Improved Voltage Profile

Highly utilized feeders with high reactive power demands have poor voltage profiles and experience large voltage variations when loading levels change significantly. In a power system, it is desirable to regulate bus voltages within a narrow range (e.g., + 5%, –8% of their nominal values) and to have a balanced load on all three phases (eliminating negative- and zero-sequence currents with undesirable consequences such as equipment heating and torque pulsations in generators and turbines). However, loads of the power system fluctuate and can result in voltage deviations beyond their acceptable limits. In view of the fact that the internal impedance of the AC system is mainly inductive, it is the reactive power change of the load that has the most adverse effect on voltage regulation.

Traditionally, line voltage drop is compensated with distributed capacitors along a feeder, and by switching them according to the reactive power demand a rather constant voltage profile is achieved. Furthermore, by keeping the power factor close to unity along the section in question, the number of expensive voltage regulators can be minimized.

Impacts of active and reactive current variations on the terminal load voltage (V_t) are demonstrated in Fig. 10.1. An increase of ΔQ in the lagging VArs drawn by the load (indicated by quantities with primes in Fig. 10.1b) causes the reactive current component to increase to (I_q + ΔI_q) while the active current component I_p is assumed to be unchanged. This will result in a terminal voltage drop of ΔV_t and decreases the real power. Figure 10.1c shows the phasor diagram where the active current component increases to (I_p + ΔI_p) with ΔI_p = ΔI_q (of Fig. 10.1b) while I_q is assumed to be unchanged. Comparison of phasor diagrams indicates that variations in reactive power (Fig. 10.1b) have more impact on voltage regulation as compared with active power variations (Fig. 10.1c).

Reduced Power System Losses

Correcting the power factor can substantially reduce network losses in between the point in the network where the correction is performed and the source of generation. This can result in an annual gross return of as much as 15% on the capacitor investment [48]. To maximize benefits, power factor correction should be implemented as close to the customer load as possible. However, due to economies of scale, installation of capacitors at the low-voltage (LV) level is very costly when compared with high-voltage (HV) options.

In most industrial plant power distribution systems, the I²R losses vary from 2.5 to 7.5% of the load kWh. This depends on the hours of full-load and no-load plant operation, wire size, and length of the main and branch feeder circuits. Capacitors are effective in reducing only that portion of the losses that is due to the reactive current. Losses are proportional to the current squared, and since current is reduced in direct proportion to power factor improvement, the losses are inversely proportional to the square of the power factor. Hence, as the power factor is increased with the addition of capacitor banks to the system, the amount of the losses is reduced. The connected capacitor banks have losses, but they are relatively small. Losses account for approximately one-third of one percent of the reactive power rating [48].

Released Power System Capacity

When capacitors are placed in a power system, they deliver reactive power and furnish magnetizing current for motors, transformers, and other electromagnetic devices. This increases the power factor and it means that for a lower current (or apparent power) level a higher real power utilization occurs. Therefore, capacitor banks can be used to reduce overloading or permit additional load to be added to the existing system. Release of system capacity by power factor improvement due to capacitors is becoming extremely important because of the associated economic and system benefits.

Reactive power compensation improves network utilization; that is, capacity is released and it allows more customers to be connected to what could have been a highly utilized feeder, without going to the costly extreme of adding an additional feeder.

Increased Plant Ratings

Another undesirable effect of a network with poor power factor is a possible reduction in plant ratings. This is often noticeable when transformer ratings are limited by overvoltage available from tapping. Correcting power factor to unity at the zone substation will maximize the real power rating of transformers by providing access to thermal emergency ratings.

Capital Deferment

As ratings are based on apparent power rather than active power, reduced apparent power utilization reduces operating risks and can therefore defer expenditure related to the network expansion. This deferment can be applied from the distribution feeder, back to the zone substation, subtransmission, terminal station, transmission network, and generation.

Internally Fused Open-Rack Capacitor Bank

These banks have internal fuses for the current-limiting action; one fuse is connected in series with each element within the capacitor. They are designed and coordinated to isolate internal faults at the capacitor-element level and allow continued operation of the remaining elements of that capacitor unit. A very small part of the capacitor is being disconnected (self-healing), and therefore the capacitor bank remains in service. The designs are customized per customer specifications and generally include racks, capacitors, insulators, switches, imbalance protection scheme, and current-limiting reactors. An internally fused open-rack capacitor bank provides the following main advantages:

• cost-effective,

• utilizes large kVAr size units,

• few live parts are exposed,

• well-known technology (over 50 years of experience), and

• decreased installation and maintenance time and cost with proven reliability in the field.

Externally Fused Open-Rack Capacitor Bank

These open-rack capacitor banks [49] are typically used at substations and utilize an external fuse (current limiting) as a means of unit protection. The designs are customized per customer specifications and generally include racks, capacitors, insulators, fuses, elevation structures, switches, imbalance protection scheme, and current-limiting reactors. An externally fused open-rack capacitor bank provides the following main advantages:

• visual indication of blown fuses,

• utilizes small kVAr size units, and

• well-known technology (over 70 years of experience).

Fuseless Open-Rack Capacitor Bank

These types of open-rack substation banks [49] utilize an imbalance protection scheme as the primary means of protection. There are no internal or external fuses in this design. The “string” configuration provides a very reliable and cost-effective design for banks rated 35 kV and above. A fuseless open-rack capacitor bank provides the following main advantages:

• cost-effective and compact size,

• utilizes large kVAr size units,

• few live parts are exposed,

• reduced installation and maintenance time as well as reduced cost, and

• proven reliability in the field.

Drawbacks of Analytical Methods

One drawback of all analytical methods is the assumption of continuous variables. The calculated capacitor sizes may not match the available standard sizes and the optimized placements may not coincide with the physical node locations in the distribution system. The results would need to be rounded up or down to the nearest practical value and may result in an overvoltage situation or a loss saving less than the calculated one.

Therefore, the earlier works provide a rough rule of thumb for capacitor placement on feeders with evenly distributed loads. More sophisticated analytical methods are suitable for distribution systems of considerable sizes, but they require more distribution system information and more time to implement.

Drawbacks of Numerical Programming Methods

The level of sophistication and the complexity of the numerical programming models increase in chronological order of their publication time. This progression concurs with the advancement in computing capability. At the present time, computing is relatively inexpensive and many general numerical optimization packages are available to implement any of the above algorithms. Some of the numerical programming methods have the advantage of considering feeder node locations and capacitor sizes as discrete variables, which is an advantage over analytical methods. However, the data preparation and interface development for numerical techniques may require more time than for analytical methods. The main drawback of all numerical programming approaches is the possibility of arriving at a local optimal solution. Therefore, one must determine the convexity of the capacitor placement problem to determine whether the result yielded by a numerical programming technique is either a local or global extreme.

Drawbacks of Heuristic Methods

Heuristic methods are intuitive, easier to understand, and simpler to implement than other optimization techniques such as analytical and numerical programming approaches. However, the results produced by heuristic algorithms are not guaranteed to be optimal. Other disadvantages associated with the use of a heuristic search algorithm are as follows:

• The criterion of selecting the nodes with the highest impact on the losses in a feeder section is not totally representative for the state of the power losses of the whole system and may result in inaccurate results;

• Compensation is based on loss reduction in the system rather than net dollar savings;

• It requires a large number of power flow runs;

• It only considers fixed capacitors; and

• It does not have the flexibility to control the balance between the global and local exploration of the search space, and its results may not be globally optimal.

Drawbacks of AI-Based Methods

The user may encounter convergence problems with GAs, SA, and ANNs methods. For on-line applications, ANNs can only be used for a particular load pattern. ANNs would need to be retrained for every different set of load curves that are characteristic of the distribution system. However, the use of ESs is better suited for on-line and dynamic applications.

Genetic Representation

Genetic algorithms are derived from a study of biological systems. In biological systems evolution takes place on organic components used to encode the structure of living beings. These organic components are known as chromosomes. A living being is only a decoded structure of the chromosomes. Natural selection is the link between chromosomes and the performance of their decoded structures. In genetic algorithms the design variables or features that characterize an individual are represented in an ordered list called a string. Each design variable corresponds to a gene and the string of genes corresponds to a chromosome. A group of chromosomes is called a population. The number of chromosomes in a population is usually selected to be between 30 and 300. Each chromosome is a string of binary codes (genes) and may contain substrings. The merit of a string is judged by the fitness function, which is derived from the objective function and is used in successive genetic operations. During each iterative procedure (referred to as generation), a new set of strings with improved performance is generated using three GA operators (namely, reproduction, crossover, and mutation).

Structure of Chromosomes

Figure 10.2 shows an example illustrating the basic structure of a chromosome. Bit strings consisting of 0’s and 1’s have been found to be most effective in representing a wide variety of information in the problem domain involving function optimization. Many other representation schemes such as ordered lists, embedded lists, and variable-element lists have been developed for specific industrial applications of genetic algorithms.

Initialization

Genetic algorithms operate with a set of strings instead of a single string. This set of strings is known as a population and is put through the process of evolution to produce new individual strings. To begin with, the initial population could be seeded with heuristically chosen strings or at random. In either case, the initial population should contain a wide variety of structures to ensure the complete problem space is covered.

Evaluation (Fitness) Function

The evaluation function is a procedure to determine the fitness of each chromosome in the population and is very much application orientated. Since genetic algorithms proceed in the direction of evolving the fittest chromosomes, and the fitness value is the only information available to the algorithm, the performance is highly sensitive to the fitness values. In the case of optimization routines, the fitness is the value of the objective function to be optimized. Genetic algorithms are basically unconstrained search procedures in the given problem domain. Any constraints associated with the problem can be incorporated into the objective function as penalty functions.

Genetic Operators

Genetic operators are the stochastic transition rules applied to each chromosome during each generation procedure to generate a new improved population from an old one. A genetic algorithm usually consists of reproduction, crossover, and mutation operators.

• Reproduction is a probabilistic process for selecting two parent strings from the population of strings on the basis of roulette-wheel mechanism, using their fitness values. This process is accomplished by calculating the fitness function for each chromosome in the population and normalizing their values. A roulette wheel is constructed where each section of the wheel represents a certain chromosome and its relative fitness value. This ensures that the expected number of times a string is selected is proportional to its fitness relative to the rest of the population. Therefore, strings with higher fitness values have a higher probability of contributing offspring and are simply copied into the next generation.

• Crossover is the process of selecting a random position in the string and swapping the characters either left or right of this point with another similarly partitioned string. This random position is called the crossover point. For example, if two parent strings are x₁ = 1010 : 11 and x₂ = 1001 : 00 and the crossover point has been selected as shown by the colon, then if swapping is implemented to the right of this point, the resulting offspring structures would be y₁ = 1010 : 00 and y₂ = 1001 : 11 (e.g., the two digits “11” to the right of the colon in x₁ are swapped with the two digits “00” to the right of the colon in x₂). The probability of crossover occurring for two parent chromosomes is usually set to a large value in range of 0.7 to 1.0.

• Mutation is the process of random modification of a string position by changing “0” to “1” or vice versa, with a small probability. It prevents complete loss of genetic material through reproduction and crossover by ensuring that the probability of searching any region in the problem space is never zero. The probability of mutation is usually assumed to be small (e.g., between 0.01 and 0.1).

Genetic Parameters

Genetic parameters are the entities that help to regulate and improve the performance of a genetic algorithm. Some of the parameters that can characterize the genetic space are as follows:

• Population size affects the efficiency and performance of the genetic algorithm. A smaller population would not cover the entire problem space, resulting in poor performance. A larger population would cover more space and prevent premature convergence to local solutions; however, it requires more evaluations per generation and may slow down the convergence rate.

• Crossover rate affects the rate at which the process of crossover is applied. In each new population, the number of strings that undergo the process of crossover can be depicted by a chosen probability (crossover rate). A higher crossover rate introduces new strings more quickly into the population; however, if the rate is too high, high-performance strings are eliminated faster than selection can produce improvements. A low crossover rate may cause stagnations due to the lower exploration rate, and convergence problems may occur.

• Mutation rate is the probability with which each bit position of each string in the new population undergoes a random change after the selection process. A low mutation rate helps to prevent any bit position from getting stuck to a single value, whereas a high mutation rate can result in essentially random search.

Convergence Criterion

The iterations (regenerations) of genetic algorithms are continued until all generated chromosomes become equal or the maximum number of iterations is achieved. Due to the randomness of the GA method, the solution tends to differ for each run, even with the same initial population. For this reason, it is suggested to perform multiple runs and select the “most acceptable” solution (e.g., with most benefits, within the permissible region of constraints).

Flowchart of Genetic Algorithm for Capacitor Placement and Sizing

Figure 10.3 shows a simplified flowchart of an iterative genetic algorithm for optimal placement and sizing of capacitor banks in distribution system under sinusoidal operating conditions. An initial population is randomly selected, genetic operators are applied, and the old population is replaced with the new generation. This process is repeated until the algorithm has converged.

Drawbacks of Genetic Algorithms

There are disadvantages associated with the use of a genetic algorithm:

• It does not have flexibility to control the balance between global and local exploration of search space;

• It requires relatively large computational time and effort; and

• Convergence problems may occur, which may be overcome by solving the problem a number of times with different initial populations and selecting the best solution.

Drawbacks of Artificial Neural Networks

ANNs may not be suitable for a large distribution system since many smaller subsystems are required and the training time becomes excessive. However, once networks are trained, iterative calculations are no longer required and a fast solution for a given set of inputs can be provided. Gu and Rizy [21] followed the method of [20] and included an additional functionality of regulator control.

Fuzzy Logic and Fuzzy Inference System

Fuzzy logic in terms of fuzzy sets is about mapping an input space to an output space. The main operation to achieve this end is through the use of if–then statements called rules. These rules are used to refer to variables and the adjectives that describe those variables. This is the main theory behind the operation of a fuzzy inference system. For the system to interpret these rules, they must first be defined in terms of the data they will control and the adjectives that will be used. A good example of this is a system defining how hot the water is coming out of a hot water system. We need to define the range of the expected water temperatures, as well as what is meant by the word “hot.”

Fuzzy Sets

A fuzzy set is a set without a crisp, clearly defined boundary. It can contain elements with partial, full, or no membership. This differs quite dramatically from the notion of a classical set, which is defined by the inclusion or exclusion of certain elements.

Fuzzy logic is based on the fact that the truth of any statement becomes a matter of degree. The true power of using fuzzy logic is not that it is an extremely complex tool which can give a direct answer, rather that it can model human perception, language, and thinking, thus being able to use linguistic descriptors to answer a problem that would normally need or expect a yes–no answer with a not-quite-yes–no answer. This is done by expanding the familiar Boolean logic of 1 for yes and 0 for no. This means that if absolute truth is defined by 1 and that absolute falsehood is defined by 0, fuzzy logic will permit alternative logical answers between the range of [0 1] such as 0.35 or 0.6521 and by doing so be able to recreate such linguistic descriptors in relation to an element’s relationship to a set as medium, high, low, very low, very high, and so on. Now with the possibility of a range of logical answers a more descriptive set can be created. A plot mapping an input space to an output space is considered to be a membership function.

Membership Function

A membership function is a curve that defines how each point in the input space is mapped to a membership value (or degree of membership) between 0 and 1. The main requirement for a membership function is that it varies between 1 and 0. The shapes of membership functions are created from such requirements as simplicity, convenience, speed, and efficiency.

A fuzzy set is an extension of a classical set. If X is the universe of discourse and its elements are denoted by x, then a fuzzy set A in X is defined as a set of ordered pairs

$A=\left\{(,x,{\mu }_A\left(x\right)\left)\right|x\in X\right\}\text{,}$

where μ_A(x) (called the membership function of x in A) maps each element of X to a membership value between 0 and 1.

Many types of membership functions have been proposed and implemented. They can be created from scratch or even tailored or trained to conform to a set of data. There are many types of membership functions such as triangular, trapezoidal, symmetric Gaussian (defines a f(x; σ,c) = exp(–(x – c) ²/2σ²))) and sigmoidal. The possibility of creating membership functions based on already existing mathematical formulas and functions is endless, such as membership functions created from the difference and/or product of two sigmoidal functions.

Fuzzy Logic Operators

The logical operations associated with fuzzy logic can be described as a superset of standard Boolean logic. The operations in Boolean known as AND, OR, and NOT can all be repeated in fuzzy logic. Shown in Table 10.4 are these three integral Boolean operations with their related logical operations in fuzzy.

Fuzzy If–Then Rules

If–then rules are used to define the relationships between variables based on defined logic and membership functions. An if–then rule is created in the form

$\mathrm{If}x\mathrm{is}A\mathrm{then}y\mathrm{is}B\text{.}$

A and B are linguistic values defined by fuzzy sets on the ranges (universes of discourse) X and Y, respectively. The if-part of the rule is called the antecedent (or premise), whereas the then-part of the rule (y is B) is called the consequent (or conclusion).

Interpreting if-then rules is a three-part process:

• Fuzzify inputs: Resolve all fuzzy statements in the antecedent to a degree of membership between 0 and 1. If there is only one part to the antecedent, this is the degree of support for the rule.

• Apply fuzzy operator to multiple part antecedents: If there are multiple parts to the antecedent, apply fuzzy logic operators and resolve the antecedent to a single number between 0 and 1. This is the degree of support for the rule.

• Apply implication method: Use the degree of support for the entire rule to shape the output fuzzy set. The consequent of a fuzzy rule assigns an entire fuzzy set to the output. This fuzzy set is represented by a membership function that is chosen to indicate the qualities of the consequent. If the antecedent is only partially true (i.e., is assigned a value less than 1), then the output fuzzy set is truncated according to the implication method.

Implication

Implication is the method used to create a final consequent function or set. The if–then rule is often not sufficient enough and more rules may be necessary to create a satisfying result. Consider an extension to this rule shown below,

$\mathit{If}x\mathrm{is}A\mathrm{OR}y\mathrm{is}B\mathrm{then}z\mathrm{is}C\text{.}$

This expanded if–then statement now has two rules involved that are controlled by the logical operation OR. Implication becomes the means by which the consequent is created from this rule, which is the “z is C” part.

Two main methods of implication are often used. These are entitled minimum (similar to the AND operation previously discussed) and the product methods. The minimum method will truncate the output fuzzy set, and the product method will scale the output fuzzy set deriving an aggregated output from a set of triggered rules.

Fuzzy Inference

The Mamdani implication methods [50] are usually used for inferencing. Two implication methods are relied on to determine the aggregated output from a set of triggered rules:

• The Mamdani max–min implication method of truncating the consequent membership function of each discarded rule (at the minimum membership value of all the antecedents) and determining the final aggregated membership function.

• The Mamdani max–prod implication method of scaling the consequent membership function of each discarded rule (to the minimum membership value of all the antecedents) and determining the final aggregated membership function.

Defuzzification

Defuzzification is the process of obtaining a single number from the output of the aggregated fuzzy set. It is used to transfer fuzzy inference results into a crisp output. In other words, defuzzification is realized by a decision-making algorithm that selects the best crisp value based on a fuzzy set. There are several forms of defuzzification including center of gravity (COG), mean of maximum (MOM), and center average methods. The COG method returns the value of the center of area under the curve and the MOM approach can be regarded as the point where balance is obtained on a curve.

Fuzzy Expert System for Capacitor Placement and Sizing

A fuzzy expert system based on approximate reasoning is defined and relied on to determine suitable candidate buses in a distribution system for capacitor placement. Voltages and power loss reduction indices of distribution system buses are modeled by fuzzy membership functions. A fuzzy expert system (FES) containing a set of heuristic rules is then employed to determine the capacitor placement suitability of each bus in the distribution system. Capacitors are then placed on the nodes with the highest suitability using the following steps:

• A load flow program calculates the power-loss reduction (PLR) of the system with respect to each bus when the reactive load current at that bus is compensated. The PLRs are then linearly normalized into a [0,1] range with the highest loss reduction being 1 and the lowest being 0.

• The PLR indices along with the per-unit nodal voltages (before PLR compensation) are the two inputs that are used by the fuzzy expert system. The FES then determines the most suitable node for capacitor installation by fuzzy inferencing.

• A savings function relating to the annual cost of the capacitor (determined by size) is defined and the savings achieved from PLR is maximized to determine the optimal sizing of capacitor bank to be placed at that node.

• The above process will be repeated, finding another node for capacitor placement, until no more financial savings can be realized (Fig. 10.4).

Drawbacks of Fuzzy Set Theory

The fuzzy optimization approach is very simple and naturally fast as compared to other optimization methods. However, for some problems the procedure might get trapped in a local optimal point and fail to converge to the global (or near global) optimal solution.

Drawbacks of Graph Search Algorithm

Many assumptions and constraints are required to obtain an optimal solution. This is not always possible for the operating conditions of actual distribution systems. Therefore, solutions based on graph search algorithms may not be very accurate.

Drawbacks of Particle Swarm Algorithm

The performance of particle swarm optimization is not competitive with respect to some problems. In addition, representation of the weights is difficult and the genetic operators have to be carefully selected or developed.

Drawbacks of Tabu Search Algorithm

The main disadvantage of tabu approach is its low solution speed. In addition, it uses certain control parameters that may be system dependent and difficult to determine.

Drawbacks of Sequential Quadratic Programming

Large numbers of computations are involved in sequential quadratic programming. Many mathematical assumptions have to be made in order to simplify the problem; otherwise it is very difficult to calculate the variables.

Case 1: High Harmonic Distortion

The nonlinear load in Fig. E10.3.1 is a six-pulse rectifier with active and reactive powers of 0.3 pu (3 MW) and 0.226 pu (2.26 MVAr), respectively. Results based on harmonic power flow show for this system a maximum voltage THD_v of 8.48%. The MSS and MSS-LV algorithms are applied to this system for optimal placement and sizing of capacitor banks.

Results of the MSS method show a yearly benefit of $20360 (last row of Table E10.3.1) and different locations and sizes of capacitor banks before (column 2 of Table E10.3.1) and after (column 3 of Table E10.3.1) optimization. The maximum voltage THD_v is only reduced to 6.37% (row 19 of Table E10.3.1 and Fig. E10.3.2). Therefore, considerable yearly benefit is achieved but power quality constraints are not satisfied.

By applying the MSS-LV algorithm, we can decrease the level of voltage THD_v to 4.72% (row 19 of Table E10.3.1). As expected (due to power quality improvement), yearly benefit is slightly decreased (to $18162). If in highly harmonic distorted systems the MSS-LV algorithm does not satisfy power quality constraints, the application of passive filters and active filters (Chapter 9) before capacitor placement is recommended.

Case 2: Medium Harmonic Distortion

The size of the nonlinear load is reduced such that its active and reactive powers are 0.12 pu and 0.166 pu, respectively. After optimal placement and sizing of capacitor banks by the MSS method, voltage THD_v decreases from 5.18 to 5.00% and a yearly benefit of $11650 is achieved. Applying the hybrid MSS-LV algorithm causes minor changes (e.g., eliminating one capacitor unit). Therefore, the performances of the two algorithms are similar for this case. This means that the output of MSS algorithm is near the system global solution.

Case 3: Low Harmonic Distortion

The size of the nonlinear load is further decreased such that its active and reactive powers are 0.02 pu and 0.046 pu, respectively. After optimal placement and sizing of capacitor banks by the MSS method, voltage THD_v increases from 2.1 to 2.95% (which is lower than the desired limit of 3%), total capacitance of capacitor banks decreases from 1.005 to 0.63 pu, and a yearly benefit of $12608 results. By applying the MSS-LV algorithm, considerable benefit is achieved ($44787), while voltage THD_v is at 2.87%.

Sensitivity of Objective Function

Sensitivities of objective function (Eq. 10-8) to capacitance values at compensation buses can be computed using Eqs. 10-10 and 10-11.

Sensitivity of THD_v

Sensitivities of THD_v (Eq. 10-6) to capacitance values at compensation buses is computed using the partial derivatives

$S_{\mathit{THDv},j}=\frac{\partial TH{D}_{v,j}}{\partial Q}={\displaystyle \sum _{h=1}^L{W}_h}\frac{\partial TH{D}_{v,j}}{\partial Q^{\left(h\right)}}\text{,}$

where W_h = 1/h is the proposed weighting function for harmonic order h. The partial derivatives of THD_v,j are computed as follows:

$\left[\begin{array}{l}\frac{\partial TH{D}_{v,j}}{\partial P^{\left(h\right)}}\hfill \\ \frac{\partial TH{D}_{v,j}}{\partial Q^{\left(h\right)}}\hfill \\ \frac{\partial TH{D}_{v,j}}{\partial I_r^{\left(h\right)}}\hfill \\ \frac{\partial TH{D}_{v,j}}{\partial I_i^{\left(h\right)}}\hfill \end{array}\right]={\left[\begin{array}{llll}\frac{\partial P^{\left(\mathrm{h}\right)}}{\partial {\theta }^{\left(h\right)}}\hfill & \frac{\partial Q^{\left(\mathrm{h}\right)}}{\partial {\theta }^{\left(h\right)}}\hfill & \frac{\partial I_r^{\left(h\right)}}{\partial {\theta }^{\left(h\right)}}\hfill & \frac{\partial I_i^{\left(h\right)}}{\partial {\theta }^{\left(h\right)}}\hfill \\ \frac{\partial P^{\left(\mathrm{h}\right)}}{\partial V^{\left(h\right)}}\hfill & \frac{\partial Q^{\left(\mathrm{h}\right)}}{\partial V^{\left(h\right)}}\hfill & \frac{\partial I_r^{\left(h\right)}}{\partial V^{\left(h\right)}}\hfill & \frac{\partial I_i^{\left(h\right)}}{\partial V^{\left(h\right)}}\hfill \end{array}\right]}^{-1}\left[\begin{array}{l}\frac{\partial TH{D}_{v,j}}{\partial {\theta }^{\left(h\right)}}\hfill \\ \frac{\partial TH{D}_{v,j}}{\partial V^{\left(h\right)}}\hfill \end{array}\right]\text{,}$

where all right-hand-side matrix entries are easily computed using results from the application of a harmonic power flow program.

Fuzzification of Sensitivities

For the sensitivities of objective function (Eqs. 10-10 and 10-11) and constraints (Eqs. 10-12 and 10-13) with respect to the reactive power injection at each bus, the exponential fuzzifier functions ${\mu }_F^{\mathrm{sensitivity}},{\mu }_{TH{D}_{v,j}}^{\mathrm{sensitivity}},\text{and}{\mu }_{{}_{v,j}}^{\mathrm{sensitivity}}$ are employed. Figure 10.7a shows the fuzzifier for the sensitivity of THD_v,j. S_THD^max is a large number, that is, buses with higher sensitivities have higher values of membership.

Fuzzification of Objective Function and Constraints

For the objective function (Eq. 10-8) a membership function (μ_F) is defined as depicted in Fig. 10.7b. F_o is the initial value of objective function and F₁ is the value of objective function after compensating the total reactive (inductive) loads at every bus of the distribution system. High objective values are given a low membership function value, whereas low objective values are assigned a high membership function value.

For voltage constraints, it is suitable to use a membership function (μ_v,j) with trapezoidal form, as shown in Fig. 10.7c. V^min and V^max correspond to Eq. 10-5 such that buses with voltages values between the two limit values V^min and V^max have full membership. $V_0^{-}$ and $V_0^{+}$ correspond to initial values of the bus effective voltage. Therefore, a bus with high rms voltage deviation (Eq. 10-5) is given a low value and a bus with low rms voltage deviation is given a high value.

For THD_v,j a membership function (μ_THDv,j) is defined as illustrated in Fig. 10.7d. THD_v^max and THD_v0 are the limit and the initial value of THD_v,j corresponding to Eq. 10-6, respectively. Buses are given a high value (e.g., μ_THDv,j = 1) unless their THD_v are higher than the IEEE-519 limits.

Fuzzy Combination of Membership Functions

The algebraic product of the membership functions is used such as the t-norm for THD_v,j:

${\mu }_{\mathit{THDv}}=t_1\left({\mu }_{\mathit{THDv},1}\cdot {\mu }_{\mathit{THDv},2}\dots {\mu }_{\mathit{THDv},n}\right)={\displaystyle \prod _{j=1}^n{\mu }_{\mathit{THDv},j}\text{.}}$

The same formulations apply to t-norms of the bus voltages, sensitivity of THD_v,j and sensitivity of voltage constraints, the combination of objective function and constraints, as well as their sensitivities to reactive power injection:

${\mu }_{\mathit{total}}^{\mathrm{sensitivity}}=t_2\left({\mu }_{\mathit{THDv}}^{\mathrm{sensitivity}},{\mu }_V^{\mathrm{sensitivity}},{\mu }_F^{\mathrm{sensitivity}}\right)$

${\mu }_{\mathrm{total}}=t_3\left({\mu }_{\mathit{THDv}},{\mu }_V,{\mu }_F\right)\text{.}$

Alpha-Cut Process

In order to direct the algorithm toward the permissible region bordered by constraints, the α-cut process is used at each iteration. To do this, the values of α for F, THD_v, and V are set to their corresponding membership values in the previous iteration (e.g., ${\alpha }_F^{\ell +1}={\alpha }_F^{\ell }$ where ℓ is the iteration number).

Analysis

The initial starting point of Eq. 10-8 (e.g., the initial compensation buses) does not usually reside inside the permissible solution region. Therefore, some type of criteria (based on simultaneous improvements of F, THD_v, and V in consecutive iterations) is required to direct the solution toward the permissible region and to select the most appropriate buses for capacitor placement. This section uses the fuzzy set theory to measure individual improvements, generate a single criterion to direct the solution, and guarantee the monotonous convergence of the solution at each iteration. Other approaches with similar characteristics (e.g., analytical hierarchy process) could also be used.

Note that the locations of candidate buses as indicated by the sensitivities of F, THD_v, and V are usually different. A single criterion – a fuzzy combination – is employed to determine the optimal locations of candidate buses. The exponential sensitivity functions of Fig. 10.7a are solely used for ranking of candidate buses. Other methods could be employed for this ranking (e.g., weighted average, statistical methods). The selected method will not have any impact on the capacitor placement algorithm. However, the selected trapezoidal membership functions of Fig. 10.7b–d for F, V, and THD_v have major impacts on the solution process (e.g., improving or deteriorating convergence).

The inclusion of sensitivity functions and fuzzification of objective and power quality constraints will automatically eliminate all solutions generating extreme values of voltages and/or currents and prevents fundamental and harmonic parallel resonances.

Case 1: High Harmonic Distortion

The nonlinear load in Fig. E10.3.1 is a six-pulse rectifier with active and reactive powers of 0.3 pu (3 MW) and 0.226 pu (2.26 MVAr), respectively. Results of harmonic power flow indicate a maximum voltage THD_v of 8.48% for this system (column 2 of Table E10.4.1).

Application of the MSS method to Fig. E10.3.1 shows that optimal capacitor placement results in considerable yearly benefit (e.g., $20653 per year) but it does not limit THD_v to the desired level of 5% (column 3 of Table E10.4.1). This is expected from the MSS method dealing with rich harmonic configurations, where capacitor placement is not the primary solution for harmonic mitigation.

The fuzzy algorithm of Fig. E10.3.1 is also applied to this system for optimal placement and sizing of capacitor banks. Results show a yearly benefit of $14732 per year (last row of Table E10.4.1) and different locations and sizes of capacitor banks before (column 2 of Table E10.4.1) and after (column 4 of Table E10.4.1) the fuzzy optimization. In addition, maximum voltage THD_v is reduced to 4.89% (row 16 of Table E10.4.1) and total capacitance is decreased by 24%.

Case 2: Low Harmonic Distortion

The size of the nonlinear load is decreased such that its active and reactive powers are 0.02 pu and 0.046 pu, respectively. After optimal placement and sizing of capacitor banks with MSS method, THD_v increases from 2.1 to 2.95% (which is lower than the desired limit of 5%), total capacitance of capacitor banks is decreased from 1.005 pu to 0.630 pu and a yearly benefit of $13271 is achieved (column 6 of Table E10.4.1).

Application of the fuzzy algorithm results in THD_v increase from 2.1 to 2.63% (which is in the permitted region) and also total capacitance decreases from 1.005 to 0.675 pu (column 7 of Table E10.4.1). In addition, annual savings are increased to $16861 (e.g., 27% greater savings than that of the MSS method). These results demonstrate the fine capability of the fuzzy method for capacitor placement in the presence of harmonics.

Structure of Chromosomes

Different types of chromosomes for GAs have been proposed and implemented. This section employs the chromosome structure of Fig. 10.9 consisting of MC substrings of binary numbers, where MC denotes the number of possible compensation (candidate) buses for capacitor placement in the entire feeder. The binary numbers indicate the size of the installed capacitor at the bus under consideration.

Fitness Functions

The inverse algebraic product of penalty functions (Fig. 10.10) is used as the fitness function combining objective and constraint functions:

$\begin{array}{l}{F}_{\mathit{fitness}}=1/\left(F_F\cdot F_{\mathit{THDv}}\cdot F_V\cdot F_U\right)\\ F_{\mathit{THDv}}={\displaystyle \prod _{i=1}^n{F}_{\mathit{THDv},i}}\\ F_v={\displaystyle \prod _{i=1}^n{F}_{v,i}}\end{array}$

where penalty functions F_F, F_THDv,j, F_v,i, and F_U are shown in Fig. 10.10.

Genetic Operators

Three genetic operators including reproduction (based on the roulette-wheel mechanism), crossover (with crossover probability of 0.6 to 1.0), and mutation (with mutation probability of 0.01 to 0.1) are established.

Convergence Criterion

The iterations (regenerations) of the genetic algorithm are continued until all generated chromosomes become equal or the maximum number of iterations (N^max = 40) is reached. Due to the randomness of the GA method, the solution tends to differ for each run, even with the same initial population. Therefore, multiple runs are performed and the most acceptable solution (with most benefits within the permissible region of constraints) is selected.

Computing Time

Unfortunately, the application of GAs often looks like a never-ending process. In fact, only a compromise between population size, mutation rate, and so on can lead to an adequate algorithm that finds good solutions in a relatively short time. However, capacitor bank placement is usually considered a planning problem and its computing time is not of great concern. The computational burden associated with the genetic algorithm is included and compared with those of the MSS, MSS-LV, and fuzzy methods as shown in row 15 of Table E10.5.1 and row 19 of Table E10.6.1. As expected, the computing times of the genetic algorithm are much longer.