An Optimization Approach for Distribution Network Planning Strategy Based on Combined Improved Cloud Model and Evidence Theory

With the continuous promotion of China’s power reform, higher requirements and standards have been put forward for distribution network investment decisions. However, the current work of distribution network project optimization in China has the problems of high subjectivity of investment decisions, fuzzy evaluation method, and high uncertainty of project optimization. In response to the above problems, this paper proposes a distribution network planning scheme optimization method based on the improved cloud model and evidence theory. Firstly, the optimal index system of the planning scheme is constructed from the dimensions of load transfer capacity, grid structure level, load capacity, investment benefit, social benefit, and life-cycle cost. Secondly, for subjectivity and ambiguity problems, the traditional degree of the membership function is replaced by a cloud model optimized by cloud entropy. Thirdly, the problem of uncertainty is addressed through evidence theory, while the game theory approach is used to identify conflicting evidence for evidence revision and fusion in the case of conflicting evidence arising from traditional evidence theory. Moreover, by improving the TOPSIS method and comparing the distances between the basic probability assignments of each planning scheme and the positive and negative virtual optimal solutions, the final decision results can be determined. Finally, the effectiveness and superiority of the proposed method are verified by experiments.


Introduction
In recent years, with the acceleration of the construction of new power systems and the continuous increase of load, the investment and construction demand for distribution networks is constantly increasing.In China, for example, as the level of transmission line construction is becoming saturated, the investment and construction focus of power grid companies are gradually shifting to the reconstruction of typical problems of grid structure improvement, power supply quality, and other directions in distribution networks [1].However, the distribution network investment planning scheme is difficult to quantify economic benefits and accurately compare the technical indicators between schemes, etc.Therefore, it is crucial for power grid companies to select the optimal investment scheme from various planning schemes.
Currently, there are two types of optimal planning decisions for distribution networks.The first type of research mainly focuses on building a multi-dimensional evaluation index system and carrying out the ranking by assigning weight to quantify the score.Between the two types, the establishment of index weight system often adopts combination methods, also combining subjective weight and objective weight, including utilising improved analytic hierarchy process [2], fuzzy comprehensive evaluation method, entropy weight method [3], etc.However, the traditional index weight distribution is usually in the form of expert scoring or triangular fuzzy function.This method normally has strong subjectivity, also, it cannot consider the randomness and fuzziness in multi-attribute evaluation decisions.
The second type of research is based on investment optimization models and obtaining solutions using related improvement methods [4], such as particle swarm optimization [5] and envelope analysis method [6] to obtain optimal ranking results according to different objective functions and boundary conditions.Dušan Božič et al. [7] calculated the improvement range of active power loss and other indicators of the project planning scheme and analysed respectively based on the overall investment maximization and the highest score of a single project.The above research methods mainly focus on the economic benefits and investment return of the project by utilizing the distribution network investment optimization model.However, it cannot integrate the multi-dimensional membership degree for decision-making, and cannot solve the uncertainty problem in decision-making.
Therefore, this paper firstly constructs the optimal index system that is suitable for actual construction projects based on the distribution network planning scheme decision.Meanwhile, to solve the problems of subjectivity, fuzziness, and randomness of the evaluation scheme, the improved cloud model based on cloud entropy optimization is used instead of the traditional membership generating function to improve the credibility of the evaluation results.In order to solve the uncertainty problem of multi-source information fusion, dynamic weights and static weights are combined in the form of game theory, and the weight improvement, conflict identification and probability assignment correction of traditional evidence theory.Then, the decision sequence is obtained by comparing the closeness degree of each planning scheme with the positive and negative virtual optimal solutions.Finally, a line group in a region of China is selected for the case study, the investment sequence is obtained by calculation, and the feasibility of the proposed method is verified by comparison with the traditional method.

Distribution network planning scheme decision preference index system
The index system used in this paper is shown in Figure 1, which is divided into six categories, including load transfer capability, network structure level, load capacity, investment efficiency, social benefit, and whole life cost.Each category is subdivided into various sub-indicators.
The index system not only refers to the representative indexes [8][9][10], but also refers to the actual planning process and evaluation indexes of Power Grid Company in China.The index system selects the accessible and representative technical indexes in the distribution network planning scheme, therefore, the calculation results are more satisfied with the actual application scenarios.The number in the upper right corner of each index represents the metric number.

Definition of cloud model
The basic definition of the cloud model is as follows [11].Let U be a quantitative domain U = {x} and C be a qualitative concept belonging to U.An element x in U is a random realization of the qualitative concept c whose degree of membership is a randomly generated number.If the range of values takes values in the interval [0, 1], then the distribution of x over U is called a cloud, and x is called a cloud droplet.The greater the number of cloud drops, the more it reflects the randomness and ambiguity of the model.The expected value E x reflects the center of distribution of the degree of membership cloud and represents the central value of different levels of fuzzy information transformed into the quantitative assessment.E n reflects the degree of dispersion and fluctuation interval of cloud droplets, which is a description of E x uncertainty.H e reflects the degree of stability of the cloud droplet and indicates the uncertainty of the entropy.
The degree of membership curve is a curve composed of the distance between the object to be evaluated and the qualitative concept.Taking the Gaussian cloud-dropping diagram as an example, the membership curve is shown in Figure 2. In the figure, the maximum, minimum and expected degree of membership curves are green, blue and red curves, respectively.The intersection of a single evaluation index with the three curves is the calculated minimum correlation ( min k ), maximum correlation ( max k ), and expected correlation ( exp k ).

Cloud Entropy Optimization Algorithm
The traditional cloud entropy calculation methods are mainly based on the "3E n " rule and the "50% correlation" rule.Based on the "3E n " rule, the cloud drops fall mainly in the interval of and the cloud drops in other areas are ignored.The rule based on "50% correlation" is characterized by a fuzzy classification of levels, where the values at the boundary of the levels and the correlation of two adjacent levels are equal, both being 50%.In this paper, the advantages of two traditional schemes are combined simultaneously, and cloud entropy optimization is used to calculate the method as follows.
It is assumed that the system index data to be evaluated has a total of m indicators, and each indicator is divided into 5 rank ranges.Let 'i' be one of the indicators and 'j' be one of the ranks.E x , E n and H e denote the expected value, super entropy and entropy sets of this method, respectively.Based on the "3E n " and "50% correlation" rules, the traditional cloud entropy sets where max

C
denote the maximum and minimum values of each rank range of the traditional cloud model, respectively, and the expected values E x of the traditional method and this method are the same.
According to the correlation curve in Equation 1, the minimum degree of membership ' min k is obtained by solving data m based on the "3E n " rule, and the maximum degree of membership '' max k is obtained by the "50% correlation" rule.Let the desired affiliation obtained by the cloud entropy optimization method be k.Then, the correlation deviation max k  and the cloud entropy optimization set E n of the three methods with each other can be expressed by the following equation.When max k  is at a minimum, the cloud entropy optimized set yields the optimal solution. (3)

Degree of membership matrix determination
In the process of building the degree of membership of the cloud model, we can divide each indicator into {Ⅰ, Ⅱ, Ⅲ, Ⅳ, Ⅴ} five grade ranges, with Ⅰ indicating the highest evaluation, and then establish the corresponding (E x , E n , H e ) for each indicator respectively, so as to obtain the degree of membership situation.The numerical characteristics of the cloud model such as E x , E n , H e and degree of membership U are calculated by the formula in reference [11].
The sum of degree membership of the cloud model solution is not 1, which conflicts with the definition of the basic distribution probability function of the evidence theory.Therefore, it is necessary to introduce additional indicator rank assignment uncertainty probabilities ( ) M  to transform the degree of membership obtained from the cloud model into a basic probability assignment consistent with the definition of evidence theory.
  ) where  denotes the uncertainty probability.

Improving Evidence Theory and Evidence Modification
Evidence theory can fuse the degree of membership degrees obtained from the cloud model with multiple sources of information and calculate the degree of degree membership that they jointly contribute to each evaluation level, thus effectively solving the uncertainty problem in the preference problems [12] .
The shortcomings of the traditional theory of evidence will cause conflicts between evidence.In this regard, this paper proposes a combination assignment method based on the game idea, which fuses the dynamic and static weights of the evidence to get the combination weight, and then identifies and corrects the conflicting evidence based on the combination weight.This method fully takes into account the important difference between the evidence and only corrects the conflicting evidence, thus improving the accuracy of evidence fusion.

Definition of Evidence Theory
The set of C independent elements in the set contains the number of all subsets and lies in the range [0, 1].The function M satisfies the sum of the basic probabilities of all elements in 2 C is 1,  , then M is called the distribution probability function, F is called the focal elements of the basic probability distribution function, and M(F) is called the basic probability number of F. The calculation of the degree ( ) M  in evidence theory can be found in Wu et al.'s work [13] .
where CT

S
indicates the degree of conflict between evidence, and denotes the degree of membership degree after fusion.

Determination of weights based on game theory
Dynamic weights are established with reference to the traditional evidence-theoretic approach and are not described too much here.Static weights are influenced only by the importance of the evaluation object itself and can be calculated using the improved CRITIC assignment method [14].When the calculation of dynamic and static weights is completed, the Nash equilibrium point is found for both.Then the deviations of static, dynamic and combined weights are minimized.The specific calculation method is as follows.
where 1  denotes static weights, 2  denotes dynamic weights, s  denotes the portfolio weights to reach the Nash equilibrium point, and a and b are weighting coefficients at the time of combination.

Evidence Identification and Amendment
When conflicts arise between evidence, data that are not in conflict will be affected if corrections are made directly.Therefore, in this paper, we first identify and judge the evidence, and if there are m sets of evidence, when the combination weight , the data is judged as non-conflicting evidence.When , it is judged as conflicting evidence and corrected by the correction factor, which is calculated as follows. where i M  denotes the corrected degree of membership after evidence conflict, and   M ψ ' denotes the result of the fusion of conflicting evidence, which can achieve conflict mitigation.

Decision selection based on improved TOPSIS method
In the TOPSIS method, a solution is the best of all solutions when it is closest to the virtual optimal solution in the decision and at the same time furthest from the virtual worst solution.However, the drawback of the TOPSIS method is that it is not possible to determine the ranking results of the solutions in the same rank.Referring to the TOPSIS idea and incorporating the principle of degree membership, this paper determines the optimal choice by calculating the average distance between the basic probability assignments formed by each solution after evidence fusion and the corresponding basic probability assignments of the virtual optimal and inferior solutions.The steps are shown below.
In Step 1, the degree of membership matrix of the positive and negative ideal clouds corresponding to each index is calculated, and the fused virtual optimal solution i M  and virtual worst solution i M  are obtained by improved evidence theory.
In Step 2, the difference between the basic probability assignment of each planning solution and the basic probability assignment of the positive and negative optimal solutions is calculated and the mean value is found.
In Step 3, the difference between the planning solution and the maximum and minimum average fit is calculated based on the mean value obtained in the previous step.The larger the difference, the closer the solution is to the optimal solution Through the elaboration of the above model, the overall framework of this paper is shown in Figure 3.

Case study
A line in QY City in South China was selected as a case study.The feeder, which is expected to report a capacity of 2,230 kVA this year, requires investment in renovation because it has problems with heavy overloads and the inability to transfer power during outages.The white lines in Figure 4 indicate the planning cases.Both Case 1 and Case 2 are connected to the main line, and Cases 3, 4, and 5 are connected to the branch line.Case 6 indicates the simultaneous construction of Case 1 and Case 2. Different planning options bring different economies and benefits, so they need to be selected for decision making.Table 1 indicates not only the index data of the six cases in different dimensions, but also the various index levels of different indicators according to the relevant documents of power grid companies, where level 1 is the best index and level 5 is the worst index.As an example, the seventh indicator is used to solve the indicator data of six programs by improving the model.The expected value E x for the five evaluation levels {I, II, III, IV, V} under this index are {31.5, 24.5, 17.5, 10.5, 3.5}, respectively.According to calculations, the cloud entropy based on the "3E n " and "50% correlation" principles is calculated as fixed values of 3.333 and 8.493, and the super entropy of the six planning schemes is 0.3333 and 0.8493, respectively.From Equation ( 5), it can be seen that the cloud entropy calculated based on the cloud entropy optimization method is a fluctuating value that will change with changes in the planning scheme, so the calculation results will be more accurate and reliable.The three evaluation benchmark clouds for index seven drawn using the cloud generator algorithm are shown in Figure 5 and Figure 6.
From Figure 5 and Figure 6, we can see that both traditional algorithms have limitations.The model ranking based on the 3E n rule is too strict and clear, which ignores the small probability events.The model ranking based on the "50%" correlation principle is too vague, which makes the calculation error larger.The calculation method based on cloud entropy optimization adopted in this study considers both strictness and fuzziness, overcomes the defects of the traditional algorithm, and finds the grade classification interval through the changing cloud entropy to make the evaluation results more accurate.In Case 1, for example, the membership matrix generated by the cloud model is transformed into a basic probability allocation matrix based on game combination weights, and then evidence recognition and correction are carried out.The calculation results of the combination weights for each planning scenario are shown in Figure 7.  Finally, with reference to the TOPSIS idea, the virtual optimal and inferior values in each index are used as parameters.The basic probability distribution of positive and negative virtual optimal solutions is calculated by substituting them into the aforementioned method.The results of evidence fusion and the difference mean values of positive and negative average fit of each scheme are shown in Table 1.Meanwhile, the degree of conflict between the traditional evidence theory and the improved evidence theory is shown in Table 2


As can be seen from Table 6, the improved evidence theory greatly reduces the degree of evidence conflict among the options and makes the evaluation results more reliable.According to the principle that the larger the average fit difference is, the closer the user is to the most ideal result.From Table 5, it can be seen that the final order of planning Case s is determined as Case 2 > Case 6 > Case 1 > Case 5 > Case 4 > Case 3, respectively.The ranking result also matches the actual situation of Guangdong Power Grid Company.

Case study
Considering the subjectivity, fuzziness and uncertainty of multi-attribute decision making for distribution network planning schemes, a distribution network planning scheme decision preference method based on the improved cloud model and improved evidence theory is proposed.The proposed model has the following characteristics: 1) In the evaluation index system, six dimensions, such as load transfer capability, network structure level, and load capacity, are used as perspectives.Combing with the data obtained from the actual project, precisely quantify the investment efficiency of each project, which achieves usefulness in the actual construction project.
2) By using cloud models to calculate the membership between different schemes and benchmark clouds, the ambiguity and subjectivity of traditional membership functions are effectively solved.And this article has made improvements to the traditional cloud model, improving the accuracy and reliability of optimization decisions.
3) Evidence conflicts possibly exist in the traditional evidence theory.The experimental results the improved method proposed in this paper show that the degree of evidence conflict under each planning scenario is smaller than the result calculated by the traditional evidence theory, which proved the method can solve the issue of evidence conflict and the uncertainty of multi-attribute decision making more effectively.

Figure 2 .
Figure 2. Cloud model membership function curve.In the figure, the maximum, minimum and expected degree of membership curves are green, blue and red curves, respectively.The intersection of a single evaluation index with the three curves is the calculated minimum correlation ( min k ), maximum correlation ( max k ), and expected correlation ( exp k ).

Table 1 .
. Results of evidence convergence across cases.

Table 2 .
Degree of conflict of evidence.