Multi-objective reactive power optimization for low voltage distribution networks based on improved marine predator algorithm

A large number of distributed photovoltaic generators are constructed in the 0.4 kV distribution network, which causes the distribution network’s power flow and voltage amplitude to change, and brings new challenges to the reactive voltage control. Firstly, a reactive power adjustment model is invented, and the objective of the optimization is to minimize the distribution network line losses and node voltage shifts. Secondly, the standard marine predator algorithm is improved by using Bernoulli mapping, nonlinear attenuation weight function, and elite replacement strategy, and the model is solved by the improved algorithm. The results of the example optimization of a 21-node low-voltage distribution network show that the presented reactive power adjustment method can control the voltage point within a reasonable range and reduce the line loss rate. The improved marine predator algorithm has better global optimality finding and faster convergence than traditional intelligent algorithms.


Introduction
Because of the desire to improve and solve the climate and environmental problems caused by conventional energy sources, China's energy structure is shifting from traditional fossil fuel energy to clean energy such as solar and nuclear energy [1].A distributed photovoltaic generator (DPG) is a small investment with less pollution, which can be installed directly on the low-voltage distribution network [2].Observing the latest data published by the National Energy Agency up to the first half of 2023, the cumulative installed capacity of DPGs in China has reached 198, 228, 000 kW.A large number of DPGs are integrated into the 0.4 kV distribution network, which will affect the three-phase imbalance rate, tidal current distribution, and nodal voltage [3].How to revise the reactive power of a 0.4 kV distribution network containing DPGs, then reduce the line loss of 0.4 kV distribution network, and improve the quality of power consumption of users, has become an important research problem.
In [4], the reactive power adjustment problem of 0.4 kV distribution network containing DPGs takes the network active loss as the objective function and performs a conversion on the type of question references by using the branch-current method.In [5], cone planning is used to convert the problem to a conic problem of two order models and adopt the method of reactive power partitioning to reduce the difficulty of solving.In [6][7], the non-dominated sequential genetic algorithm (NSGA-II) is used and particle swarm optimization algorithm (PSO) and other swarm intelligence algorithms are improved to solve the model, and the optimality and convergence have been improved compared with the traditional exhaustive algorithms.However, these intelligent algorithms depending on parameter settings and the optimization results are not stable enough.The marine Predators Algorithm (MPA) is a new type of meta-heuristic intelligence method invented by Faramarzi et al. in 2020 [8], which originates from the simplification and simulation of the foraging behavior of ferocious predators in the ocean, and has the advantages of fewer parameters, simpler setup, easy implementation, and more accurate calculation.It shows better optimization and convergence in solving nonlinear planning problems compared with the particle swarm algorithm and genetic algorithm.
Synthesizing these studies mentioned above, this paper establishes a reactive power adjustment model for low-voltage distribution networks with DPGs.Focusing on the defects of the genetic algorithm, the method of composition of a group of particles, and other intelligent algorithms that are easily captured by a small number of bad solutions, the model in this paper adopts the improved multiobjective marine predator algorithm to solve the problem and finally verifies the validity of the recommendation invented in this study through the example of 21-node low-voltage distribution network.

Adjustment program for reactive power
Considering the economy of distribution network operation and the effect of voltage values, this paper builds an adjustment program for reactive power with the objective function of the most insignificant network losses and the most inconspicuous voltage movement in a low-voltage network.

The objective function
Low voltage distribution network loss Ploss can be calculated from the current calculation.
where N is the count of nodes in the 0.4 kV distribution network; Gij is the line conductance of this distribution network; Ui and Uj are the magnitude of the voltage value; θij is the difference in the phase angle of the voltages.
The node voltage deviation ΔU can be expressed as: where Ui is the magnitude of the voltage value; Uspec is the magnitude of the voltage reference value; Ui.min and Ui.max are the limit values of the node i voltage movement, respectively.

Restrictive condition
The methods of limiting formulas are active and reactive power logarithmic formulas for the low voltage distribution network can be expressed by the following formula.
Since there is a DPG intervention in the 0.4 kV distribution network, the node-injected power is calculated as: .. ..
where Pi.DPG and Qi.DPG are the power output of the DPGs connected at Node i; Pi.D and Qi.D are the load power at Node i.
The inequality constraints mainly include state variable constraints such as node voltage, branch current, and lossy and lossless power output constraints of DPGs.
The magnitude of the voltage value and branch current constraints are: .min .max .max where Ui.min and Ui.max are the limit values of the voltage movement at Node i, respectively; Iij.max is the limit value that can flow in the line between Nodes i and j.
The capacitor bank casting constraint is: where where Ps.DPG and Qs.DPG are the power output values of the DPGs connected to Node s; Ps.DPG.min,Ps.DPG.max,Qs.DPG.min, and Qs.DPG.max are the power output limits of the DPGs connected to Node s; Ss.PV is the nominal capacity of the DPG connected at Node s, which is usually 1.1 times the nominal capacity power of the DPG.

Improved MPA optimization algorithm
The core of the MPA algorithm lies in the combination of Brownian and Levy motions with each other and the introduction of the ocean vortex phase effect, which possesses the predator storage memory property.To improve the convergence of the intelligent method, this paper proposes a revised marine predator algorithm (MO-MPA), which improves the initialization phase, the iterative computation phase, and the phase of jumping out of the small fraction optimization of the standard MPA algorithm, respectively.

Initialization based on Bernoulli mapping
The MPA algorithm first generates an initial model randomly in the search field, and the standard MPA population initialization formula is: where X0 is the initial solution; Xmin and Xmax are the limit values of the initial solution.
To improve the uniformity of the Primitivized group distribution in the search field, this paper introduces Bernoulli mapping into the initialization stage, and the use of Bernoulli mapping for population initialization often achieves better results than pseudo-random numbers, which can effectively enrich the number of populations and improve the uniformity of the distribution of the initial populations in the high-dimensional space.The mapping formula is as follows: ,, ,1 ,, / (1 ), 0 1 ( 1 ) / , 1 where u=1, 2,..., n is the count of individuals in the initial solution; v=1, 2,..., m is the dimension of the solution; λ is the mapping constant, which is set to be 0.4; Yu,v is the message from the location of the u th individual in the v th dimension.The population is initialized by Bernoulli mapping, and the standard MPA algorithm initialization formula is improved as: The initial population is defined as the Prey, and the adjusted solution matrix is defined as the predator matrix (Elite), which are denoted as follows:

Elite=
where n is the number of populations; m is the number of variables in the reactive power optimization model.

Nonlinear weights and elite strategies introduced iteratively
After obtaining the initialized population, the population position needs to be adjusted to align the Prey and predator matrix, i.e., we enter the iterative computation phase of the MPA algorithm.
1) At the first stage of the iteration, the size of the step update formula and the predator matrix update formula are respectively: ( Prey ), where RB is the m-dimensional random vector generated by using Brownian random wandering; Si is the pacemaker matrix; R is the m-dimensional random number vector; P has a value of 0.5.
2) At the second stage of the iteration, the population needs to be divided into two parts for the update operation.The first one update formula the prey matrix update formula are respectively: ( Prey ), 1, 2, , / 2 Prey Prey where RL is the m-dimensional random vector generated by using Brownian random walk.
The second update formula and prey matrix update formula are respectively: ( Prey ), / 2, , where CF is the automatic change of parameters for updating distances Si, which is calculated as ; t and T are the iteration rounds at this moment and limit iteration rounds, respectively.
To improve the big-picture optimization of the intelligent method in the pre-iteration and miditeration in the optimization, the nonlinear weight function  Thus, the prey matrix update formula in Formula (11) and Formula (12) can be improved as: At the third stage of the iteration, an elite replacement strategy is introduced to avoid looking for a small percentage of bad solutions near the end of the iteration.The automatic change of parameters update formula and the prey matrix update formula are respectively: ( We calculate the fitness value of the solution at this point and apply the elite replacement mechanism to the solution vectors that are lagging in fitness with the following formula.
where Preyi,ineffective is the 20% of individuals with the most backward adaptation; Preyi,effective is the 20% of individuals with the best adaptation; Rstar is the Euclidean distance between the optimal individual and the individual closest to the optimal individual.

The escape from a small part of the bad solution process
To avoid looking for a small percentage of bad solutions, the environmental effects of eddy currents and fish aggregation effects (FADs) are introduced, i.e., the predator wanders near the prey in 80% of the search space, and the remaining 20% wanders in different regions, thereby improving the big-picture optimization and enhancing the convergence of the intelligent method, which is denoted as: where r is an unpredictable number; FADs are constants with a value of 0.2; r1 and r2 are numbers of markers not predicated in the prey matrix (1 < r1 and r2 < 2); U is an m-dimensional binary vector.

Reactive power adjustment process based on MO-MPA
For the above reactive power adjustment model containing DPGs, the MO-MPA method is used to solve tricky models, and the process methodology is demonstrated in Figure 2.

Example
The wiring diagram of the 21-node 0.4 kV distribution network is demonstrated in Figure 3.The full load is 116.272+j46.462kVA.Distributed photovoltaic power sources are installed at Nodes 2, 7, 10, and 14, and the rated active power is 10 kW.Capacitor banks are installed at Nodes 12, 13, and 17, and the capacity is 5 kVar×4.The magnitude of the voltage amplitude at the node is controlled to be 0.95-1.05p.u.The extreme maximum number of iteration rounds was set to 160, and the reactive power adjustment model and the improved marine predator algorithm established in this research are used to explore new avenues.The superiority of the proposed scheme presented in this study can be concluded from the magnitude of different node voltage amplitudes and network loss of the distribution network without optimization regulation and the optimization method in this paper.
(1) Comparative analysis of node voltage As can be seen from Figure 4, most of the node voltages of this distribution network before optimization are lower than 0.94 p.u. Through the optimization of distributed photovoltaics and capacitor bank output, from the representation of the curve, we can learn in Figure 4 that the model and solution wisdom method proposed in this paper can effectively inhibit the voltage overrun, and the optimized node voltage is higher than 0.96 p.u., which is in line with the requirements of residents' daily use or industrial production.The number and capacity of capacitor bank casting are shown in Table 1.From the presentation and results in Table 1, it can be understood that the capacitor bank sends out inductive reactive power and reduces the inductive reactive power delivered on the line, which reduces the voltage drop on the line and improves the load node voltage.
2) Comparative analysis of line loss As can be seen from Figure 5, after the reactive power adjustment and regulation of this 0.4 kV network through the model proposed in the research, the high branch loss is reduced from the original 5.06 kWh to 3.44 kWh, and the line loss rate is reduced by 1.29 percentage points, which is due to the optimization of the node voltage magnitude.It returns to a reasonable range and reduces the network loss with it.
By comparing the MO-MPA, Genetic Algorithm (GA), and Particle Swarm Algorithm (PSO) for solving the optimization model of the arithmetic example in this paper, we can verify the optimality and convergence of the MO-MPA proposed in this paper.The comparison of optimization results obtained by the three methods is displayed in Table 2. From the presentation and results in Table 2, it can be understood that compared with the traditional the iterative computation in the first two-thirds of the stage, and the comparison curve of its iterative convergence ability with that of CF is shown in Figure1.Compared with CF in the original algorithm, the nonlinear weight function W has better adaptive convergence ability in the first and middle stages of iteration, and the convergence speed is faster.

Figure 1 .
Figure 1.Comparison of convergence ability of the weight function W and CF.

Figure 4 .
Figure 4. Comparison of node voltage.Figure 5. Comparison of line loss.

Figure 5 .
Figure 4. Comparison of node voltage.Figure 5. Comparison of line loss.
Ni.CB is the number of capacitor casting groups; Ni.CB.max is the maximum number of capacitor casting groups; Qi.CB is the capacitor group casting capacity; Qi.CB.step is the single group capacitor capacity.Distributed photovoltaic power output conventional constraints are:

Table 2 .
Comparison of optimization results of different algorithms.21-node distribution network Node voltage deviation Line loss rate (%) Line loss reduction