau}_{LPF}^{f}$ | 0.050 | s |

${\tau}_{LPF}^{RoCoF}$ | 0.100 | s | $\rho $ | 0.050 | s |

${M}_{VI}$ | 750 | Ws${}^{2}$ | ${D}_{VI}$ | 500 | Ws |

${\tau}_{LPF}^{power}$ | 0.001 | s | ${R}_{s}$ | 2.200 | Ω |

${R}_{r}$ | 1.940 | Ω | ${L}_{m}$ | 271.780 | mH |

${L}_{s}$ | 288.780 | mH | ${L}_{r}$ | 288.780 | mH |

J | 0.0127 | kgm${}^{2}$ | B | 0.080 | Nms |

f | 50 | Hz | ${\tau}_{i}$ | 0.003 | s |

${i}_{sq0}$ | 2 | A | ${i}_{sd0}$ | 3.420 | A |

${v}_{sq0}$ | 41.667 | V | ${i}_{mr}$ | 3.420 | A |

${\mathit{T}}_{\mathit{o}}$ (°C) | $\Delta {\mathit{\omega}}_{\mathit{max}}$ (rpm) | $\Delta {\mathit{P}}_{{\mathit{ref}}_{\mathit{max}}}$ (W) | $\Delta {\mathit{KE}}_{\mathit{max}}$ (Ws) |
---|---|---|---|

5 | 113 | 237.5 | 50.356 |

0 | 358.140 | 500 | 109.100 |

−5 | 608 | 761 | 187.534 |

−10 | 843 | 1022 | 285.711 |

${\mathit{T}}_{\mathit{o}}$ (°C) | $\Delta {\mathit{P}}_{{\mathit{ref}}_{\mathit{max}}}$ (W) | $\Delta {\mathit{KE}}_{\mathit{max}}$ (Ws) | ${\mathit{D}}_{\mathit{VI}}$ (Ws) | ${\mathit{M}}_{\mathit{VI}}$ (W s${}^{2}$) |
---|---|---|---|---|

5 | 237.500 | 50.360 | 118.750 | 169.110 |

0 | 500 | 109.100 | 250 | 359.100 |

−5 | 76 | 187.530 | 380.500 | 568.030 |

−10 | 1022 | 285.710 | 511 | 796.710 |

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

${P}_{frac},{Q}_{frac}$ | Fraction of total active or reactive load represented by the static model | 1, 1 |

${K}_{pi},{K}_{qi}$ | Per unit of active or reactive load that is constant current | 0.100, 0.100 |

${K}_{pc},{K}_{qc}$ | Per unit of active or reactive load that is constant power | 0.200, 0.200 |

${K}_{pz},{K}_{qz}$ | Per unit of active or reactive load that is constant impedance | 0.200, −0.252 |

${K}_{p1},{K}_{q1}$ | Per unit of active or reactive load that is voltage and frequency sensitive (Term 1) | 0.100, 0.384 |

${K}_{p2},{K}_{q2}$ | Per unit of active or reactive load that is voltage and frequency sensitive (Term 2) | 0.400, 0.572 |

${n}_{pv1},{n}_{qv1}$ | Voltage sensitivity exponent (Term 1) | 1, 3 |

${n}_{pv2},{n}_{qv2}$ | Voltage sensitivity exponent (Term 2) | 0.100, 0.500 |

${n}_{pf1},{n}_{qf1}$ | Frequency sensitivity (Term 1) | 1, −2.800 |

${n}_{pf2},{n}_{qf2}$ | Frequency sensitivity (Term 2) | 1.900, 1.200 |

Parameters | Values | Units | Parameters | Values | Units |
---|---|---|---|---|---|

${P}_{gen}^{nom}$ | 2 | MW | D | 0.940 | MWs |

${P}_{base}$ | 2 | MW | R | −0.800 | MWs |

${P}_{load}$ | 1.210 | MW | ${\tau}_{t}$ | 0.020 | s |

M | 4 | MWs${}^{2}$ | ${\tau}_{g}$ | 0.010 | s |

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## Share and Cite

**MDPI and ACS Style**

Ibrahim, I.; O’Loughlin, C.; O’Donnell, T.
Virtual Inertia Control of Variable Speed Heat Pumps for the Provision of Frequency Support. *Energies* **2020**, *13*, 1863.
https://doi.org/10.3390/en13081863

**AMA Style**

Ibrahim I, O’Loughlin C, O’Donnell T.
Virtual Inertia Control of Variable Speed Heat Pumps for the Provision of Frequency Support. *Energies*. 2020; 13(8):1863.
https://doi.org/10.3390/en13081863

**Chicago/Turabian Style**

Ibrahim, Ismail, Cathal O’Loughlin, and Terence O’Donnell.
2020. "Virtual Inertia Control of Variable Speed Heat Pumps for the Provision of Frequency Support" *Energies* 13, no. 8: 1863.
https://doi.org/10.3390/en13081863