# Determination of Maximum Acceptable Standing Phase Angle across Open Circuit Breaker as an Optimisation Task

^{*}

## Abstract

**:**

## 1. Introduction

- The possibility of damage of the breaker due to exceeding its switching capacity;
- The possibility of unnecessary excitation of distance-sensitive protections;
- Hazards related to the possibility of damaging the windings of transformers or synchronous generators by the action of dynamic forces caused by the high value of peak switching current;
- Stresses in the shafts of generating units, contributing to fatigue and reduction in their service life;
- Threats related to the loss of stability of the power system (concerns connecting subsystems operating asynchronously and eliminating disturbances in the auto-reclosing cycle).

## 2. Literature Review

## 3. Description of the Problem—Details

- Change in load distribution among generating units to increase the ${\theta}_{\mathrm{ab}}$ angle by a small value $\Delta {\theta}_{\mathrm{ab}}$;
- Calculation of new values of electromotive forces ${\underset{\_}{E}}_{\mathrm{G}i}^{"}$ of individual generators in the changed state of load distribution;
- Calculation of power surges due to switching on;
- If the calculated power surges are less than the criterion value for return to point (a), and if the criterion value for any generator is met for an active power stroke, then complete the calculation and take the last value ${\theta}_{\mathrm{ab}}$ as the limit.

## 4. Assumptions for Calculations

**x**= [P

_{G1}, …, P

_{Gm}]—vector of active powers generated by the considered generation sources (vector of decision variables);

**y**= [P

_{L1}, …, P

_{Li}, Q

_{L1}, …, Q

_{Li}, P

_{Gn1}, …, P

_{Gnn}]—vector of uncontrolled active and reactive loads and generations (independent variable vector);

**z**= [U

_{1}, …, U

_{i}, δ

_{1}, …, δ

_{i}]—vector of phasors of nodal voltages (vector of dependent variables).

**x**,

**y**, and

**z**, the optimisation task can be formulated in the following way:

- Objective function—the vector of control variables x providing the maximum function described as (10) is sought:$$F(x,y,z)\to \mathrm{max}$$

- Vector of equality constraints
**g**ensuring, in particular, the fulfillment of the power balance for all network nodes:$$g\left(x,y,z\right)=0$$

- Vector of inequality constraints
**h**, ensuring that the elements of the vector of state variables and elements of the vector of control variables are kept within the range specified by the technical requirements:$$h\left(x,y,z\right)\ge 0$$

- Limitation of decision variables (vector
**x**) presented in Table 1; - Limits of dependent variables (vector
**z**), which constitute permissible values of voltage at buses (generally maintained within the range from 0.9U_{n}to 1.1U_{n}); - Equal restrictions, which are nodal equations of power flow as well as equations of power balance in the power system;
- Inequality restrictions, which constitute the permissible load capacity of power lines and transformers;
- The limitation related to the power stroke criterion, expressed in relation (1).

**z**for the known values of

**x**and

**y**vectors is one of the basic computational problems included in the computer analysis of power systems. This problem, known as load flow analysis (LF), is described in textbooks such as [30,31].

**z**(state variables) are determined from a system of nonlinear equations of the form

## 5. Description of the Optimisation Method

_{l1}, …, g

_{lp}and their arguments ${\xi}_{1},\dots ,{\xi}_{p}$ can be pointed out, which provide an effective solution to many optimisation tasks treated as benchmark models.

_{1}and g

_{2}of the same specific form, i.e.,

- 1.
- Drawing of initial values of the components of the decision vector $\begin{array}{l}\left[{x}_{1}^{0}\right]=\left[{x}_{11}^{0},{x}_{12}^{0},\dots ,{x}_{1n}^{0}\right]\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{first}\hspace{0.17em}\mathrm{start}\hspace{0.17em}\mathrm{vector}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\dots \\ \left[{x}_{m}^{0}\right]=\left[{x}_{m1}^{0},{x}_{m2}^{0},\dots ,{x}_{mn}^{0}\right]\hspace{0.17em}\hspace{0.17em}m-\mathrm{th}\hspace{0.17em}\mathrm{start}\hspace{0.17em}\mathrm{vector}\end{array}$;
- 2.
- Finding the target function values for the start vector of the control variables, ${F}_{c}\left({x}_{1}^{0}\right)\dots {F}_{c}\left({x}_{m}^{0}\right)$, ${F}_{\mathrm{c}\_\mathrm{best}}=\mathrm{min}\left[{F}_{c}\left({x}_{1}^{0}\right)\dots {F}_{c}\left({x}_{m}^{0}\right)\right]$;
- 3.
- Setting the iteration counter to i = 1, substitution ${x}_{1}^{i}={x}_{1}^{0}\dots \hspace{0.17em}{x}_{m}^{i}={x}_{m}^{0}$;
- 4.
- Determining the scope of ${\alpha}_{\mathrm{max}}^{i}$ and ${\beta}_{\mathrm{max}}^{i}$;
- 5.
- Correction of the decision vector components by means of correction factors g(α), g(β):solution j = 1, drawing of “correction angles” ${\alpha}_{11}^{i}\dots {\alpha}_{1n}^{i}$ and ${\beta}_{11}^{i}\dots {\beta}_{1n}^{i}$ according to a defined probability distribution, correction of the components of the decision vector${x}_{11}^{\left(i+1\right)}={x}_{11}^{\left(i\right)}\cdot g\left({\alpha}_{11}^{i}\right)\cdot g\left({\beta}_{11}^{i}\right)\hspace{0.17em}\hspace{0.17em}\dots \hspace{0.17em}\hspace{0.17em}{x}_{1,n}^{\left(i+1\right)}={x}_{1n}^{\left(i\right)}\cdot g\left({\alpha}_{1n}^{i}\right)\cdot g\left({\beta}_{1n}^{i}\right)$solution j = m, drawing of “correction angles” ${\alpha}_{m1}^{i}\dots {\alpha}_{mn}^{i}$ and ${\beta}_{m1}^{i}\dots {\beta}_{mn}^{i}$ according to a defined probability distribution, correction of the components of the decision vector${x}_{m1}^{\left(i+1\right)}={x}_{m1}^{\left(i\right)}\cdot g\left({\alpha}_{m1}^{i}\right)\cdot g\left({\beta}_{m1}^{i}\right)\hspace{0.17em}\hspace{0.17em}\dots \hspace{0.17em}\hspace{0.17em}{x}_{mn}^{\left(i+1\right)}={x}_{mn}^{\left(i\right)}\cdot g\left({\alpha}_{mn}^{i}\right)\cdot g\left({\beta}_{mn}^{i}\right)$;
- 6.
- Determining the values of target function ${F}_{c}\left({x}_{m}^{i+1}\right)$ for subsequent solutions 1…m, checking if the target function for the next solution is better than the solution from the previous iteration (if so, replace this solution with the current, better solution, and if not, leave it), identification of the q solution for which the function ${F}_{\mathrm{c}\_\mathrm{best}}={F}_{\mathrm{c}}\left({x}_{q}^{i+1}\right)=\mathrm{min}\left[{F}_{\mathrm{c}}\left({x}_{1}^{i+1}\right)\dots {F}_{\mathrm{c}}\left({x}_{m}^{i+1}\right)\right]$ reaches the minimum value;
- 7.
- Checking whether for the identified solution q the relation ${F}_{\mathrm{c}}\left({x}_{q}^{i+1}\right)<{F}_{\mathrm{c}}\left({x}^{i}\right)$ occurs; if so, then accepting it as the next decision vector in the iterative process, i.e., ${x}_{\mathrm{best}}^{i+1}={x}_{q}^{i+1}$;
- 8.
- Verification of the stopping criterion (i + 1 = i
_{max}), end, or subsequent iteration (i = i + 1); - 9.
- End of calculation.

## 6. Test System

## 7. Results and Discussion

- Case 1—the TRA-1 transformer located between B4H211 and B4L112 nodes is switched on; the connection point is located on the side of the B4H211 node, and the breaker at the B4L112 node is closed; the maximum safe value of the SPA was determined as 45.3° (global maximum value is equal to 61.1°). Table 3 shows the list of sources with the values of power generated to ensure this safe maximum;

**Figure 6.**The process of determining the maximum SPA for case 1; (

**a**) changing the generation distribution to determine the maximum (global) SPA; (

**b**) a graph of changes in the value of the objective function (maximum SPA); (

**c**) changing the generation distribution to determine a maximum safe SPA; (

**d**) graph of changes in the value of the objective function with the penalty function (maximum safe SPA).

**Table 3.**List of sources with optimal capacity and expected values of power surges calculated for each generator.

Name | Bus | P_{opt}, MW | ΔP/P_{n}, % |
---|---|---|---|

B01-G1 | B01112 | 135 | 49 |

B05-G1 | B05211 | 210 | 13 |

B06-G1 | B06211 | 170 | 8 |

B07-G1 | B07211 | 206 | <5 |

B3H-G1 | B3H211 | 219 | 24 |

B4H-G1 | B4H211 | 292 | 28 |

- Case 2—line LIN4 between B3H211 and B02211 nodes is switched on; connection point is located on the side of B02211 node, and the breaker at B3H211 node is closed; the maximum safe value of the SPA was determined as 35.7° (global maximum value is equal to 77.0°). Table 4 shows the list of sources and the values of power generated to ensure this maximum;

**Table 4.**List of sources with optimal capacity and expected values of power surges calculated for each generator.

Name | Bus | P_{opt}, MW | ΔP/P_{n}, % |
---|---|---|---|

B01-G1 | B01112 | 80 | 8 |

B05-G1 | B05211 | 196 | <5 |

B06-G1 | B06211 | 384 | <5 |

B07-G1 | B07211 | 96 | <5 |

B3H-G1 | B3H211 | 143 | 49 |

B4H-G1 | B4H211 | 521 | <5 |

**Figure 7.**The process of determining the maximum SPA for case 2; (

**a**) changing the generation distribution to determine the maximum (global) SPA; (

**b**) a graph of changes in the value of the objective function (maximum SPA); (

**c**) changing the generation distribution to determine a maximum safe SPA; (

**d**) graph of changes in the value of the objective function with the penalty function (maximum safe SPA).

- Case 3—line LIN7 between B06211 and B4H211 nodes is switched on; the connection point is located on the side of B06211 node, and the breaker at B4H211 node is closed; the maximum safe value of the SPA was determined as 48.6° (global maximum value is equal to 63.9°). Table 5 shows the list of sources with the values of power generated to ensure this maximum;

**Table 5.**List of sources with optimal capacity and expected values of power surges calculated for each generator.

Name | Bus | P_{opt}, MW | ΔP/P_{n}, % |
---|---|---|---|

B01-G1 | B01112 | 82 | 6 |

B05-G1 | B05211 | 102 | 14 |

B06-G1 | B06211 | 281 | 49 |

B07-G1 | B07211 | 215 | 14 |

B3H-G1 | B3H211 | 206 | 6 |

B4H-G1 | B4H211 | 235 | 28 |

**Figure 8.**The process of determining the maximum SPA for case 3; (

**a**) changing the generation distribution to determine the maximum (global) SPA; (

**b**) a graph of changes in the value of the objective function (maximum SPA); (

**c**) changing the generation distribution to determine a maximum safe SPA; (

**d**) graph of changes in the value of the objective function with the penalty function (maximum safe SPA).

_{max}, P

_{min}) can be changed. This is described below.

- Case 4—It was assumed that the LIN21 line between B11112 and B01112 nodes would be switched on. The connection point is located on the B11112 node side, and the breaker on the B01112 node is closed. The control range of the B01-G1 generator was changed from 60 to 350 MW. The maximum safe value of the SPA was determined as 32.3° (global maximum value is equal to 52.3°). Table 6 shows the list of sources with their optimal power generation values;

**Figure 9.**The process of determining the maximum SPA for case 4; (

**a**) changing the generation distribution to determine the maximum (global) SPA; (

**b**) a graph of changes in the value of the objective function (maximum SPA); (

**c**) changing the generation distribution to determine a maximum safe SPA; (

**d**) graph of changes in the value of the objective function with the penalty function (maximum safe SPA).

**Table 6.**List of sources with optimal capacity and expected values of power surges calculated for each generator.

Name | Bus | P_{opt}, MW | ΔP/P_{n}, % |
---|---|---|---|

B01-G1 | B01112 | 321 | 49 |

B05-G1 | B05211 | 101 | 9 |

B06-G1 | B06211 | 170 | <5 |

B07-G1 | B07211 | 115 | <5 |

B3H-G1 | B3H211 | 170 | 26 |

B4H-G1 | B4H211 | 251 | 26 |

- Case 5—the TRA-2 transformer located between B3H211 and B3L112 nodes is switched on; the connection point is located on the side of the B3L112 node, and the breaker at the B3H211 node is closed; the maximum safe value of the SPA was determined as 24.6° (global maximum value is equal to 52.3°). Table 7 shows the list of sources with the values of power generated to ensure this maximum.

**Figure 10.**The process of determining the maximum SPA for case 5; (

**a**) changing the generation distribution to determine the maximum (global) SPA; (

**b**) a graph of changes in the value of the objective function (maximum SPA); (

**c**) changing the generation distribution to determine a maximum safe SPA; (

**d**) graph of changes in the value of the objective function with the penalty function (maximum safe SPA).

**Table 7.**List of sources with optimal capacity and expected values of power surges calculated for each generator.

Name | Bus | P_{opt}, MW | ΔP/P_{n}, % |
---|---|---|---|

B01-G1 | B01112 | 63 | 49 |

B05-G1 | B05211 | 100 | 14 |

B06-G1 | B06211 | 274 | 9 |

B07-G1 | B07211 | 113 | <5 |

B3H-G1 | B3H211 | 115 | 24 |

B4H-G1 | B4H211 | 490 | 34 |

_{nG}) is not exceeded in any case. Therefore, it can be said that the solution of the optimisation task has fulfilled its goal: the maximum values of SPA were determined for which the power surges in the generators do not exceed the critical value of 0.5 of the rated power.

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Oscillations of the output power of the synchronous generator caused by switch-on operation—interpretation of $\Delta {P}_{\mathrm{G}i\_\mathrm{AV}}$.

**Figure 3.**Model for determining the initial switching current; (

**a**) substitute scheme, (

**b**) phasor’s diagram.

**Figure 4.**General scheme of the organisation of the computational process used in solving the optimisation task.

Name | Bus | P_{gmin}, MW | P_{gmax}, MW |
---|---|---|---|

B01-G1 | B01112 | 60 | 135 |

B02-G1 | B02211 | 170 | 385 |

B05-G1 | B05211 | 100 | 230 |

B06-G1 | B06211 | 170 | 385 |

B07-G1 | B07211 | 96 | 220 |

B3H-G1 | B3H211 | 96 | 220 |

B4H-G1 | B4H211 | 235 | 530 |

Line | Terminal Nodes | U_{n}, kV | |
---|---|---|---|

LIN2 | B3H | B9 | 220 |

LIN4 | B3H | B2 | |

LIN6 | B9 | B4H | |

LIN7 | B4H | B6 | |

LIN9 | B4H | B10 | |

LIN10 | B9 | B8 | |

LIN11 | B8 | B6 | |

LIN13 | B10 | B2 | |

LIN20 | B3L | B1 | 110 |

LIN21 | B1 | B11 | |

LIN22 | B11 | B15 | |

LIN24 | B4L | B12 | |

LIN25 | B12 | B14 | |

LIN26 | B14 | B13 | |

LIN27 | B13 | B3L |

Case | ΔP_{B01}/P_{nB01} | ΔP_{B05}/P_{nB05} | ΔP_{B06}/P_{nB06} | ΔP_{B07}/P_{nB07} | ΔP_{B3H}/P_{nB3H} | ΔP_{B4H}/P_{nB4H} |
---|---|---|---|---|---|---|

1 | 49% | 13% | 8% | <5% | 24% | 28% |

2 | 8% | <5% | <5% | <5% | 49% | <5% |

3 | 6% | 14% | 49% | 14% | 6% | 28% |

4 | 49% | 9% | <5% | <5% | 26% | 26% |

5 | 49% | 14% | 9% | <5% | 24% | 34% |

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**MDPI and ACS Style**

Kacejko, P.; Miller, P.; Pijarski, P. Determination of Maximum Acceptable Standing Phase Angle across Open Circuit Breaker as an Optimisation Task. *Energies* **2021**, *14*, 8105.
https://doi.org/10.3390/en14238105

**AMA Style**

Kacejko P, Miller P, Pijarski P. Determination of Maximum Acceptable Standing Phase Angle across Open Circuit Breaker as an Optimisation Task. *Energies*. 2021; 14(23):8105.
https://doi.org/10.3390/en14238105

**Chicago/Turabian Style**

Kacejko, Piotr, Piotr Miller, and Paweł Pijarski. 2021. "Determination of Maximum Acceptable Standing Phase Angle across Open Circuit Breaker as an Optimisation Task" *Energies* 14, no. 23: 8105.
https://doi.org/10.3390/en14238105