The traditional ant colony algorithm is applied to the optimization and reconstruction of distribution network, which is easy to fall into the local optimal solution. In addition, the convergence speed of the above traditional ant colony algorithm needs to be improved. How to avoid prematurely falling into the local optimal solution and improve the convergence speed of the algorithm is the problem to be solved by the improved ant colony algorithm. The maximum-minimum ant colony algorithm (MMAs) solves both problems well.

The improvement of the traditional ant colony algorithm by MMAS is mainly reflected in the pheromone, mainly for the following three points:

(1) Setting the upper and lower limits of the pheromone. When the update of a pheromone causes the value of the tributary pheromone to be greater than the upper limit, the pheromone value is equal to the upper limit, and the value of the pheromone is less than the value of the pheromone due to the volatilization of the pheromone. The lower limit causes the pheromone value to be equal to the lower limit.

(2) The initial pheromone concentration is set to a maximum value, and the pheromone volatilization coefficient is set to a smaller value;

(3) A pheromone update method combining local pheromone update method and global pheromone update method are adopted.

The above three points will be described in detail below.

In order to avoid the case where the ant colony algorithm converges to the local optimal solution due to pheromone accumulation, the MMAS forcibly limits the maximum and minimum values of the pheromone concentration, and makes ${\tau}_{min}<{\tau}_{{i}^{\prime}{j}^{\prime}}(t)<{\tau}_{max}$for each path. After each iteration, it ensures that the value of the pheromone concentration is still within this range. When ${\tau}_{{i}^{\prime}{j}^{\prime}}(t)>{\tau}_{max}$, let ${\tau}_{{i}^{\prime}{j}^{\prime}}(t)={\tau}_{max};$when ${\tau}_{{i}^{\prime}{j}^{\prime}}(t)<{\tau}_{max},$let ${\tau}_{{i}^{\prime}{j}^{\prime}}(t)={\tau}_{min}.$Also it makes sure that *τ*_{min} > 0, so that the probability that each path is selected is not zero.

Under the MMAS algorithm, the pheromone *τ*_{max} is set to a progressive maximum limit. Each time, a new solution is obtained through the population iteration, *τ*_{max} is updated to obtain a value that dynamically changes with the number of iterations (Zhou & Littler, 2016).

In order to determine the value of *τ*_{min}, the following assumptions need to be made:

(1) The optimal solution will be searched before the search stagnation occurs. Under this assumption, the probability of finding the global optimal solution after the iteration is completed will be much greater than 0, which will ensure that a better solution is found until the optimal solution is found.

(2) The ability to find the optimal solution depends mainly on the selection of the upper and lower limits of the pheromone, rather than the selection of the heuristic function.

The rationality of the first hypothesis is determined by the size of the search space, which means that there is a possibility of finding a better solution near the good solution. The second hypothesis is based on the following set of methods for setting the pheromone’s minimum value of *τ*_{min}. Since the distribution network reconstruction takes ${\eta}_{{i}^{\prime}{j}^{\prime}}=1/{R}_{{i}^{\prime}{j}^{\prime}},$which is a constant, the influence of the heuristic function on the transition probability can be ignored. In the MMAS algorithm, the value of parameter *β* is taken very small or simply no heuristic function is used.

Under the premise of the above assumptions, a relatively good value of *τ*_{min} can be obtained in a progressive manner. When the MMAS algorithm has converged, the probability value of the resulting optimal solution is a value *P*_{b} that is much larger than zero. In this case, the ant that got the optimal solution made a “correct” choice on each node, and the pheromone concentration on the selected path is a maximum of *τ*_{max}. In fact, at each node, the probability *P*_{d} of selecting the optimal path accordingly depends directly on *τ*_{max} and *τ*_{min}. For the sake of simplicity, assuming that *P*_{d} is constant on each node facing the selection. An ant must make the *n*ʹ times of correct choice and the probability of finding the optimal solution is ${P}_{d}^{{n}^{\prime}}.$

Then,

$${P}_{d}^{{n}^{\prime}}={P}_{b}$$(19)which is:

$${P}_{d}=\sqrt[{n}^{\prime}]{{P}_{b}}$$(20)Given a value of *P*_{b}, it can reasonably set the value of *τ*_{min}. On average, at each node, the ant needs to make a choice among the *avg* branches. Ignoring the heuristic function, the optimal path pheromone concentration is *τ*_{max}, and the pheromone concentration on other paths is *τ*_{min} in the extreme case, then the branch transition probability of selecting the optimal path on each node is:

$${P}_{b}=\frac{{\tau}_{max}}{{\tau}_{max}+(avg-1){\tau}_{min}}$$(21)The above equation is solved to get:

$${\tau}_{min}=\frac{{\tau}_{max}(1-{P}_{d})}{(avg-1){P}_{d}}=\frac{{\tau}_{max}(1-\sqrt[{n}^{\prime}]{{P}_{b}})}{(avg-1)\sqrt[{n}^{\prime}]{{P}_{b}}}$$(22)Analysis of the above equation shows that if *P*_{b} = 1 is satisfied, then *τ*_{min} = 0.

By the above formula, it is assumed that given a *P*_{b}, the value of *τ*_{min} can be determined. When the MMAS converges, the selection of the value of *P*_{b} is directly related to the number of solutions found by the algorithm. According to the statistical law, if the number of ants in the ant colony is large enough, the macroscopic manifestation of *P *_{b} is the proportion of the ant that obtains the optimal solution in a certain iteration to the total ants in the ant colony [1]. Generally, it is set as:

$${\tau}_{min}=\frac{{\tau}_{max}}{2{n}^{\prime}}$$(23)Where *n*ʹ indicates the number of system nodes.

In order to speed up the convergence of the algorithm, it is hoped that the pheromone of all branches will be updated to *τ*_{max}(1) after the first iteration is completed. By arbitrarily setting a relatively large initial value of *τ*(0), after the first iteration of the MMAS, the value of the pheromone is forcibly limited to the given range. In general, the setting is:

$$\tau (0)={\tau}_{max}(1)$$(24)In the first few iterations, choosing this pheromone initialization method will speed up the convergence of the algorithm.

Different from the traditional ant colony algorithm, MMAS adopts the method of local pheromone update and global pheromone update. The local pheromone update method is the pheromone update after the completion of an ant traversal in the iteration. The global pheromone update is a pheromone update to the global after an iteration is completed. The two pheromone update methods are introduced as follows:

Supposing when the program proceeds to the DD iteration, after the *κ*th ant traversal is completed, if the ant-derived network is subjected to load flow calculation constraints, and the network loss of the network is smaller than the current optimal solution, then the selected branch pheromone by this ant will be updated [12]. The update formula is as follows:

$${\tau}_{{i}^{\prime}{j}^{\prime}}^{\kappa}=(1-\rho \cdot {\tau}_{{i}^{\prime}{j}^{\prime}}^{\kappa -1})D(D+\mathrm{\Delta}{\tau}_{{i}^{\prime}{j}^{\prime}}^{\kappa}DD)$$(25)$$\mathrm{\Delta}{\tau}_{{i}^{\prime}{j}^{\prime}}^{\kappa}DD=Q/p{s}_{\kappa}$$(26)Where ${i}^{\prime}{j}^{\prime}$denotes a branch ${i}^{\prime}{j}^{\prime},\rho $denotes a pheromone’s volatilization coefficient (0 ≤ *ρ* < 1), denotes the number of ants, $\mathrm{\Delta}{\tau}_{{i}^{\prime}{j}^{\prime}}^{\kappa}DD$denotes a change amount of pheromone on the branch *i*ʹ*j*ʹ, and 8 denotes a local pheromone’s update parameter, that is, *Q* = *τ*_{0}, *ps* is the *κ*th ant traversing the network loss of the resulting network.

When the program proceeds to the *DD*th iteration and is ready for the DD+1th iteration, if the network of this iteration is better than the current optimal solution by the power flow calculation, the pheromone of all the branches will be updated once. The update formula is as follows:

$${\tau}_{{i}^{\prime}{j}^{\prime}}(DD+1)=(1-\rho \cdot {\tau}_{{i}^{\prime}{j}^{\prime}})D(D+1)\mathrm{\Delta}{\tau}_{{i}^{\prime}{j}^{\prime}}DD$$(27)$$\mathrm{\Delta}{\tau}_{{i}^{\prime}{j}^{\prime}}DD=\sum _{\kappa =1}^{m}\mathrm{\Delta}{\tau}_{{i}^{\prime}{j}^{\prime}}DL$$(28)$$\mathrm{\Delta}{\tau}_{{i}^{\prime}{j}^{\prime}}^{\kappa}(DD)={Q}^{\prime}/p{s}_{\kappa}$$(29)In the traditional ant colony algorithm, each tributary is updated with pheromone after the iteration is completed, and after the iteration is completed in the MMAS, only the tributary pheromone on the optimal network is updated. The revised pheromone update rules are given below:

$${\tau}_{{i}^{\prime}{j}^{\prime}}(DD+1)=(1-\rho ){\tau}_{{i}^{\prime}{j}^{\prime}}(DD)+\mathrm{\Delta}{\tau}_{{i}^{\prime},{j}^{\prime}}^{b}(DD)$$(30)Where, $\mathrm{\Delta}{\tau}_{{i}^{\prime},{j}^{\prime}}^{b}(DD)={Q}^{\prime}/f({s}_{b})$represents the objective function value of the optimal solution *s *_{b} found in the last iteration or the previous solution 3 *s*_{gb}, and obviously, in the high-voltage distribution network reconstruction, there are:

$$f({s}_{b})=p{s}_{b}$$(31)Combining with the pheromone update method, the disadvantages of the traditional ant colony algorithm are analyzed. The idea of using the maximum and minimum ant colony algorithm for distribution network reconfiguration is put forward. Finally, the detailed description and analysis of the pheromone setting method are given and the pheromone update method of the traditional ant colony algorithm is improved.

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.