# Generalized Distribution Feeder Switching with Fuzzy Indexing for Energy Saving

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Fundamental Theory

- The loss increment ΔP obtained by switching is a quadratic function of the equivalent load current in the corresponding power supply area;
- The increase in loss ΔP is a convex function, so there is an optimal value I
_{opt}to minimize ΔP, and this optimal value I_{opt}represents the optimal equivalent load current that can be transferred;

- 3.
- All feeders are radially structured;
- 4.
- Closing of a tie switch should be followed by the opening of a sectionalized switch.

_{loop}is the impedance of the loop when connecting the two feeders. For the total load transfer I

_{tot}from feeder B to A [2], the loss will increase by

_{Loop}) is always positive, $\left|{\mathrm{E}}_{m}\right|$ should be less than $\left|{\mathrm{E}}_{n}\right|$ to make ΔP negative, that is, needing a higher $\left|{\mathrm{E}}_{n}\right|\mathrm{voltage}.$Reducing the feeder loss can be achieved by transferring the terminal with higher voltage to the terminal with lower voltage. Omitting the imaginary part, (1) becomes

_{tot}as a function of x, i.e., the point of load transfer. To obtain the most effective transfer current I(x), differentiating ΔP with respect to I(x), we have

_{opt}where the increase in loss ΔP is the lowest.

_{tot}= I

_{x}is the total load transfer. The transfer loads are n and n−1 in Figure 1, i.e., opening section switch No. 2. The total current transfer is I

_{2}= I

_{n}+ I

_{n−1}, and Δ1 is the current difference between I

_{2}and I

_{opt}. Similarly, Δ2 is the current difference between I

_{3}= I

_{n}+ I

_{n−1}+ I

_{n−2}and I

_{opt}by opening switch No. 3. Since ∆

_{1}< ∆

_{2}in Figure 2, current transfer by opening section switch No. 2 is a better choice.

## 3. The Fuzzy Index Feeder Switching

#### 3.1. Tie Switch Strategy

#### 3.1.1. Large Loss to Small Loss

_{L}(r_i) =$\frac{{\mathrm{P}}_{\mathrm{L}}\left(\mathrm{i}\_\mathrm{n}\right)}{{\mathrm{P}}_{\mathrm{L}}\left(\mathrm{i}\_\mathrm{m}\right)}$= power loss ratio of two feeders connected by switch (i). That is, the power loss ratio P

_{L}(r_i) is defined for tie switch (i) connecting two feeder ends n and m, with both feeder losses compared, where

_{L}(i_n) is the light feeder loss;

_{L}(i_m) is the heavy feeder loss.

_{d}is defined in (5) and called the “loss severity factor” for calculating the severity of tie switch (i) for transfer, as shown in Figure 3. The term $\tilde{u}$

_{d}is defined as

_{L}(r_max) is the maximum ratio of P

_{L}(r_i) among all tie switches {(i)}.

#### 3.1.2. High Voltage to Low Voltage

_{a}is defined in (6) as the “voltage indication factor” to calculate the severity of voltage difference ΔE across the tie switch (i). Figure 4 indicates the urgency of needing a transfer. The term $\tilde{u}$

_{a}is defined by

_{n}(i) − E

_{m}(i) is the voltage difference across both ends of tie switch (i).

_{n}(i) is the end with higher voltage drop;

_{m}(i) is the end with lower voltage drop;

#### 3.1.3. Determination of the Candidate Tie Switch

_{d}and $\tilde{u}$

_{a}, a “tie candidate index” membership $\tilde{u}$

_{t}(i) can be defined for each tie using

_{t}(i) shows the need of tie switch (i) to close: the larger the value, the more urgently the tie needs to be closed. The switch with the maximum $\tilde{u}$

_{t}(max) is the candidate switch.

#### 3.2. Sectionalized Switch Strategy

#### 3.2.1. Calculation of Optimal Load

_{opt}. For a specific tie switch (i), a membership function $\tilde{u}$

_{b}(i,j) is defined for the load point (j) as the “optimal current transfer factor” to calculate the closeness of the load (j) transfer current to I

_{opt}. With (i) given, Figure 5 shows the membership function $\tilde{u}$

_{b}(i,j) of load (j) defined by

#### 3.2.2. The Effect of Excessive Transfer

_{opt}, i.e., transferring the feeder with low current to high current, a negative impact on reducing losses is caused. A membership function $\tilde{u}$

_{c}(i,j) is defined similar to $\tilde{u}$

_{b}(i,j) in Figure 6 for the “effect of excessive current switching” to forbid the move by

_{x}(i, j) from being greater than I

_{opt}(i), i.e., forbid the switching operation.

#### 3.3. A Complete Switching Strategy

#### 3.3.1. Transfer for Multiple Feeders

#### 3.3.2. Transfer for Single Feeder

_{a}(i), $\tilde{u}$

_{b}(I,j) and $\tilde{u}$

_{c}(i,j). The largest value $\tilde{u}$s(max) is the optimal switching solution, such as a multiple feeder.

## 4. A Layered Feeder Switching Scheme

#### 4.1. For Multiple Feeders

- Step 1:
- Read system load flow data and compute

_{Loss}(feeder), ∆E(tie), I

_{x}(bus), R

_{loop}(tie), P

_{L}(r_max), ∆E(max);

- Step 2:
- For layer l, search for the tie switch (i) with (8);
- Step 3:
- Find a sectionalized switch with (11) or (12);
- Step 4:
- Find complete switching strategy with (13);

- Step 5:
- Execute the load flow program.

- Step 6:
- Repeat until the remaining tie switches are 0.

#### 4.2. For a Single Feeder

## 5. Test Results and Discussion

- The exhausted search enumeration:

- 2.
- del_P loss formula method [2]:

#### 5.1. Multiple Feeder Transfer

#### 5.1.1. Case 1: A Three-Feeder System

_{t}(21) = 0.8088 is the largest. Tie 21 is the switch to close, and we can calculate I

_{opt}. Following the calculation, one switch of (16, 17, 22, 24) needs to open. Figure 2 shows the evaluation process and switch 17 is chosen. A similar process follows in all the examples below; $\tilde{u}$

_{s}(21,17) = 0.4174 is the largest. Therefore, we have a switch pair (21, 17) as the solution to the first layer. Similarly, $\tilde{u}$

_{s}(15, 19) = 0.423 is the solution to the second layer. Since the third search can no longer reduce the loss, the optimal switching sequences stopped with (21, 17) and (15, 19).

_{t}and $\tilde{u}$

_{s}for each path. The normally open switches become 17, 19 and 26. For this small system, so-called “optimal” results exist, which are verifiable using the exhausted search. For a large-scale network, we try to find the optimal or sub-optimal solution since there is no way to verify it. Table 1 shows the comparison charts where all three methods yield the same optimal results for this simple network. The proposed method works successfully, as well as the other two methods.

#### 5.1.2. Case 2: Five Feeder System with 33 Switches

#### 5.2. Single Feeder Transfer

#### Case 3: Single Feeder with 37 Switches

## 6. Conclusions

- It deals with the large-scale mixed-integer combinatorial problem where conventional techniques would generally fail;
- The fuzzy indexing simplifies the optimization process with easy numeric calculations instead of large-scale sorting or a large amount of computation;
- It executes the load flow only once after finding the switch (i,j); this is a great advantage compared with other methods requiring heavy computation;
- The solution quality is high. It shows that for a small system, the “optimal” results exist and are verifiable by exhausted search. However, for large-scale networks, the solution will be optimal or sub-optimal;
- The method is suitable for real-time applications even for a large distribution system;
- It is applicable to all feeder configurations, including the multiple-feeder and single-feeder systems;
- It can get the best configuration with less switching operations and save on costs;
- The first switching is the most significant to reduce the loss and balance the load;
- Proper switching can solve the transformer load management and terminal voltage problems.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Layer | Optimal Switch Configuration | del_P Loss Formula | Fuzzy Index Algorithm | ||||||
---|---|---|---|---|---|---|---|---|---|

(on,off) | Loss (p.u) | Red (%) | (on,off) | Loss (p.u.) | Red (%) | (on,off) | Loss (p.u) | Red (%) | |

1 | (21,17) | 0.004839 | 5.400 | (21,17) | 0.004839 | 5.400 | (21,17) | 0.004839 | 5.400 |

2 | (15,19) | 0.004662 | 8.860 | (15,19) | 0.004662 | 8.860 | (15,19) | 0.004662 | 8.860 |

Original (Tie Switch) | Exhausted Search | del_p Formula | Fuzzy index Algorithm |
---|---|---|---|

5,6,7,8,14,15, 16,21,22,28 | 5,7,8,11,15,16, 21,22,26,28 | 6,7,8,13,14,15, 16,21,26,28 | 5,7,8,11,15,16, 21,22,26,28 |

Action switch | (14,11),(6,26) | (22,26),(5,13) | (14,11),(6,26) |

Loss red. (p.u.) | 0.000510 (p.u) | 0.000504 (p.u) | 0.000510 (p.u) |

Reduction (%) | 6.104% | 6.032% | 6.104% |

Search Layer | Goswami (Method I) | Goswami (Method II) | Goswami (Method III) | Baran (Method I) | Fuzzy Index Algorithm | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

(on,off) | Loss (p.u) | Red. (%) | (on,off) | Loss (p.u) | Red. (%) | (on,off) | Loss (p.u) | Red. (%) | (on,off) | Loss (p.u) | Red. (%) | (on,off) | Loss (p.u) | Red. (%) | |

1 | (35,8) | 0.01535 | 24.270 | (37,28) | 0.01751 | 13.593 | (33,7) | 0.01584 | 21.852 | (33,6) | 0.01633 | 19.435 | (35,7) | 0.01565 | 22.770 |

2 | (37,28) | 0.01477 | 27.107 | (33,7) | 0.01581 | 21.976 | (34,9) | 0.01579 | 22.104 | (35,11) | 0.01450 | 28.439 | (33,11) | 0.01445 | 28.686 |

3 | (36,32) | 0.01462 | 27.847 | (35,11) | 0.01443 | 28.809 | (35,14) | 0.01422 | 29.855 | (36,31) | 0.01544 | 23.826 | (36,32) | 0.01432 | 29.352 |

4 | (34,14) | 0.01460 | 27.980 | (34,14) | 0.01432 | 29.366 | (36,32) | 0.01396 | 31.148 | (37,28) | 0.01598 | 21.137 | (34,14) | 0.01412 | 30.334 |

5 | (8,9) | 0.01446 | 28.666 | (36,32) | 0.01416 | 30.121 | * | * | * | (6,33) | 0.01463 | 27.827 | (11,9) | 0.01396 | 31.148 |

6 | (33,7) | 0.01400 | 30.935 | (28,37) | 0.01412 | 30.334 | * | * | * | * | * | * | * | * | * |

7 | (28,37) | 0.01396 | 31.148 | (11,9) | 0.01396 | 31.148 | * | * | * | * | * | * | * | * | * |

Original Tie Switch | Goswami (I) | Goswami (II) | Goswami (III) | Baran. (I) | Baran. (II/III) | Fuzzy Index |
---|---|---|---|---|---|---|

33 | 7 | 7 | 7 | 11 | 6 | 7 |

34 | 9 | 9 | 9 | 28 | 11 | 9 |

35 | 14 | 14 | 14 | 31 | 31 | 14 |

36 | 32 | 32 | 32 | 33 | 34 | 32 |

37 | 37 | 37 | 37 | 34 | 37 | 37 |

Number of operations | 7 | 7 | 4 | 5 | 3 | 5 |

Loss reduction (%) | 31.148% | 31.148% | 31.148% | 27.83% | 23.83% | 31.148% |

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**MDPI and ACS Style**

Lin, W.-M.; Tsai, W.-C.
Generalized Distribution Feeder Switching with Fuzzy Indexing for Energy Saving. *Processes* **2023**, *11*, 1572.
https://doi.org/10.3390/pr11051572

**AMA Style**

Lin W-M, Tsai W-C.
Generalized Distribution Feeder Switching with Fuzzy Indexing for Energy Saving. *Processes*. 2023; 11(5):1572.
https://doi.org/10.3390/pr11051572

**Chicago/Turabian Style**

Lin, Whei-Min, and Wen-Chang Tsai.
2023. "Generalized Distribution Feeder Switching with Fuzzy Indexing for Energy Saving" *Processes* 11, no. 5: 1572.
https://doi.org/10.3390/pr11051572