Optimal Sizing and Placement of Capacitor Banks in Distribution Networks Using a Genetic Algorithm
Abstract
:1. Introduction
- Loss reductions;
- Voltage profile improvement;
- Loading improvement in existing branches.
- Capital costs of capacitor banks (investment);
- Energy losses cost (variable cost);
- Penalty cost for the power factor infringements (variable cost).
- Definition at the same time of the optimal location (node) and the most suitable size for each capacitor unit. Additionally, in the algorithm it is possible to specify in an easy way a bus list where it is not desirable to place capacitors. Not all the approaches available in the literature solve the two different problems (siting and sizing of capacitor units) simultaneously.
- Adoption of suitable daily profiles for loads and generators to obtain a better representation of the electrical conditions in all time intervals of the day/year. In particular, different typical days for the different year quarters are adopted. This particular approach to represent the electrical model (the accuracy in the objective function calculation increase), involving the optimization method, is not adopted in the most common methodologies.
- Definition of a daily program to manage the capacitors appropriately to obtain the best benefits for each time interval without overcompensation situations. The definition of the optimal control pattern is obtained, including a suitable constraint referred to the switches aging in the capacitors. As discussed regarding the previous novelty, the definition of an optimal pattern control for capacitor units is not adopted in the most common methodologies.
- The proposed approach can be used also in a meshed network, while many methods discussed are limited to the radial network only.
2. Genetic Algorithm
- Individual. The point for which an objective function (OF) is implemented can be treated as an individual. This is a set of values of variables for which function is to be optimized. The OF value for an individual is called its score. The vector entries related to the genes of a genome are considered individuals.
- Population. The population is an array of individuals. From a mathematical point of view, if N is the size of the population and NV the number of variables in the OF, the population can be represented by a matrix of the size N × NV in which each row corresponds to an individual.
- Generation. At each iteration, the genetic operators (described in the next sections) are used to perform a series of computations on the current population to produce a new population. The successive populations produced are known as the new generation.
- Parents and children. To create the next generation, the GA selects a number of individuals in the existing population, called parents. This generation is used to create individuals in the next generation, known as children.
2.1. General Framework
- Definition of the OF, according to the specific optimization;
- Definition of GA representation, by a suitable coding, which consists normally in the characterization of a string useful to operate with the genetic operators;
- Definition of GA operators;
- Production of the initial population: the creation of a suitable initial population is important to have a good evolution of population during the different generation steps;
- By applying genetic operators (described in the next section) on selected people, the children population is produced.
2.2. Genetic Operators
- Selection. This selects the fittest individuals in the current population to be used in generating the next population. People with higher fitness should have greater opportunities to produce children. For selection by the roulette wheel, race or elitism methods can be used in the production of children. Generally, the OF value is considered a fitness number anywhere.
- Crossover. This operator causes pairs of individuals to exchange genetic information with each other. These children are known as crossover children. Two random chromosomes in the middle generation are selected. Then a random number (n) between 1 and the length of a chromosome is selected, and pairs of selected chromosomes from the n-th gene to later are swapped with each other to produce new chromosomes.
- Mutation. The mutation causes individual genetic representations required to be changed following a set of some probabilistic rules. To test each element for fitness and to avoid algorithm stopping at a local optimum, some solutions are also randomly modified. Therefore, a chromosome is selected randomly. Then some of its genes are replaced with another random number. Random variation of a gene in each category is called mutation.
2.3. GA Schematic Representation
3. The Proposed Methodology for the Optimal Sizing and Placement of Capacitor Banks
3.1. Solution Coding
- The first element, with an integer number (between 0 and S), identifies the eventual capacitor unit allocated in node i (Figure 2). This gene detects no capacitor in the node with the null value, while a positive number indicates a size code between the S sizes set included in the optimization (Figure 2).
- A subarray identifies the optional definition of a daily profile for the capacitor bank unit if the performed analysis includes the pattern control optimization. In particular, only in the typical day d fixed in the first array element (day 2 in the example of Figure 2), the other two genes are the day time when the unit is turned on and turned off, respectively (Figure 2). The fundamental assumption of this representation confines the switch operation of the capacitor bank unit in a unique representative day of the total D days. Anyway, this assumption is not a limit of the proposed approach but denotes an important real operating condition that avoids a frequently switching use, preserving its lifetime and consequently obtaining a maintenance cost reduction. The occurrence of the single representative day returns the yearly occurrence for the switching operations performed for the capacitor bank unit.
3.2. Genetic Operators
3.3. Objective Function
- Costs of capacitor banks;
- Energy losses;
- Power factor infringements, with a reference to a prefixed threshold based on the voltage level.
- Voltage nodes Vi,h,d, for each node i of the network and time interval h of the representative day d;
- Current branches Ib,h,d, for each branch b (lines or transformers) of the network and time interval h of the representative day d;
- Power factor PFb,h,d, for each branch b (lines or transformers), calculated at the first node, in the time interval h of the representative day d;
- Energy losses for the whole network eh,d, in the time interval h of the representative day d. This quantity is calculated taking into account the joule losses in the electrical lines and transformers according to their specific resistance values.
3.4. Constraints
4. Case Study
5. Results and Discussion
- (a)
- Optimal capacitor bank placement without pattern control optimization: in this analysis, the banks are always connected to their nominal value without any control in the daily pattern. In this case, the solution coding is limited, for each candidate node to placement, to the first gene in Figure 2.
- (b)
- Optimal capacitor bank placement with pattern control optimization, where the daily schedule for the capacitors can be controlled to improve the OF, as discussed in the paragraph dedicated to the solution coding.
6. Conclusions
Funding
Institutional Review Board Statement
Informed Consent Statement
Conflicts of Interest
References
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Node i | Node j | Branch Type | Rij (Ω) | Xi,j (Ω) | Smax (MVA) | Pj (MW) | Qj (MVar) |
---|---|---|---|---|---|---|---|
1 | 2 | 69 kV line | 3.361 | 9.808 | 103 | 8.123 | 3.714 |
1 | 3 | 69 kV line | 9.094 | 17.282 | 40 | 6.275 | 2.658 |
1 | 4 | 69/34.5 kV transformer | 0.595 | 15.859 | 12 | 8.016 | 4.039 |
1 | 5 | 69 kV line | 0.480 | 14.616 | 92 | --- | --- |
5 | 6 | 69 kV line | 15.654 | 30.261 | 40 | 6.632 | 2.554 |
6 | 7 | 69 kV line | 0.338 | 0.652 | 40 | --- | --- |
7 | 8 | 69/34.5 kV transformer | 0.714 | 19.031 | 10 | 6.528 | 1.579 |
5 | 9 | 69 kV line | 0.162 | 0.439 | 46 | 14.192 | 4.719 |
5 | 10 | 69 kV line | 16.449 | 31.794 | 40 | --- | --- |
10 | 11 | 69/34.5 kV transformer | 0.298 | 5.917 | 20 | 5.296 | 1.003 |
11 | 12 | 34.5/13.8 kV transformer | 0.435 | 7.923 | 6 | 1.638 | 1.011 |
5 | 13 | 69 kV line | 21.715 | 21.696 | 24 | --- | --- |
13 | 14 | 69/34.5 kV transformer | 0.298 | 5.917 | 20 | 7.059 | 1.146 |
14 | 15 | 34.5/13.8 kV transformer | 0.435 | 7.923 | 6 | 1.468 | 0.706 |
Type of Day | Quarters | |
---|---|---|
Working day | Q1 (January ÷ March) | 64 |
Weekend | 26 | |
Working day | Q2 (April ÷ June) | 65 |
Weekend | 26 | |
Working day | Q3 (July ÷ September) | 66 |
Weekend | 26 | |
Working day | Q4 (October ÷ December) | 66 |
Weekend | 26 |
Voltage Level | Inductive Mode | (Capacitive Mode) |
---|---|---|
Vnom < 69 kV | 0.92 ÷ 1.00 | 0.92 ÷ 1.00 |
Vnom ≥ 69 kV | 0.95 ÷ 1.00 | --- |
Yearly Power Factor Violations | |
---|---|
< 10% | 0.100 |
10% ≥ > 20% | 0.125 |
20% ≥ > 40% | 0.175 |
40% ≥ > 60% | 0.225 |
60% ≥ > 80% | 0.350 |
≥ 80% | 0.600 |
Size Code | Size (MVar) | Lifespan | Switching Limits | Unit Cost | |
---|---|---|---|---|---|
(Vnom < 69 kV) | (Vnom ≥ 69 kV) | ||||
0 | No capacitor bank unit in the node () | ||||
1 | 1.2 MVar | 5 years | 30 op/year | 100 k$ | 200 k$ |
2 | 2.4 MVar | 5 years | 30 op/year | 200 k$ | 440 k$ |
3 | 3.6 MVar | 5 years | 30 op/year | 300 k$ | 690 k$ |
4 | 5.4 MVar | 5 years | 30 op/year | 400 k$ | 1000 k$ |
(M$/Year) | (M$/Year) | (M$/Year) | (M$/Year) | |
---|---|---|---|---|
Existing case | --- | 10.51 | 0.31 | 10.81 |
(a) Without pattern control optimization | 0.28 | 9.40 | 0.00 | 9.68 (−10.5%) |
(b) With pattern control optimization | 0.36 | 9.03 | 0.10 | 9.49 (−12.3%) |
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Soma, G.G. Optimal Sizing and Placement of Capacitor Banks in Distribution Networks Using a Genetic Algorithm. Electricity 2021, 2, 187-204. https://doi.org/10.3390/electricity2020012
Soma GG. Optimal Sizing and Placement of Capacitor Banks in Distribution Networks Using a Genetic Algorithm. Electricity. 2021; 2(2):187-204. https://doi.org/10.3390/electricity2020012
Chicago/Turabian StyleSoma, Gian Giuseppe. 2021. "Optimal Sizing and Placement of Capacitor Banks in Distribution Networks Using a Genetic Algorithm" Electricity 2, no. 2: 187-204. https://doi.org/10.3390/electricity2020012
APA StyleSoma, G. G. (2021). Optimal Sizing and Placement of Capacitor Banks in Distribution Networks Using a Genetic Algorithm. Electricity, 2(2), 187-204. https://doi.org/10.3390/electricity2020012