# Reactive Power Optimization Based on the Application of an Improved Particle Swarm Optimization Algorithm

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Problem Statement

#### 1.2. Research Contribution

#### 1.3. Manuscript Organization

## 2. Literature Review

#### 2.1. Common Optimization Objectives in Smart Grids

- (a)
- Generation cost (GC);
- (b)
- Real power losses (RPL);
- (c)
- Transient stability (TST);
- (d)
- Voltage profile improvement (VPI);
- (e)
- Emissions (EM);
- (f)
- Grid resiliency (GR).

#### 2.2. Optimization Algorithms for Smart Grid Optimization

#### 2.3. Smart Grid Trends

_{2}emissions, which have also been subject to the SDG discussed by the United Nations, following the notion of net-zero emissions. As a result, the services related to energy can be categorized into two main groups, namely: (i) the availability of data on electrical energy consumption, and (ii) the management of energy data. The objective is to improve environmental conditions and enhance the sustainability of future cities. With the support of the infrastructure resulting from the deployment and advancement of SG, smart cities are now within reach for everyone, allowing consumers to access detailed information about their home’s electricity usage, compare their consumption with similar users, and receive personalized recommendations to reduce energy consumption in a smart way. These services enable consumers to act, plan and manage their energy consumption, interact with controllable loads, and make autonomous and intelligent decisions. Furthermore, energy service providers are expected to offer their customers a wider range of services and systems to actively manage their energy demand [42].

- More efficient network activity monitoring;
- More efficient mitigation of distribution service interruptions and reduction of the total number of affected customers;
- Faster and more reliable malfunction management;
- Reconfiguration of network structure in near real-time;
- Provision of new services toward better quality of service and user experience,

- Demand response initiatives: Engaging users in the energy supply system has become a necessity for ensuring service availability and quality during high-demand periods. Smart grids leverage the widespread adoption of intelligent appliances to exert greater control over demand, which in turn facilitates the provision of more economic services to customers [45].
- Smart metering: Deploying advanced metering infrastructure empowers these algorithms to swiftly detect service disruptions and exert better control over energy demand. Additionally, users gain access to more appealing energy rates, encouraging them to adjust their consumption patterns accordingly, and thus to lower energy bills [46].
- Residential energy management: The proliferation of the Internet of Things has extended to household electrical appliances, enabling their administration through applications that offer users pertinent information about their energy consumption, rates, and connected devices [47].
- Renewable energies: The algorithms outlined in this section incorporate local power generation sources into the primary grid, leveraging service users who have the capability to inject renewable energy. By employing various compensation mechanisms, both smart grids and customers reap the advantages of this collaboration [48].

- They provide models to detect malicious nodes in the SG and utilize outlier denial and outlier mining scenarios;
- They perform extensive big data analytics, with power electronics providing added value for both renewable energy sources and SGs;
- They provide additional functionalities for monitoring active devices and network traffic in home area networks (HAN), neighborhood area networks (NAN) and wide area networks (WAN);

- Increased technology implementation costs: lack of low-cost controllers suitable for:
`o`- smart metering;
`o`- prediction of energy utilization patterns;
`o`- monitoring of energy demand;
`o`- energy conservation.

- Lack of legislation and pricing schemes for energy storage and energy sharing in SGs.
- Lack of suitable and empirical models for sensor networks in order to conduct more detailed/accurate simulations of the physical network systems.
- The need for high communication network requirements to transmit, sense, and control data while ensuring the QoS (Quality of Service) requirements of SGs.

#### 2.4. Reactive Power Injection

## 3. Smart Grid Modelling

## 4. Particle Swarm Optimization—PSO

#### 4.1. Objective Function

_{loss}corresponds to the reactive power, i.e., the power loss between the nodes i and j. The final objective function for the PSO algorithm is presented in Equation (2):

_{Gen,i}) is bound by the minimum and maximum allowed values:

#### 4.2. Inertia Weight Strategy

Algorithm 1. Pseudocode for PSO inertia weight calculation |

Pseudocode for inertia weight calculation |

For i = 1 to 1000 |

Rand(z), $\forall z\in (0,1)$, select a random value for z |

$z=r\ast z(1-z)$, calculate |

$w=\left({w}_{1}-{w}_{2}\right)\ast \frac{{MAX}_{iteration}-iteration}{{MAX}_{iteration}}+{w}_{2}\ast z$ |

EndFor |

#### 4.3. PSO Acceleration Coefficients

_{1}and c

_{2}are utilized in an attempt to simulate the socialization and the instincts of the birds. Specifically, c

_{1}in PSO is used in order to define the ability of the swarm to be influenced by the best solution identified by a single particle. Equation (8) expresses the value of c

_{1}:

_{1}and ${c}_{1,e}$ is the ending value. Similarly, c

_{2}is used in order to define the ability of the swarm to be influenced by the global best solution of the problem, and is expressed in Equation (9):

_{2}and ${c}_{2,e}$ is the ending value, and m is the control factor in both Equations (5) and (6).

Algorithm 2. Main particle swarm optimization pseudocode |

Improved PSO inertia pseudocode |

Initialize PSO |

Set particles to 20 |

Set max number of iterations MAX_{iteration} = 1000 |

Set acceleration coefficients |

While i < MAX_{iteration} do |

For each particle n in particles N |

$w=\left({w}_{1}-{w}_{2}\right)\ast \frac{{MAX}_{iteration}-iteration}{{MAX}_{iteration}}+{w}_{2}\ast z$ |

Update particle position |

Check and/or update bests |

EndFor |

Check for convergence |

Update iteration i = i+1 |

EndWhile |

Return best solution |

End |

## 5. Experimental Setup

## 6. Software Tool Implementation

## 7. Conclusions and Outlook

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

ACO | Ant Colony Optimization |

AI | Artificial Intelligence |

BFA | Bacteria Foraging Algorithm |

BSA | Bat Search Algorithm |

CPS | Cyberphysical System |

DER | Distributed Energy Resources |

DFA | Dragonfly Algorithm |

DV | Design Variables |

EM | Emissions |

GC | Generation Cost |

GR | Grid Resiliency |

HAN | Home Area Networks |

IDE | Integrated Development Environment |

LVRT | Low-Voltage Ride-Through |

MDA | Modified Dragonfly Algorithm |

MIP | Mixed Integer Programming |

NAN | Neighborhood Area Networks |

NIST | National Institute for Standards and Technology |

OPF | Optimal Power Flow |

PSO | Particle Swarm Optimization |

QoS | Quality of Service |

RES | Renewable Energy Sources |

RPL | Real Power Losses |

SDG | Sustainable Development Goals |

SG | Smart Grid |

SGIP | Smart Grid Interoperability Panel |

TST | Transient Stability |

VPI | Voltage Profile Improvement |

WAN | Wide Area Networks |

WOA | Whale Optimization Algorithm |

## Appendix A

Algorithm A1. Improved PSO algorithm, code implementation in Python |

Improved PSO inertia pseudocode |

# Modules Declaration |

import numpy as np |

# Setup of Swarm Object |

class Swarm(object): |

# Initialization method for the swarm properties |

def __init__(self, function, search_space, num_particles = 10, w = 0.9, max_error = 0.005): |

self.w_value = w |

self.num_particles = num_particles |

print(num_particles) |

self.max_error = max_error |

self.search_space = search_space |

self.dimensions = len(search_space) |

self.function = function |

self.V, self.fitness, self.local_best = [], [], [] |

self.w_min = 0.2 |

self.w_max = 1.2 |

# Particle Position Initialization Process |

self.X = self.generate_particles(num_particles) |

# Particle Initialization Process |

self.V, self.w = self.init_particles(self.X) |

current = self.evaluate(self.X) |

# Set the global best particle |

self.global_best = current.index(min(current)) |

# Set the local best score of the particles |

self.local_best = list(zip(current, self.X)) |

# Function to create particles of the swarm |

def generate_particles(self, num_particles): |

return [np.array([np.random.uniform(low, high) |

for _, (low, high) in self.search_space]) |

for _ in range(num_particles)] |

def evaluate(self, particles): |

return [self.function( |

**{k [0]: v for k, v in zip(self.search_space, particle)}) |

for particle in particles] |

def init_particles(self, particles): |

num_particles = len(particles) |

V = np.random.uniform(0, 1, (num_particles, len(self.search_space))) |

w = [self.w_value for _ in range(num_particles)] |

return V, w |

# Particle velocity function |

def velocity(self, velocities): |

return [self.w[p_i] * p_v + self.c_1 * np.random.uniform() * |

self.local_diff(p_i) + |

self.c_2 * np.random.uniform() * self.global_diff(p_i) |

for p_i, p_v in enumerate(velocities)] |

def local_diff(self, p_i): |

return self.local_best[p_i][1] − self.X[p_i] |

def global_diff(self, p_i): |

return self.local_best[self.global_best][1] − self.X[p_i] |

def location(self, locations, velocities): |

new_locations = [] |

for x, v in zip(locations, velocities): |

new_x = x + v |

for i in range(self.dimensions): |

search_var = self.search_space[i][1] |

dim_min, dim_max = search_var [0], search_var [1] |

if new_x[i] > dim_max: |

new_x[i] = dim_max |

elif new_x[i] < dim_min: |

new_x[i] = dim_min |

new_locations.append(new_x) |

return new_locations |

def get_local_best(self, current): |

return [(cur_score, self.X[p_i]) |

if cur_score < self.local_best[p_i][0] |

else self.local_best[p_i] |

for p_i, cur_score in enumerate(current)] |

def new_w(self, mean_score, score, min_score): |

if score <= mean_score and min_score < mean_score: |

return self.w_min + (((self.w_max − self.w_min) * (score − min_score))/ |

(mean_score − min_score)) |

else: |

return self.w_max |

# Logistic Function for Chaotic Inertia Weight Calculation |

def logistic_function(self, particle, mins, maxs): |

cxs = self.part_to_cx(particle, mins, maxs) |

logistic = [4 * cx * (1 − cx) for cx in cxs] |

return self.cx_to_part(logistic, mins, maxs) |

def part_to_cx(self, particle, lows, highs): |

return [(x − low)/(high − low) |

for x, (low, high) in zip(particle, zip(lows, highs))] |

def cx_to_part(self, cxs, lows, highs): |

return [low + cx * (high − low) |

for cx, (low, high) in zip(cxs, zip(lows, highs))] |

def pso(self, iterations = 50): |

error, i, similar = 1, 0, False |

while error > self.max_error or i < iterations and not \ |

self.same_particles(): |

current = self.evaluate(self.X) |

self.local_best = self.get_local_best(current) |

self.V = self.velocity(self.V) |

self.X = self.location(self.X, self.V) |

best_index = current.index(min(current)) |

self.global_best = best_index |

error = self.local_best[self.global_best][0] |

i += 1 |

def same_particles(self): |

if len(set(tuple(p) for p in self.X)) == 1: |

return True |

return False |

def decrease_search_space(self, particle, r = 0.25): |

mins = [var [0] for name, var in self.search_space] |

maxs = [var [1] for name, var in self.search_space] |

xmins = [max(mins[i], particle[i] − (r * (maxs[i] − mins[i]))) for i in |

range(self.dimensions)] |

xmaxs = [min(maxs[i], particle[i] + (r * (maxs[i] − mins[i]))) for i in |

range(self.dimensions)] |

return [(k [0], (xmins[i], xmaxs[i])) for i, k in |

enumerate(self.search_space)] |

def new_generation(self, old, amount): |

new_particles = self.generate_particles(amount) |

self.X = [*old, *new_particles] |

new_V, new_w = self.init_particles(new_particles) |

self.V = [*self.V[:len(old)], *new_V] |

self.w = [*self.w[:len(old)], *new_w] |

def run(self): |

error, i = 1, 0 |

while error > self.max_error or i < 100: |

self.pso() |

top = sorted(self.local_best, key = lambda x: x [0])[:int(self.num_particles/5)] |

self.local_best[self.global_best] = top [0] |

self.search_space = self.decrease_search_space(top [0][1]) |

num_new = int((4 * self.num_particles)/5) |

self.new_generation([p [1] for p in top], num_new) |

error = self.local_best[self.global_best][0] |

i += 1 |

return self.X[self.global_best] |

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**Figure 4.**IEEE 30-bus model adapted from [55].

Algorithm Name | Advantages | Disadvantages |
---|---|---|

Artificial Bee Colony | Easy to implement, good at exploring a wide search space | Slow convergence rate, may converge to suboptimal solutions |

Bat Algorithm | Good for continuous optimization problems, adaptable to different functions | Inefficient for discrete optimization problems, requires detailed calibration of parameters |

Cuckoo Search | Simple implementation, good for large-scale optimization | Slow rate of convergence, may converge to suboptimal solutions |

Differential Evolution | Fast convergence, good for high-dimensional optimization | Can get stuck in local optima, may require fine-tuning of parameters |

Firefly Algorithm | Good for multimodal optimization, scalable to large problems | Slow rate of convergence, requires detailed calibration of parameters |

Genetic Algorithm | Versatile, good for a wide range of problems, can handle noisy data | Slow rate of convergence rate, may converge to suboptimal solutions |

Particle Swarm Optimization (PSO) | Fast convergence, easy to implement, good for multimodal optimization | May converge to suboptimal solutions, requires detailed calibration of parameters |

Simulated Annealing | Good for complex optimization problems, can handle noise in the objective function | Slow rate of convergence, requires detailed calibration of parameters |

Whale Optimization Algorithm | Good for multimodal optimization, can handle noisy data | Slow rate of convergence, requires fine-tuning of parameters |

Optimization Stage | Network Loss (MW) | Loss Reduction (MW) | Loss Reduction Percentage |
---|---|---|---|

Not optimized | 9.259 | - | - |

Standard PSO | 7.683 | 0.369 | −7.66% |

Improved PSO | 7.432 | 0.693 | −10.67% |

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**MDPI and ACS Style**

Mourtzis, D.; Angelopoulos, J.
Reactive Power Optimization Based on the Application of an Improved Particle Swarm Optimization Algorithm. *Machines* **2023**, *11*, 724.
https://doi.org/10.3390/machines11070724

**AMA Style**

Mourtzis D, Angelopoulos J.
Reactive Power Optimization Based on the Application of an Improved Particle Swarm Optimization Algorithm. *Machines*. 2023; 11(7):724.
https://doi.org/10.3390/machines11070724

**Chicago/Turabian Style**

Mourtzis, Dimitris, and John Angelopoulos.
2023. "Reactive Power Optimization Based on the Application of an Improved Particle Swarm Optimization Algorithm" *Machines* 11, no. 7: 724.
https://doi.org/10.3390/machines11070724