# Steady-State Stand-Alone Power Flow Solvers for Integrated Transmission-Distribution Networks: A Comparison Study and Numerical Assessment

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Power Flow Problem in Single-Phase and Three-Phase Representation

## 3. Integration Methods

#### 3.1. Unified Methods

#### 3.1.1. The Full Three-Phase Method

#### 3.1.2. The Interconnected Method

#### Using Current Injections

#### Using Power Injections

#### 3.2. Master–Slave Splitting Methods

#### 3.2.1. Algorithmic Approach

Algorithm 1 The Master-Slave Splitting Iterative Scheme. |

1: Set iteration counter $\nu =0$. Initialize the voltage ${V}_{B}^{0}$ of the distribution system. |

2: Solve the distribution system. Output: ${S}_{B}^{\nu +1}$. |

3: Inject ${S}_{B}^{\nu +1}$ into the transmission system. |

4: Solve the transmission system. Output: ${V}_{B}^{\nu +1}$. |

5: Is $|{V}_{B}^{\nu +1}-{V}_{B}^{\nu}{|}_{1}>{\epsilon}_{MSS}$? Repeat Steps 2 to 5. |

#### 3.2.2. The MSS Homogeneous Method

#### 3.2.3. The MSS-Hybrid Method

#### 3.3. Advantages and Disadvantages

## 4. Numerical Experiments

#### 4.1. Test Case Description

- 1:
- Test case T9-D13
- 2:
- Test case T118-D123
- 3:
- Test case T3120-D8500

#### Connection Bus

#### 4.2. Numerical Performance

#### 4.3. Accuracy

#### 4.4. Voltage Unbalance

#### 4.5. Multiple Distribution Networks

#### 4.6. Distributed Generation

#### 4.7. Speeding Up the Master–Slave Iterative Scheme

## 5. Large Integrated Electricity Systems: Relative Time and Speedup Comparison

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Appendix A. List of Mathematical Symbols, Subscripts, and Superscripts

Symbol | Description | Symbol | Description |
---|---|---|---|

a | Rotation variable, $a={e}^{\frac{2}{3}\pi \iota}$ | P | Active power, real part of S |

b | Line susceptance, imaginary part of y | Q | Reactive power, imaginary part of S |

$\delta $ | Voltage angle | S | Complex power, $S=P+\iota Q$ |

$\u03f5$ | Tolerance value | $\mathbf{T}$ | Transformation matrix |

$\mathbf{F}\left(x\right)$ | Mismatch vector in the Newton–Raphson method | V | Voltage, with angle $\delta $ and magnitude $\left|V\right|$ |

I | Current | $\left|V\right|$ | Voltage magnitude |

$\mathcal{I}$ | Amount of iterations | $\mathbf{x}$ | Vector of variables |

$\iota $ | Imaginary unit | y | Line admittance |

N | Amount of buses in a network | Y | Admittance |

$\pi $ | Exact number | Z | Impedance |

Subscript | Name | Superscript | Name |
---|---|---|---|

B | Boundary bus | a | First phase |

i | Bus index | b | Second phase |

j | (Different) bus index | c | Third phase |

k | Bus index at transmission side | $\nu $ | Iteration counter |

m | Bus index at distribution side | T | Transpose |

M | Master | ||

$MSS$ | Master–slave splitting | ||

p | Phase index | ||

q | (Different) phase index | ||

s | Specified | ||

S | Slave | ||

U | Unified |

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**Figure 1.**The substation transformer in an integrated network connecting a single-phase transmission bus k and a three-phase distribution bus m.

**Figure 3.**Representation of the relative norms of ${\left|F\right|}_{\infty}$, per iteration, of the interconnected and full three-phase methods for three test cases. It is shown that quadratic convergence is preserved (after the first iteration for some of the test cases). The horizontal black dotted line is the tolerance value. Note that the results of the IC methods are not clearly visible because the graphs are below the F3P results.

**Figure 4.**Representation of the relative norms of $|{V}_{B}^{\nu +1}-{V}_{B}^{\nu}{|}_{1}$, per iteration, of the splitting methods for three different test cases. It is shown that the iterative scheme has linear convergence. The horizontal black dotted line is the tolerance value.

**Figure 5.**Per-unit voltage profile of the three phases of the Interconnected and full three-phase method. The represented network is the T9-D13 network.

**Figure 6.**Per-unit voltage profile of the three phases of the master–slave hybrid and homo splitting methods. The represented network is the T9-D13 network.

Parameter | Transmission | Distribution |
---|---|---|

${S}_{i}$ | ${\left[{S}^{a}\right]}_{i}$ | ${\left[{S}^{a}\phantom{\rule{4pt}{0ex}}{S}^{b}\phantom{\rule{4pt}{0ex}}{S}^{c}\right]}_{i}^{T}$ |

${V}_{i}$ | ${\left[{V}^{a}\right]}_{i}$ | ${\left[{V}^{a}\phantom{\rule{4pt}{0ex}}{V}^{b}\phantom{\rule{4pt}{0ex}}{V}^{c}\right]}_{i}^{T}$ |

${Y}_{ij}$ | ${\left[\begin{array}{cc}\underset{{\scriptscriptstyle 1\times 1}}{{Y}_{11}^{a}}& \underset{{\scriptscriptstyle 1\times 1}}{{Y}_{12}^{a}}\\ \underset{{\scriptscriptstyle 1\times 1}}{{Y}_{21}^{a}}& \underset{{\scriptscriptstyle 1\times 1}}{{Y}_{22}^{a}}\end{array}\right]}_{ij}$ | ${\left[\begin{array}{cc}\underset{{\scriptscriptstyle 3\times 3}}{{Y}_{11}^{abc}}& \underset{{\scriptscriptstyle 3\times 3}}{{Y}_{12}^{abc}}\\ \underset{{\scriptscriptstyle 3\times 3}}{{Y}_{21}^{abc}}& \underset{{\scriptscriptstyle 3\times 3}}{{Y}_{22}^{abc}}\end{array}\right]}_{ij}$ |

Integrated Approach | ||
---|---|---|

Network Model | Unified | Splitting |

Hybrid | Interconnected (IC)Transform substation Solve as a whole | MSS-hybridTransform substation Extra iterative scheme |

Homogeneous | Full three-phase (F3P)Transform Transmission Solve as a whole | MSS-homoTransform Transmission Extra iterative scheme |

Advantages | Disadvantages | |
---|---|---|

Hybrid | Smaller Jacobian In line with current separated models | Balanced substation bus |

Homogeneous | Intuitive physical approach Suitable for unbalanced transmission conditions | Larger system |

Unified | One outer iteration | Same solver (NR) must be used for complete system |

Splitting | Limited data sharing between system operators Allows for continuation of separate developments | Extra iterative scheme |

**Table 4.**Comparison on number of iterations (for the MSS method: ${\mathcal{I}}_{MSS}$ of the MSS scheme and ${\mathcal{I}}_{M}$ and ${I}_{D}$, the average number of iterations per sub domain) and CPU time of the integration methods, of three test cases. The top table displays methods applied to hybrid and the bottom one on homogeneous networks. The slowest CPU times are printed in bold.

IC | MSS-Hybrid | |||||
---|---|---|---|---|---|---|

${\mathcal{I}}_{\mathit{U}}$ | CPU | ${\mathcal{I}}_{\mathit{MSS}}$ | ${\mathcal{I}}_{\mathit{M}}$ | ${\mathcal{I}}_{\mathit{S}}$ | CPU | |

Test Case | # | s | # | # | # | s |

T9-D13 | 3 | 0.016 | 3 | 4 | 4 | 0.901 |

T118-D123 | 4 | 0.025 | 3 | 7 | 5 | 0.807 |

T3120-D8500 | 4 | 0.367 | 3 | 6 | 5 | 2.569 |

F3P | MSS-Homo | |||||

${\mathcal{I}}_{\mathit{U}}$ | CPU | ${\mathcal{I}}_{\mathit{MSS}}$ | ${\mathcal{I}}_{\mathit{M}}$ | ${\mathcal{I}}_{\mathit{S}}$ | CPU | |

Test Case | # | s | # | # | # | s |

T9-D13 | 3 | 0.015 | 3 | 4 | 4 | 1.071 |

T118-D123 | 4 | 0.039 | 3 | 4 | 5 | 1.173 |

T3120-D8500 | 4 | 0.612 | 3 | 6 | 5 | 3.697 |

**Table 5.**Per-unit voltage profiles of the connection bus in different networks and the differences between hybrid and homogeneous network models.

Unified | Splitting | ||||||
---|---|---|---|---|---|---|---|

IC | F3P | Hybrid | Homo | ||||

Test Case | Phase | $\left|\mathit{V}\right|$ | $\left|\mathit{V}\right|$ | ${\left|\mathit{V}\right|}_{\mathbf{IC}}-{\left|\mathit{V}\right|}_{\mathbf{F}\mathbf{3}\mathbf{P}}$ | $\left|\mathit{V}\right|$ | $\left|\mathit{V}\right|$ | ${\left|\mathit{V}\right|}_{\mathbf{Hy}}-{\left|\mathit{V}\right|}_{\mathbf{Ho}}$ |

T9-D13 | A | 1.0075 | 1.0076 | $1.00\times {10}^{-4}$ | 1.0074 | 1.0073 | $-1.00\times {10}^{-4}$ |

B | 1.0075 | 1.0076 | $1.00\times {10}^{-4}$ | 1.0074 | 1.0073 | $-1.00\times {10}^{-4}$ | |

C | 1.0075 | 1.0074 | $-1.00\times {10}^{-4}$ | 1.0074 | 1.0075 | $1.00\times {10}^{-4}$ | |

T118-D123 | A | 0.9651 | 0.9651 | $0.00\times {10}^{0}$ | 0.9662 | 0.9662 | $0.00\times {10}^{0}$ |

B | 0.9651 | 0.9652 | $1.00\times {10}^{-4}$ | 0.9662 | 0.9661 | $-1.00\times {10}^{-4}$ | |

C | 0.9651 | 0.9651 | $0.00\times {10}^{0}$ | 0.9662 | 0.9661 | $-1.00\times {10}^{-4}$ | |

T3120-D8500 | A | 1.0716 | 1.0716 | $0.00\times {10}^{0}$ | 1.0722 | 1.0722 | $0.00\times {10}^{0}$ |

B | 1.0716 | 1.0715 | $-1.00\times {10}^{-4}$ | 1.0722 | 1.0722 | $0.00\times {10}^{0}$ | |

C | 1.0716 | 1.0716 | $0.00\times {10}^{0}$ | 1.0722 | 1.0722 | $0.00\times {10}^{0}$ |

**Table 6.**The amount of maximum and average voltage unbalance of the distribution feeder for the three different test cases and the number of iterations of the three different methods.

Unified | Splitting | |||||
---|---|---|---|---|---|---|

Imbalance | IC | F3P | Hybrid | Homo | ||

Max | Avg | ${\mathcal{I}}_{\mathit{U}}$ | ${\mathit{I}}_{\mathit{U}}$ | ${\mathcal{I}}_{\mathit{MSS}}$ | ${\mathcal{I}}_{\mathit{MSS}}$ | |

Test Case | % | % | # | # | # | # |

T9-D13 | 4.24 | 2.02 | 3 | 3 | 3 | 3 |

T118-D123 | 1.96 | 1.00 | 4 | 4 | 3 | 3 |

T3120-D8500 | 5.87 | 3.23 | 4 | 4 | 3 | 3 |

**Table 7.**Comparison on number of iterations and CPU time of the integration methods, applied to the test cases having multiple distribution feeders connected. The change in iteration number, compared to the original networks, are bold.

IC | MFS-Hybrid | |||||
---|---|---|---|---|---|---|

${\mathcal{I}}_{\mathit{U}}$ | CPU | ${\mathcal{I}}_{\mathit{MSS}}$ | ${\mathcal{I}}_{\mathit{M}}$ | ${\mathcal{I}}_{\mathit{S}}$ | CPU | |

Test Case | # | s | # | # | # | s |

T9-3D13 (7–9) | 3 | 0.020 | 3 | 4 | 5 | 1.494 |

D33-2D37 (30–31) | 10 | 0.048 | 13 | 5 | 6 | 4.974 |

T118-5D123 (108–112) | 4 | 0.060 | 3 | 7 | 4 | 1.691 |

T3120-10D8500 (2700–2709) | 5 | 3.015 | 3 | 6 | 4 | 12.51 |

F3P | MFS-Homo | |||||

${\mathcal{I}}_{\mathit{U}}$ | CPU | ${\mathcal{I}}_{\mathit{MSS}}$ | ${\mathcal{I}}_{\mathit{M}}$ | ${\mathcal{I}}_{\mathit{S}}$ | CPU | |

Test Case | # | s | # | # | # | s |

T9-3D13 (7–9) | 3 | 0.017 | 3 | 4 | 5 | 1.791 |

D33-2D37 (30–31) | 12 | 0.065 | 13 | 5 | 6 | 6.833 |

T118-5D123 (108–112) | 4 | 0.073 | 3 | 4 | 4 | 1.973 |

T3120-10D8500 (2700–2709) | 4 | 3.675 | 3 | 6 | 4 | 14.53 |

**Table 8.**Influence of PV buses on the number of iterations. The changes in iteration number compared to the original network are bold.

Original | Distr. Generation | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

PV | IC | MSS-Hybrid | PV | IC | MSS-Hybrid | |||||

Test Case | Buses | ${\mathcal{I}}_{\mathit{U}}$ | ${\mathcal{I}}_{\mathit{MSS}}$ | ${\mathcal{I}}_{\mathit{M}}$ | ${\mathcal{I}}_{\mathit{S}}$ | Buses | ${\mathcal{I}}_{\mathit{U}}$ | ${\mathcal{I}}_{\mathit{MSS}}$ | ${\mathcal{I}}_{\mathit{M}}$ | ${\mathcal{I}}_{\mathit{S}}$ |

T9-D13 | 0 | 3 | 3 | 4 | 4 | 4 | 3 | 5 | 4 | 5 |

T118-D123 | 0 | 4 | 3 | 7 | 5 | 5 | 4 | 6 | 7 | 5 |

T3120-D8500 | 0 | 4 | 3 | 6 | 5 | 5 | 4 | 3 | 6 | 4 |

Original | Distr. Generation | |||||||||

PV | F3P | MSS-Homo | PV | F3P | MSS-Homo | |||||

Test Case | Buses | ${\mathcal{I}}_{\mathit{U}}$ | ${\mathcal{I}}_{\mathit{MSS}}$ | ${\mathcal{I}}_{\mathit{M}}$ | ${\mathcal{I}}_{\mathit{S}}$ | Buses | ${\mathcal{I}}_{\mathit{U}}$ | ${\mathcal{I}}_{\mathit{MSS}}$ | ${\mathcal{I}}_{\mathit{M}}$ | ${\mathcal{I}}_{\mathit{S}}$ |

T9-D13 | 0 | 3 | 3 | 4 | 4 | 4 | 3 | 6 | 4 | 5 |

T118-D123 | 0 | 4 | 3 | 4 | 5 | 5 | 4 | 6 | 4 | 5 |

T3120-D8500 | 0 | 4 | 3 | 6 | 5 | 5 | 4 | 3 | 6 | 4 |

**Table 9.**The number of master iterations per MSS iteration and the CPU time of the three different test cases when the idea of speeding up the splitting methods is applied.

MSS-Hybrid | |||||||
---|---|---|---|---|---|---|---|

${\mathcal{I}}_{\mathit{MSS}}$ | ${\mathcal{I}}_{\mathit{M}}^{\mathbf{1}}$ | ${\mathcal{I}}_{\mathit{M}}^{\mathbf{2}}$ | ${\mathcal{I}}_{\mathit{M}}^{\mathbf{3}}$ | ${\mathcal{I}}_{\mathit{M}}^{\mathbf{4}}$ | ${\mathcal{I}}_{\mathit{S}}$ | CPU | |

Test Case | # | # | # | # | # | # | s |

T9-D13 | 3 | 4 | 2 | 1 | 4 | 5 | 0.785 |

T118-D123 | 3 | 7 | 1 | 1 | - | 5 | 0.831 |

T3120-D8500 | 3 | 6 | 2 | 1 | - | 5 | 2.227 |

MSS-Hybrid | |||||||

${\mathcal{I}}_{\mathit{MSS}}$ | ${\mathcal{I}}_{\mathit{M}}^{\mathbf{1}}$ | ${\mathcal{I}}_{\mathit{M}}^{\mathbf{2}}$ | ${\mathcal{I}}_{\mathit{M}}^{\mathbf{3}}$ | ${\mathcal{I}}_{\mathit{M}}^{\mathbf{4}}$ | ${\mathcal{I}}_{\mathit{S}}$ | CPU | |

Test Case | # | # | # | # | # | # | s |

T9-D13 | 3 | 4 | 2 | 1 | 4 | 5 | 1.028 |

T118-D123 | 3 | 4 | 2 | 1 | - | 5 | 1.118 |

T3120-D8500 | 3 | 6 | 2 | 1 | - | 5 | 3.651 |

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**MDPI and ACS Style**

Kootte, M.E.; Vuik, C. Steady-State Stand-Alone Power Flow Solvers for Integrated Transmission-Distribution Networks: A Comparison Study and Numerical Assessment. *Energies* **2021**, *14*, 5784.
https://doi.org/10.3390/en14185784

**AMA Style**

Kootte ME, Vuik C. Steady-State Stand-Alone Power Flow Solvers for Integrated Transmission-Distribution Networks: A Comparison Study and Numerical Assessment. *Energies*. 2021; 14(18):5784.
https://doi.org/10.3390/en14185784

**Chicago/Turabian Style**

Kootte, Maria Eliza, and Cornelis Vuik. 2021. "Steady-State Stand-Alone Power Flow Solvers for Integrated Transmission-Distribution Networks: A Comparison Study and Numerical Assessment" *Energies* 14, no. 18: 5784.
https://doi.org/10.3390/en14185784