# State Estimation for Hybrid VSC Based HVDC/AC Transmission Networks

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## Abstract

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## 1. Introduction

- New timescales requirements in data transmission [23];
- Advanced Remote Terminal Unit (RTU) and Intelligent Electronic Devices (IED) characteristics (e.g., hybrid HVDC/AC sensors/transducers) [23];
- Unified Human Machine Interface (HMI) and power system applications for AC and DC network (i.e., modified toolboxes such as the state estimator and fault detection blocks) [1];
- Cyber-security and communication overload traffic [1].

## 2. VSC and SE Toolbox: Literature Review

#### 2.1. Voltage Source Converter (VSC): Modes of Operation

- On the DC side:
- Constant active P: The converter is injecting constant active power to the AC side.
- Constant DC voltage V: The converter aims to maintain the DC bus voltage regulated by keeping the active power flow float (not constant), also referred to as DC Slack bus.
- Droop control of P and ${V}_{dc}$: The converter works on a relation between the real power injected to the DC grid and the voltage level of the same bus.

- On AC side:
- PV: The voltage level is fixed (regulated) while the reactive power is float and adapts the regulator to ensure constant AC bus voltage magnitude.
- PQ: The reactive power is fixed and injected to the AC grid regardless of the AC bus voltage magnitude state.

#### 2.2. The State Estimation Toolbox

## 3. State Estimation and Power Flow: Mathematical Background

#### 3.1. Weighted Least Squares (WLS) Algorithm

#### 3.2. AC Power Flow Calculations

- ${h}_{a{c}_{1}}\left(x\right)$ and ${h}_{a{c}_{2}}\left(x\right)$ for active ${P}_{i}$ and reactive ${Q}_{i}$ power injection at bus i respectively:$$\begin{array}{c}\hfill \begin{array}{c}\hfill {h}_{a{c}_{1}}\left(x\right)={P}_{i}=\sum _{j\u03f5{N}_{i}}{P}_{ij}={V}_{i}\sum _{j\u03f5{N}_{i}}{V}_{j}\left({G}_{{m}_{ij}}\mathrm{cos}\left({\theta}_{ij}\right)+{B}_{{m}_{ij}}\mathrm{sin}\left({\theta}_{ij}\right)\right)\end{array}\end{array}$$$$\begin{array}{c}\hfill \begin{array}{c}\hfill {h}_{a{c}_{2}}\left(x\right)={Q}_{i}=\sum _{j\u03f5{N}_{i}}{Q}_{ij}={V}_{i}\sum _{j\u03f5{N}_{i}}{V}_{j}\left({G}_{{m}_{ij}}\mathrm{sin}\left({\theta}_{ij}\right)-{B}_{{m}_{ij}}\mathrm{cos}\left({\theta}_{ij}\right)\right)\end{array}\end{array}$$
- ${h}_{a{c}_{3}}\left(x\right)$ and ${h}_{a{c}_{4}}\left(x\right)$ for active ${P}_{ij}$ and reactive ${Q}_{ij}$ power flow from bus i to j respectively:$$\begin{array}{c}\hfill {h}_{a{c}_{3}}\left(x\right)={P}_{ij}={V}_{i}^{2}\left({G}_{{m}_{ij}}+{G}_{s{h}_{i}}\right)-{V}_{i}{V}_{j}\left({G}_{{m}_{ij}}\mathrm{cos}\left({\theta}_{ij}\right)+{B}_{{m}_{ij}}\mathrm{sin}\left({\theta}_{ij}\right)\right)\end{array}$$$$\begin{array}{c}\hfill {h}_{a{c}_{4}}\left(x\right)={Q}_{ij}=-{V}_{i}^{2}\left({B}_{{m}_{ij}}+{B}_{s{h}_{i}}\right)-{V}_{i}{V}_{j}\left({G}_{{m}_{ij}}\mathrm{sin}\left({\theta}_{ij}\right)-{B}_{{m}_{ij}}\mathrm{cos}\left({\theta}_{ij}\right)\right)\end{array}$$
- ${h}_{a{c}_{5}}\left(x\right)$ for current flow ${I}_{ij}$ from bus i to j:$${h}_{a{c}_{5}}\left(x\right)={I}_{ij}=\frac{\sqrt{{P}_{ij}^{2}+{Q}_{ij}^{2}}}{{V}_{i}}$$

#### 3.3. DC Power Flow Calculations

- ${h}_{d{c}_{1}}\left(x\right)$ for real ${P}_{i}$ power injection at bus i:$$\begin{array}{c}\hfill {h}_{d{c}_{1}}\left(x\right)={P}_{i}=\sum _{j\u03f5{N}_{i}}{P}_{ij}={V}_{i}\sum _{j\u03f5{N}_{i}}{V}_{j}\left(p{G}_{{m}_{ij}}\right)\end{array}$$$${h}_{d{c}_{1}}\left(x\right)={P}_{{i}_{0}}-\frac{1}{{K}_{i}}({V}_{i}-{V}_{{i}_{0}})$$$$p=\left\{\begin{array}{cc}1,\hfill & \mathrm{for}\phantom{\rule{4.pt}{0ex}}\mathrm{monopolar}\phantom{\rule{4.pt}{0ex}}\mathrm{systems}\hfill \\ 2,\hfill & \mathrm{for}\phantom{\rule{4.pt}{0ex}}\mathrm{bipolar}\phantom{\rule{4.pt}{0ex}}\mathrm{systems}\hfill \end{array}\right.$$
- ${h}_{d{c}_{2}}\left(x\right)$ for real ${P}_{ij}$ power flow from bus i to j:$${h}_{d{c}_{2}}\left(x\right)={P}_{ij}={V}_{i}^{2}\left(p{G}_{{m}_{ij}}+{G}_{s{h}_{i}}\right)-{V}_{i}{V}_{j}\left(p{G}_{{m}_{ij}}\right)$$
- ${h}_{d{c}_{3}}\left(x\right)$ for current flow ${I}_{ij}$ from bus i to j:$${h}_{d{c}_{3}}\left(x\right)={I}_{ij}=\frac{Pd{c}_{ij}}{Vd{c}_{i}}=p{G}_{{m}_{ij}}({V}_{i}-{V}_{j})$$

#### 3.4. Converter Power Coupling

- a has a typical value of $11.033\times {10}^{-3}$ pu and it represents the no load losses of transformers and averaged axillary equipment losses, such as heating and cooling losses;
- b has a typical value of $3.464\times {10}^{-3}$ pu and it represents the switching losses of valves and freewheeling diodes;
- c represents the conduction losses of the valves and depends on the operating condition of the converter (rectifier or inverter). it is typical values are:$$c=\left\{\begin{array}{cc}4.4\times {10}^{-3}\phantom{\rule{3.33333pt}{0ex}}\mathrm{pu},\hfill & \mathrm{for}\phantom{\rule{4.pt}{0ex}}\mathrm{rectifiers}\hfill \\ 6.67\times {10}^{-3}\phantom{\rule{3.33333pt}{0ex}}\mathrm{pu},\hfill & \mathrm{for}\phantom{\rule{4.pt}{0ex}}\mathrm{inverters}\hfill \end{array}\right.$$

#### 3.5. Converter Voltage Coupling

## 4. The Unified WLS for HV-(AC/VSC-DC/AC) Transmission Network

- $zAC$ are the AC side measurements and can be in the form of $Vm,Va,{P}_{inj},{Q}_{inj},{P}_{flow},{Q}_{flow},{I}_{flow}$;
- $zDC$ are the DC side measurements and can be in the form of $Vm,{P}_{inj},{P}_{flow},{I}_{flow}$;
- $zConv$ are zero measurements and represent the right side of converter power coupling constraints;
- ${M}_{factor}$ are the ${V}_{dc}$ to ${V}_{ac}$ ratios (measurements), calculated and transmitted by the converter;
- n, m and k are the number of AC systems, DC systems and converters respectively.

- ${H}_{AC-AC}$ is the partial derivative of $hac\left(x\right)$ to $\theta $ and ${V}_{ac}$
- ${H}_{DC-AC}$ is the partial derivative of $hdc\left(x\right)$ to $\theta $ and ${V}_{ac}$↔ 0-matrix
- ${H}_{AC-DC}$ is the partial derivative of $hac\left(x\right)$ to ${V}_{dc}$↔ 0-matrix
- ${H}_{DC-DC}$ is the partial derivative of $hdc\left(x\right)$ to ${V}_{dc}$
- ${H}_{Conv-AC}$ is the partial derivative of the power coupling constraint to $\theta $ and ${V}_{ac}$
- ${H}_{Conv-DC}$ is the partial derivative of the power coupling constraint to ${V}_{dc}$
- ${H}_{M-AC}$ is the partial derivative of ${M}_{factor}$ to $\theta $ and ${V}_{ac}$
- ${H}_{M-DC}$ is the partial derivative of ${M}_{factor}$ to ${V}_{dc}$

#### 4.1. Converter Components: Power Coupling

#### 4.2. Converter Components: Voltage Coupling

## 5. Decentralized vs. Power Coupling vs. Unified: State Estimation Simulations

- Decentralized: The systems are assumed to be separated with no data/measurements exchange (no coupling). The Jacobian matrix is reduced to contain only AC and DC components, and multi-thread WLSs are run for each AC and DC system.
- Converter P-Coupling: A single WLS is run centralized with the converter power coupling constraints added to the Jacobian matrix.
- Converter PV-Coupling (unified): The power constraints and voltage coupling measurements are taken into the centralized WLS estimation, forming the unified estimation approach.

#### 5.1. Hybrid VSC-HVDC/AC Networks Test Cases

#### 5.1.1. Four(4)-AC/Four(4)-DC/Four(4)-AC Network

#### 5.1.2. Cigre B4 AC/DC/AC Network

## 6. Main Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AC | Alternating Current |

DC | Direct Current |

EMS | Energy Management Systems |

HVDC | High Voltage Direct Current |

IED | Intelligent Electronic Devices |

LAV | Least Absolute Value |

MAE | Mean Absolute Error |

PCC | Point of Common Coupling |

PMU | Phasor Measurements Unit |

RTU | Remote Terminal Unit |

SCADA | Supervisory, Control and Data Acquisition |

TFR | Transformer, Filter and Reactor |

VSC | Voltage Source Converter |

WLS | Weighted Least Squares |

## Appendix A

#### Appendix A.1. AC Components

#### Appendix A.2. DC Components

## Appendix B

#### Appendix B.1. AC SE of Case 1

Sys. # | Bus # | ${\mathit{V}}_{\mathit{True}}$ | ${\mathit{V}}_{\mathit{Est}}$ | ${\mathit{V}}_{\mathit{Err}}$ | ${\mathit{\theta}}_{\mathit{True}}$ | ${\mathit{\theta}}_{\mathit{Est}}$ | ${\mathit{\theta}}_{\mathit{Err}}$ |
---|---|---|---|---|---|---|---|

1 | 1.0 | 1.0600 | 1.0600 | −0.0000 | 0.0000 | 0.0000 | 0.0000 |

2.0 | 1.0569 | 1.0569 | −0.0000 | −0.0278 | −0.0277 | 0.0001 | |

3.0 | 1.0551 | 1.0551 | −0.0000 | −0.0601 | −0.0599 | 0.0002 | |

4.0 | 1.0540 | 1.0540 | −0.0000 | −0.0612 | −0.0610 | 0.0002 | |

2 | 5.0 | 1.0615 | 1.0615 | 0.0000 | 0.0004 | 0.0003 | 0.0001 |

6.0 | 1.0600 | 1.0600 | 0.0000 | 0.0000 | −0.0000 | 0.0000 | |

7.0 | 1.0433 | 1.0432 | 0.0001 | −0.0353 | −0.0350 | −0.0003 | |

8.0 | 1.0289 | 1.0287 | 0.0002 | −0.0721 | −0.0714 | −0.0007 |

Sys. # | Bus # | ${\mathit{V}}_{\mathit{True}}$ | ${\mathit{V}}_{\mathit{Est}}$ | ${\mathit{V}}_{\mathit{Err}}$ | ${\mathit{\theta}}_{\mathit{True}}$ | ${\mathit{\theta}}_{\mathit{Est}}$ | ${\mathit{\theta}}_{\mathit{Err}}$ |
---|---|---|---|---|---|---|---|

1 | 1.0 | 1.0600 | 1.0600 | −0.0000 | 0.0000 | 0.0000 | 0.0000 |

2.0 | 1.0569 | 1.0569 | 0.0000 | −0.0278 | −0.0278 | −0.0000 | |

3.0 | 1.0551 | 1.0551 | 0.0000 | −0.0601 | −0.0601 | −0.0000 | |

4.0 | 1.0540 | 1.0540 | 0.0000 | −0.0612 | −0.0612 | −0.0000 | |

2 | 5.0 | 1.0615 | 1.0615 | 0.0000 | 0.0004 | 0.0004 | 0.0000 |

6.0 | 1.0600 | 1.0600 | 0.0000 | 0.0000 | −0.0000 | 0.0000 | |

7.0 | 1.0433 | 1.0432 | 0.0001 | −0.0353 | −0.0351 | −0.0002 | |

8.0 | 1.0289 | 1.0287 | 0.0002 | −0.0721 | −0.0715 | −0.0006 |

Sys. # | Bus # | ${\mathit{V}}_{\mathit{True}}$ | ${\mathit{V}}_{\mathit{Est}}$ | ${\mathit{V}}_{\mathit{Err}}$ | ${\mathit{\theta}}_{\mathit{True}}$ | ${\mathit{\theta}}_{\mathit{Est}}$ | ${\mathit{\theta}}_{\mathit{Err}}$ |
---|---|---|---|---|---|---|---|

1 | 1.0 | 1.0600 | 1.0600 | −0.0000 | 0.0000 | 0.0000 | 0.0000 |

2.0 | 1.0569 | 1.0569 | 0.0000 | −0.0278 | −0.0278 | −0.0000 | |

3.0 | 1.0551 | 1.0551 | 0.0000 | −0.0601 | −0.0601 | −0.0000 | |

4.0 | 1.0540 | 1.0540 | 0.0000 | −0.0612 | −0.0612 | −0.0000 | |

2 | 5.0 | 1.0615 | 1.0615 | 0.0000 | 0.0004 | 0.0004 | 0.0000 |

6.0 | 1.0600 | 1.0600 | 0.0000 | 0.0000 | −0.0000 | 0.0000 | |

7.0 | 1.0433 | 1.0433 | 0.0000 | −0.0353 | −0.0351 | −0.0002 | |

8.0 | 1.0289 | 1.0287 | 0.0002 | −0.0721 | −0.0715 | −0.0006 |

#### Appendix B.2. AC SE of Case 2

Sys. # | Bus # | ${\mathit{V}}_{\mathit{True}}$ | ${\mathit{V}}_{\mathit{Est}}$ | ${\mathit{V}}_{\mathit{Err}}$ | ${\mathit{\theta}}_{\mathit{True}}$ | ${\mathit{\theta}}_{\mathit{Est}}$ | ${\mathit{\theta}}_{\mathit{Err}}$ |
---|---|---|---|---|---|---|---|

1 | 1.0 | 1.0000 | 1.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

2.0 | 1.0000 | 1.0000 | 0.0000 | −0.1157 | −0.1161 | 0.0004 | |

3.0 | 1.0003 | 1.0003 | 0.0000 | −0.1154 | −0.1159 | 0.0005 | |

4.0 | 0.9970 | 1.9970 | 0.0000 | −0.1187 | −0.1192 | 0.0005 | |

2 | 5.0 | 1.0000 | 1.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

6.0 | 0.9969 | 0.9969 | 0.0000 | −0.0398 | −0.0394 | −0.0004 | |

7.0 | 0.9984 | 0.9984 | 0.0000 | −0.0382 | −0.0379 | −0.0003 | |

8.0 | 0.9968 | 0.9968 | 0.0000 | −0.0329 | −0.0327 | −0.0002 | |

9.0 | 0.9985 | 0.9986 | −0.0000 | −0.0312 | −0.0310 | −0.0002 | |

10.0 | 0.9967 | 0.9967 | −0.0000 | −0.0330 | −0.0328 | −0.0002 | |

11.0 | 0.9965 | 0.9965 | −0.0000 | −0.0407 | −0.0406 | −0.0001 | |

12.0 | 0.9973 | 0.9973 | −0.0000 | −0.0399 | −0.0398 | −0.0001 | |

3 | 13.0 | 1.0000 | 1.0000 | −0.0000 | 0.0000 | 0.0000 | 0.0000 |

14.0 | 0.9995 | 0.9995 | −0.0000 | −0.0005 | −0.0005 | 0.0000 | |

15.0 | 1.0000 | 1.0000 | −0.0000 | 0.0000 | −0.0000 | 0.0000 | |

16.0 | 0.9995 | 0.9995 | −0.0000 | −0.0005 | −0.0005 | 0.0000 | |

4 | 17.0 | 1.0000 | 1.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

18.0 | 0.9990 | 0.9990 | 0.0000 | −0.0010 | −0.0010 | 0.0000 | |

5 | 19.0 | 1.0000 | 1.0000 | −0.0000 | 0.0000 | 0.0000 | 0.0000 |

20.0 | 1.0001 | 1.0001 | 0.0000 | 0.0001 | 0.0001 | 0.0000 | |

6 | 21.0 | 1.0000 | 1.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

22.0 | 0.9995 | 0.9995 | 0.0000 | −0.0005 | −0.0005 | 0.0000 |

Sys. # | Bus # | ${\mathit{V}}_{\mathit{True}}$ | ${\mathit{V}}_{\mathit{Est}}$ | ${\mathit{V}}_{\mathit{Err}}$ | ${\mathit{\theta}}_{\mathit{True}}$ | ${\mathit{\theta}}_{\mathit{Est}}$ | ${\mathit{\theta}}_{\mathit{Err}}$ |
---|---|---|---|---|---|---|---|

1 | 1.0 | 1.0000 | 1.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

2.0 | 1.0000 | 1.0000 | 0.0000 | −0.1157 | −0.1161 | 0.0004 | |

3.0 | 1.0003 | 1.0003 | 0.0000 | −0.1154 | −0.1158 | 0.0004 | |

4.0 | 0.9970 | 1.9970 | 0.0000 | −0.1187 | −0.1191 | 0.0004 | |

2 | 5.0 | 1.0000 | 1.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

6.0 | 0.9969 | 0.9969 | 0.0000 | −0.0398 | −0.0395 | −0.0003 | |

7.0 | 0.9984 | 0.9984 | 0.0000 | −0.0382 | −0.0380 | −0.0002 | |

8.0 | 0.9968 | 0.9968 | 0.0000 | −0.0329 | −0.0328 | −0.0001 | |

9.0 | 0.9985 | 0.9986 | −0.0000 | −0.0312 | −0.0311 | −0.0001 | |

10.0 | 0.9967 | 0.9967 | −0.0000 | −0.0330 | −0.0329 | −0.0001 | |

11.0 | 0.9965 | 0.9965 | −0.0000 | −0.0407 | −0.0406 | −0.0001 | |

12.0 | 0.9973 | 0.9973 | −0.0000 | −0.0399 | −0.0398 | −0.0001 | |

3 | 13.0 | 1.0000 | 1.0000 | −0.0000 | 0.0000 | 0.0000 | 0.0000 |

14.0 | 0.9995 | 0.9995 | −0.0000 | −0.0005 | −0.0005 | 0.0000 | |

15.0 | 1.0000 | 1.0000 | −0.0000 | 0.0000 | −0.0000 | 0.0000 | |

16.0 | 0.9995 | 0.9995 | −0.0000 | −0.0005 | −0.0005 | 0.0000 | |

4 | 17.0 | 1.0000 | 1.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

18.0 | 0.9990 | 0.9990 | 0.0000 | −0.0010 | −0.0010 | 0.0000 | |

5 | 19.0 | 1.0000 | 1.0000 | −0.0000 | 0.0000 | 0.0000 | 0.0000 |

20.0 | 1.0001 | 1.0001 | 0.0000 | 0.0001 | 0.0001 | 0.0000 | |

6 | 21.0 | 1.0000 | 1.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

22.0 | 0.9995 | 0.9995 | 0.0000 | −0.0005 | −0.0005 | 0.0000 |

Sys. # | Bus # | ${\mathit{V}}_{\mathit{True}}$ | ${\mathit{V}}_{\mathit{Est}}$ | ${\mathit{V}}_{\mathit{Err}}$ | ${\mathit{\theta}}_{\mathit{True}}$ | ${\mathit{\theta}}_{\mathit{Est}}$ | ${\mathit{\theta}}_{\mathit{Err}}$ |
---|---|---|---|---|---|---|---|

1 | 1.0 | 1.0000 | 1.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

2.0 | 1.0000 | 1.0000 | 0.0000 | −0.1157 | −0.1161 | 0.0004 | |

3.0 | 1.0003 | 1.0003 | 0.0000 | −0.1154 | −0.1158 | 0.0004 | |

4.0 | 0.9970 | 1.9970 | 0.0000 | −0.1187 | −0.1190 | 0.0003 | |

2 | 5.0 | 1.0000 | 1.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

6.0 | 0.9969 | 0.9969 | 0.0000 | −0.0398 | −0.0395 | −0.0003 | |

7.0 | 0.9984 | 0.9984 | 0.0000 | −0.0382 | −0.0380 | −0.0002 | |

8.0 | 0.9968 | 0.9968 | 0.0000 | −0.0329 | −0.0328 | −0.0001 | |

9.0 | 0.9985 | 0.9986 | −0.0000 | −0.0312 | −0.0311 | −0.0001 | |

10.0 | 0.9967 | 0.9967 | −0.0000 | −0.0330 | −0.0329 | −0.0001 | |

11.0 | 0.9965 | 0.9965 | −0.0000 | −0.0407 | −0.0407 | −0.0000 | |

12.0 | 0.9973 | 0.9973 | −0.0000 | −0.0399 | −0.0399 | −0.0000 | |

3 | 13.0 | 1.0000 | 1.0000 | −0.0000 | 0.0000 | 0.0000 | 0.0000 |

14.0 | 0.9995 | 0.9995 | −0.0000 | −0.0005 | −0.0005 | 0.0000 | |

15.0 | 1.0000 | 1.0000 | −0.0000 | 0.0000 | −0.0000 | 0.0000 | |

16.0 | 0.9995 | 0.9995 | −0.0000 | −0.0005 | −0.0005 | 0.0000 | |

4 | 17.0 | 1.0000 | 1.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

18.0 | 0.9990 | 0.9990 | 0.0000 | −0.0010 | −0.0010 | 0.0000 | |

5 | 19.0 | 1.0000 | 1.0000 | −0.0000 | 0.0000 | 0.0000 | 0.0000 |

20.0 | 1.0001 | 1.0001 | 0.0000 | 0.0001 | 0.0001 | 0.0000 | |

6 | 21.0 | 1.0000 | 1.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

22.0 | 0.9995 | 0.9995 | 0.0000 | −0.0005 | −0.0005 | 0.0000 |

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Measurement Type | Weight |
---|---|

voltages/angles, power coupling and zero injections constraints | $1\times {10}^{-6}$ |

power flow and ${M}_{factor}$ | $1\times {10}^{-5}$ |

±power injections | $1\times {10}^{-4}$ |

Sys. # | Meas. Type | True Meas. | Noisy Meas. | T1 Estimation | T2 Estimation | T3 Estimation |
---|---|---|---|---|---|---|

1 | ${V}_{dc}$ @ bus 1 (slack) | 1.0100 | 1.0100 | 1.0100 | 1.0100 | 1.0100 |

${P}_{inj}$ @ bus 1 | 0.111280 | 0.112209 | 0.111653 | 0.111236 | 0.111235 | |

${P}_{inj}$ @ bus 2 | −0.111202 | −0.109628 | −0.111563 | −0.111151 | −0.11116 | |

${P}_{inj}$ @ bus 3 | 0.0001 * | $9.9022\times {10}^{-5}$ | $9.4749\times {10}^{-5}$ | $9.6284\times {10}^{-5}$ | $9.6288\times {10}^{-5}$ | |

${P}_{inj}$ @ bus 4 | 0.0 | 0.0 | $-5.5783\times {10}^{-6}$ | $-4.041\times {10}^{-6}$ | $-4.038\times {10}^{-6}$ |

Sys. # | Meas. Type | True Meas. | Noisy Meas. | T1 Estimation | T2 Estimation | T3 Estimation |
---|---|---|---|---|---|---|

1 | ${V}_{ac}$ @ bus 1 (slack) | 1.0600 | 1.0600 | 1.0600 | 1.0600 | 1.0600 |

${P}_{inj}$ @ bus 1 | 0.122963 | 0.121141 | 0.122597 | 0.122877 | 0.122895 | |

${P}_{inj}$ @ bus 4 | −0.122493 | −0.123621 | −0.122159 | −0.122443 | −0.122444 | |

${Q}_{inj}$ @ bus 1 | 0.00753466 | 0.00756398 | 0.00752762 | 0.00754452 | 0.00753644 | |

${Q}_{inj}$ @ bus 4 | 0.0 | 0.0 | $-3.5946\times {10}^{-5}$ | $-1.865\times {10}^{-5}$ | $-1.0725\times {10}^{-5}$ | |

2 | ${V}_{ac}$ @ bus 6 (slack) | 1.0600 | 1.0600 | 1.0600 | 1.0600 | 1.0600 |

${P}_{inj}$ @ bus 5 | 0.1 | 0.101183 | 0.0997715 | 0.0999448 | 0.0999446 | |

${P}_{inj}$ @ bus 6 | 0.0505456 | 0.0509225 | 0.0495076 | 0.0498217 | 0.0498781 | |

${P}_{inj}$ @ bus 8 | −0.15 | −0.147212 | −0.148724 | −0.14881 | −0.14882 | |

${Q}_{inj}$ @ bus 5 | 0.06 | 0.0614694 | 0.0608747 | 0.060879 | 0.0608781 | |

${Q}_{inj}$ @ bus 6 | 0.00262294 | 0.00261144 | 0.00201452 | 0.00201872 | 0.00202918 | |

${Q}_{inj}$ @ bus 8 | −0.05 | −0.0496862 | −0.0504239 | −0.0503197 | −0.0503192 |

Conv. # | Meas. Type | True Meas. | Noisy Meas. | Estimated Meas. |
---|---|---|---|---|

1 | Power constraint | 0.0 | - | $5.7070\times {10}^{-6}$ |

${M}_{factor}$ | 0.67685604 | 0.67653596 | 0.67674968 | |

2 | Power constraint | 0.0 | - | $-2.5849\times {10}^{-6}$ |

${M}_{factor}$ | 0.67268026 | 0.671752966 | 0.67257118 |

Bus # | ${\mathit{V}}_{\mathit{True}}$ | ${\mathit{V}}_{\mathit{Est}}$ | ${\mathit{V}}_{\mathit{Err}}$ |
---|---|---|---|

1.0 | 1.0100 | 1.0100 | 0.0000 |

2.0 | 1.0084 | 1.0084 | −0.0000 |

3.0 | 1.0087 | 1.0087 | −0.0000 |

4.0 | 1.0094 | 1.0094 | −0.0000 |

Meas. Type | Count | Details |
---|---|---|

AC | 6 | voltages, 1 per AC system |

15 & 15 | active & reactive power injection | |

8 & 8 | active & reactive power flow | |

14 | zero injection | |

DC | 2 | voltages, 1 per DC system |

11 | real power injection | |

4 | real power flow | |

4 | zero injection | |

Conv. Power Coupling | 11 | power constraints, 1 per converter |

Conv. Voltage Coupling | 11 | ${M}_{factor}$, 1 per converter |

Conv. # | Meas. Type | True Meas. | Noisy Meas. | Estimated Meas. |
---|---|---|---|---|

1 | Power constraint | 0.0 | - | $-8.98316\times {10}^{-6}$ |

${M}_{factor}$ | 0.707107 | 0.707506 | 0.707281 | |

2 | Power constraint | 0.0 | - | $8.24138\times {10}^{-6}$ |

${M}_{factor}$ | 0.714178 | 0.716779 | 0.716452 | |

3 | Power constraint | 0.0 | - | $5.70049\times {10}^{-5}$ |

${M}_{factor}$ | 0.71367 | 0.712683 | 0.712631 | |

4 | Power constraint | 0.0 | - | $5.57066\times {10}^{-5}$ |

${M}_{factor}$ | 0.710751 | 0.709244 | 0.709355 | |

5 | Power constraint | 0.0 | - | $1.95246\times {10}^{-5}$ |

${M}_{factor}$ | 0.70977 | 0.709629 | 0.709617 | |

6 | Power constraint | 0.0 | - | $1.25227\times {10}^{-6}$ |

${M}_{factor}$ | 0.710009 | 0.709829 | 0.709526 | |

7 | Power constraint | 0.0 | - | $-1.54079\times {10}^{-6}$ |

${M}_{factor}$ | 0.707588 | 0.706818 | 0.707458 | |

8 | Power constraint | 0.0 | - | $7.29433\times {10}^{-6}$ |

${M}_{factor}$ | 0.714181 | 0.712509 | 0.713995 | |

9 | Power constraint | 0.0 | - | $3.10264\times {10}^{-6}$ |

${M}_{factor}$ | 0.713775 | 0.712258 | 0.713975 | |

10 | Power constraint | 0.0 | - | $4.40378\times {10}^{-6}$ |

${M}_{factor}$ | 0.711713 | 0.711656 | 0.71159 | |

11 | Power constraint | 0.0 | - | $6.37395\times {10}^{-6}$ |

${M}_{factor}$ | 0.710957 | 0.711459 | 0.71128 |

Sys. # | Bus # | ${\mathit{V}}_{\mathit{True}}$ | ${\mathit{V}}_{{\mathit{Est}}_{\mathit{T}1}}$ | ${\mathit{V}}_{{\mathit{Err}}_{\mathit{T}1}}$ | ${\mathit{V}}_{{\mathit{Est}}_{\mathit{T}2}}$ | ${\mathit{V}}_{{\mathit{Err}}_{\mathit{T}2}}$ | ${\mathit{V}}_{{\mathit{Est}}_{\mathit{T}3}}$ | ${\mathit{V}}_{{\mathit{Err}}_{\mathit{T}3}}$ |
---|---|---|---|---|---|---|---|---|

1 | 1.0 | 1.0000 | 1.0000 | 0.0000 | 1.0000 | 0.0000 | 1.0000 | 0.0000 |

2.0 | 1.0007 | 1.0007 | 0.0000 | 1.0007 | 0.0000 | 1.0007 | 0.0000 | |

2 | 3.0 | 1.0100 | 1.0100 | 0.0000 | 1.0100 | 0.0000 | 1.0100 | 0.0000 |

4.0 | 1.0100 | 1.0100 | 0.0000 | 1.0100 | 0.0000 | 1.0100 | 0.0000 | |

5.0 | 1.0094 | 1.0094 | −0.0000 | 1.0094 | −0.0000 | 1.0094 | −0.0000 | |

6.0 | 1.0079 | 1.0078 | −0.0001 | 1.0079 | −0.0000 | 1.0079 | −0.0000 | |

7.0 | 1.0073 | 1.0073 | −0.0000 | 1.0073 | −0.0000 | 1.0073 | −0.0000 | |

8.0 | 1.0062 | 1.0062 | 0.0000 | 1.0062 | 0.0000 | 1.0062 | 0.0000 | |

9.0 | 1.0061 | 1.0061 | −0.0000 | 1.0061 | −0.0000 | 1.0061 | −0.0000 | |

10.0 | 1.0020 | 1.0020 | −0.0000 | 1.0020 | −0.0000 | 1.0020 | 0.0000 | |

11.0 | 1.0006 | 1.0005 | −0.0001 | 1.0006 | −0.0000 | 1.0006 | 0.0000 | |

12.0 | 1.0006 | 1.0005 | −0.0001 | 1.0006 | −0.0000 | 1.0006 | 0.0000 | |

13.0 | 1.0034 | 1.0033 | −0.0001 | 1.0034 | −0.0000 | 1.0034 | 0.0000 | |

14.0 | 1.0054 | 1.0053 | −0.0001 | 1.0054 | −0.0000 | 1.0054 | 0.0000 | |

15.0 | 1.0065 | 1.0065 | −0.0000 | 1.0065 | −0.0000 | 1.0065 | −0.0000 |

Scenarios | Memory Allocations (in Thousands) | Memory Storage (MB) |
---|---|---|

Decentralized | 89.95 | 2.855 |

P-coupling | 180.96 | 4.720 |

Unified | 185.71 | 5.003 |

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**MDPI and ACS Style**

Ayiad, M.; Leite, H.; Martins, H.
State Estimation for Hybrid VSC Based HVDC/AC Transmission Networks. *Energies* **2020**, *13*, 4932.
https://doi.org/10.3390/en13184932

**AMA Style**

Ayiad M, Leite H, Martins H.
State Estimation for Hybrid VSC Based HVDC/AC Transmission Networks. *Energies*. 2020; 13(18):4932.
https://doi.org/10.3390/en13184932

**Chicago/Turabian Style**

Ayiad, Motaz, Helder Leite, and Hugo Martins.
2020. "State Estimation for Hybrid VSC Based HVDC/AC Transmission Networks" *Energies* 13, no. 18: 4932.
https://doi.org/10.3390/en13184932