Wind power output analysis method based on WOA-K-means

Abstract

The invention provides a wind power output analysis method based on WOA-K-means. A K-means algorithm is used as a basis, volatility and output efficiency are used as characteristic indexes to carry out dimension reduction processing on a daily output curve, an initial clustering center is optimized through global search capability of a WOA algorithm, a comprehensive difference distance suitable for processing the curve is obtained through shape and dimension fluctuation difference and generating capacity difference, and the comprehensive difference distance is used for the K-means algorithm to measure similarity of different samples. And finally, the clustering result is evaluated through the improved contour coefficient. Therefore, the problems that the traditional K-means algorithm is sensitive to the initial clustering center and the intrinsic characteristics of the daily output curve are difficult to reflect through the Euclidean distance are solved. The method has the advantages that the clustering result is high in accuracy and stability, the efficiency of the clustering algorithm is improved, the calculation speed of the clustering algorithm is increased, a systematic clustering analysis process is formed, and scientific and efficient clustering analysis of the wind power daily output data is achieved.

Description

Technical Field

[0001] The present invention relates to the technical field of wind power generation, and in particular to a wind power output analysis method based on WOA-K-means. Background Art

[0002] As an important renewable energy source, wind power generation has great volatility and uncertainty in its output data. This complexity increases the difficulty of analyzing wind power output data and also poses a challenge to traditional clustering algorithms. As a commonly used unsupervised learning method, the K-means clustering algorithm is widely used in the field of data mining and analysis. It maximizes the similarity of data within a cluster by dividing the data set into different clusters, but its sensitivity to the initial cluster center and the limitations of using Euclidean distance to measure sample similarity make it unsatisfactory when processing wind power daily output curves.

[0003] As time series data, the daily output curve of wind power is affected by many factors such as weather and season, showing significant volatility and uncertainty. In the clustering process, the traditional K-means algorithm may lead to differences in the final results or even fall into a local optimal solution due to the randomness of the initial clustering center. In addition, the Euclidean distance measurement method can only reflect the numerical similarity of the curve, and it is difficult to accurately reflect the intrinsic characteristics of the wind power output curve. Therefore, an improved algorithm is urgently needed to improve the accuracy and efficiency of clustering analysis of wind power output data. Summary of the invention

[0004] In view of the shortcomings of the prior art, the present invention provides a wind power output analysis method based on WOA-K-means, which solves the problem in the prior art that the randomness of the initial clustering centers may lead to differences in the final results or even fall into a local optimal solution, and the problem that the Euclidean distance measurement method can only reflect the numerical similarity of the curves and is difficult to accurately reflect the intrinsic characteristics of the wind power output curve.

[0005] The present invention proposes a wind power output analysis method based on WOA-K-means, comprising the following steps:

[0006] S1. Calculate the characteristic value of the output curve data of each wind power day to perform data dimension reduction processing;

[0007] S2, after reducing the dimension of the daily output data, construct a feature data set of samples to be clustered as the basic data for clustering, and obtain c cluster centers of the current c number of clusters through the whale optimization algorithm WOA optimization iteration;

[0008] S3, respectively calculating the power generation difference and shape dimension fluctuation difference between each sample and the current cluster center, and then determining the comprehensive difference between the sample and the cluster center, which is used as the distance division basis in the K-means algorithm;

[0009] S4, taking c cluster centers as the initial cluster centers of the K-means algorithm, and dividing the samples into clusters with the smallest distance; after completing the allocation of all data points, recalculating the center of each cluster; updating the iterative cluster center; repeating the allocation step and the update step until the cluster center no longer changes significantly or reaches the preset number of iterations;

[0010] S5. Calculate the clustering effect evaluation index to evaluate the current clustering results and measure the effectiveness of clustering;

[0011] S6, repeat steps S1-S5, cycle (c max -c min +1) times, and output the number of clusters that maximizes the clustering effect evaluation index and the corresponding clustering results.

[0012] The technical principle of the present invention is: the traditional use of K-means can extract typical output scenarios from massive and complex wind power output data, but there is a problem that it is sensitive to the initial cluster center, and the intrinsic characteristics of the daily output curve are difficult to reflect through the Euclidean distance. The present invention proposes to use volatility and output efficiency as characteristic indicators to reduce the dimension of the daily output curve, and then optimize the initial cluster center through the global search capability of the Whale Optimization Algorithm (WOA), and proposes to use the shape dimension fluctuation difference and power generation difference to obtain the comprehensive difference distance of the applicable processing curve for K-means to measure the similarity of different samples, and finally evaluate the clustering results through the improved silhouette coefficient.

[0013] Compared with the prior art, the present invention has the following beneficial effects:

[0014] 1. Reduce clustering sensitivity: Use the global search capability of the whale optimization algorithm to optimize the initial clustering center, thereby reducing the sensitivity of the K-means algorithm to the initial clustering center, avoiding falling into the local optimal solution, and improving the stability and accuracy of the clustering results.

[0015] 2. Improved similarity measurement: Introduce the volatility of wind power output curve and power generation difference as comprehensive difference measurement indicators to replace the traditional Euclidean distance. This can more accurately reflect the morphological characteristics of wind power output curve and enhance the practical significance and application value of clustering results.

[0016] 3. Improve clustering quality: Determine the optimal number of clusters through the improved silhouette coefficient method and evaluate the effectiveness of the clustering results. Maximizing the mean of the silhouette coefficient ensures the rationality of the number of clusters and the quality of the clustering results.

[0017] 4. Data dimensionality reduction: Before cluster analysis, data dimensionality reduction is performed based on the characteristic values ​​of the wind power daily output curve, which reduces the dimension of the data and improves the efficiency and calculation speed of the clustering algorithm.

[0018] 5. Comprehensive process optimization: Combining the above improvement methods, a systematic clustering analysis process is formed, from eigenvalue extraction, data dimension reduction, initial cluster center optimization to the final clustering result evaluation, to achieve scientific and efficient clustering analysis of wind power daily output data. BRIEF DESCRIPTION OF THE DRAWINGS

[0019] Figure 1 The figure is a flow chart of the algorithm of the present invention.

[0020] Figure 2 It is a schematic diagram of the wind power output process line and shape dimension fluctuation of the present invention. DETAILED DESCRIPTION

[0021] The technical solution of the present invention is further described below in conjunction with the accompanying drawings and embodiments.

[0022] like Figure 1 As shown, the present invention proposes a wind power output analysis method based on WOA-K-means.

[0023] As a time series data, the daily output curve data of wind power generation is greatly affected by factors such as weather and season. The volatility and uncertainty of wind power output make it difficult to analyze a large amount of wind power output data. When clustering algorithms are used to divide output characteristics, more data will increase complexity and reduce the calculation speed. Therefore, feature evaluation indicators are first used to quantitatively characterize output characteristics, enhance their internal feature expression, and perform dimensionality reduction on the data.

[0024] This embodiment sets 6 characteristic indicators to specifically analyze different attributes, namely, coefficient of variation C V , extreme volatility K pfr , average volatility coefficient K fluc , peak-to-valley difference rate K diff , output kurtosis K s , total power generation E. Among them:

[0025] Coefficient of Variation:

[0026]

[0027] average value It represents the average level of output in a period of time. The standard deviation σ is used to indicate the deviation of output in a period of time from the average output. The coefficient of variation can be used to reflect the fluctuation of the output curve. The larger the coefficient of variation, the greater the deviation of the data from the average value and the greater the volatility. Therefore, it has good adaptability as a measure of the fluctuation characteristic value of the output curve.

[0028] Extreme Volatility:

[0029]

[0030] Extreme Volatility K pfr The ratio of the number of moments exceeding or falling below the limit fluctuation rate to the total number of moments within a day, thus reflecting whether there is a situation of drastic fluctuation; where t is the moment within the cycle, T is the total number of moments within the cycle; v t is the fluctuation rate at time t, obtained by differentiating the wind power output during this period, v min and v max Indicates the minimum and maximum fluctuation rate of change.

[0031] Average volatility:

[0032]

[0033] This characteristic value can reflect the number of fluctuation reversals within the force sample period, where N t Represents the output value at time t, (N t -N t-1 )(N t -N t+1 )>0, it means that the output is reversed once and there is fluctuation. The magnitude of the fluctuation can be analyzed in combination with the extreme fluctuation power and the coefficient of variation.

[0034] Peak-to-valley rate:

[0035]

[0036] Where N max 、N min They are the peak and valley values ​​in the output curve respectively.

[0037] Output kurtosis:

[0038]

[0039] Where E is the mathematical expectation. Normal distribution is usually used as a benchmark. When the statistical data is steeper than the normal distribution, K s >0, and K s The larger the value, the steeper the slope, and vice versa.

[0040] Total power generation:

[0041]

[0042] The total power generation is the area enclosed by the daily power generation curve and the coordinate axis, where x t is the output corresponding to the time t of the output curve.

[0043] The above six wind power output characteristic indicators are combined to evaluate the output characteristics of a certain day, that is, the indicators are used to characterize the output characteristics within a day, and a time multi-dimensional problem is converted into a six-dimensional problem, forming the following characteristic attribute array:

[0044] X=(C V , K pfr , K fluc , K diff , K s ,E) (9)

[0045] Using these indicators to represent the output characteristics within a day can reduce the dimension of cluster analysis.

[0046] Wind power output scenarios are divided into different categories based on the differences and similarities between output curves. Most methods for measuring curve similarity use methods such as Euclidean distance and absolute distance, but these methods are difficult to accurately reflect the characteristics of wind power output curves. Therefore, the present invention measures the distance of daily output curves from the comprehensive difference of power generation difference and shape dimension fluctuation.

[0047] Power generation difference: The power generation difference represents the shaded area enclosed by the two curves and the coordinate axis. If the two photovoltaic output curves are similar, the power generation of the two curves must be similar, and the power generation of curves x and y must be similar.

[0048]

[0049] Shape dimension fluctuation quantity difference: Most effective indicators’ similarity measurement often uses Euclidean distance and absolute distance to describe sample characteristics (referred to as quantity dimension), which cannot accurately reflect the shape characteristics of time series (referred to as shape dimension). Shape dimension fluctuation and wind power output process line are as follows: Figure 2 As shown, t is the time node, x is the x-th wind power output scenario, when t=1 or T, the shape dimension fluctuation is the angle between the head and tail wind power output broken line segments and the horizontal line.

[0050] The shape and dimension fluctuation is characterized by the slope of the wind power output curve in each period, and the shape and dimension fluctuation value at time t is calculated as follows:

[0051]

[0052] θ xx is the shape dimension fluctuation of the x-th output scenario, where K t is the slope of the tth broken line segment.

[0053] The difference in shape dimension fluctuation between scene x and scene y is expressed as follows:

[0054]

[0055] In order to reflect the comprehensive similarity between wind power generation curves, the power generation difference and the shape dimension fluctuation difference are combined to obtain the comprehensive difference distance CD(x, y), as shown in the following formula:

[0056] CD(x,y)=ΔE(x,y)·Δθ(x,y) (13)

[0057] The smaller the comprehensive difference distance between the x and y curves, the higher the similarity between them.

[0058] K-Means algorithm is a widely used cluster analysis method, which is used to divide a data set into K predefined clusters. However, the initial cluster center matrix of the K-means algorithm is randomly generated, and the analysis results generated by different initial cluster centers may be different, or may fall into a local optimum due to improper selection of initial values. Therefore, the invention uses the Whale Optimization Algorithm (WOA) to perform global exploration and optimization in the search space, and the characteristics of fast convergence speed and not easy to fall into local optimum are combined with the characteristics of strong local convergence of the K-means algorithm to reduce the sensitivity of the K-means algorithm to the initial cluster center.

[0059] The WOA whale optimization algorithm is inspired by the hunting process of humpback whales. It solves the objective function by simulating the process of whales' bubble net attack and prey search during hunting. It has a high global search capability and a fast convergence speed. WOA consists of a mathematical model of encircling prey, bubble net attack, and prey exploration.

[0060] In the first stage of encircling the prey, humpback whales can identify the location of the prey and encircle them. Since the location of the optimal solution in the search space is unknown, the WOA algorithm assumes that the current optimal candidate solution is the target prey or close to the optimal solution. After the best search agent is defined, other search agents update their positions to the best search agent. This behavior is expressed by the following equation:

[0061]

[0062] Where D is the distance between the individual to be solved and the random whale, t represents the current iteration, and is the coefficient vector, is the current optimal solution position. For the current location, is a random vector in [0, 1], It decreases linearly from 2 to 0.

[0063] The second stage is the bubble net attack. The distance between the whale and the prey is obtained according to the spiral mathematical method. The update formula for the whale to move towards the optimal position is as follows:

[0064]

[0065] in is the best position at the tth iteration; b is a constant that determines the spiral shape, which is generally taken as 1; l is a random number in [0, 1]; D' is the optimal solution in the current state.

[0066]

[0067] Where p is a random number in the interval [0, 1].

[0068] The third phase is the search for prey. The same method based on the change of the A vector can be used to search for prey. Humpback whales search randomly based on each other's positions. Therefore, using A with a random value greater than 1 or -1 forces the search agent to update the positions of other whales in the exploration phase based on the position of the randomly selected whale, rather than updating the position based on the best whale found so far. This phase emphasizes exploration and allows the WOA algorithm to perform a global search. The global search mathematical model is as follows:

[0069]

[0070] In the formula is a random vector.

[0071] The WOA algorithm is used to determine the initial cluster center and the number of clusters c for the K-means algorithm. The solution model is constructed according to the WOA algorithm. The specific steps are as follows:

[0072] (1) Whale position coding:

[0073] Assume that the data set to be clustered is X = {x1, x2, ...x n}, where x j It is the jth (j=1, 2..., n) sample feature attribute vector obtained by dimensionality reduction of the daily force curve data, which can be expressed as:

[0074] x j =(x j1 ,x j2 ……x jm )=(C V , K pfr , K fluc , K diff , Ks ,E) (23)

[0075] Where m is the number of eigenvalues ​​of each sample, that is, m = 6; x kj is the j-th dimension eigenvalue of the k-th sample;

[0076] According to the sample feature vector, the c×m-dimensional whale position code is constructed, and the i-th whale individual is represented as:

[0077] P i =(p1,p2……p c ) (twenty four)

[0078] Where p c ∈{x j}.

[0079] As an example, if there are 300 daily power sequence curves, they need to be divided into c=3 clusters.

[0080] Based on the dimensionality reduction calculation, the six eigenvalues ​​of the three samples can be obtained according to formula (23), and three feature attribute vectors x1, x2, x3...x12 can be obtained.

[0081] The WOA algorithm is to find the most representative sample in each cluster after being divided into 3 clusters.

[0082] Assume that in this algorithm, four whale individuals are generated for optimization, and the third clusters divided in the optimization process are {x1, x3, x5, x7}, {x2, x6, x9, x10}, {x4, x8, x11, x12};

[0083] Then the positions of the four whales randomly generated can be expressed as p1 = (x1, x6, x8); p2 = (x1, x9, x12); p3 = (x7, x2, x11); p4 = (x7, x10, x4). The generation rule is that the first position of p is a random selection of elements from cluster 1, the second position is a random selection of elements from cluster 2, and the same applies to the third position.

[0084] (2) Determine the fitness function

[0085] The fitness function needs to measure the quality of the selected central curve set by using the sum of the Euclidean distances of all curves to their nearest central curve as the objective function. The specific form is as follows:

[0086]

[0087] For a whale individual, the fitness f(P i) value is smaller, indicating that its location is better and the selection of cluster center is more reasonable. That is, in a cluster, the individual with the smallest value is the cluster center.

[0088] During the operation of the algorithm, the cluster division and the position of individual whales are continuously updated until the fitness function converges to the minimum. The fitness function of this algorithm uses the Euclidean distance. The smaller the Euclidean distance, the higher the similarity between the same clusters. The sum of the Euclidean distances from the selected cluster center to its cluster is the smallest, which means it is the most representative.

[0089] During the program editing process, the convergence condition is set to stop the iteration when the fitness value is not significantly improved within a certain number of iterations. This condition can help terminate the search process that is no longer significantly improved in advance and save computing resources.

[0090] By solving the mathematical model of cluster centers constructed by the WOA algorithm, we can finally obtain c initial cluster centers that can reflect the characteristics of the group, which can scientifically and effectively solve the problem of initial value determination of the K-means algorithm.

[0091] To further explain, suppose there are 100 daily output curves that need to be analyzed. Since the purpose of this program is to extract the most representative output curve, we hope to control the number of clusters to less than 4. However, it is not certain whether it is better to divide it into 4 categories or 3 categories, and calculation and comparison are needed.

[0092] That is, the maximum value of c is 4, and then the program is started to calculate the clustering conditions when c=4, 3, and 2 respectively.

[0093] First, the WOA algorithm finds c initial cluster centers, and then inputs them into the K-means algorithm to further determine which c types of clusters to divide into and which c cluster centers to find. In each case, an optimal clustering result will be calculated. The final selection of the clustering structure depends on the evaluation index of the clustering effect.

[0094] Each clustering situation will calculate the clustering effect according to formula (29). When c = 4, it is divided into 4 clusters. After calculating a result, an evaluation index will be obtained. The same is true when c = 3 and 2. After the calculation is completed, the program will compare and select the classification with the best evaluation index as the final result output.

[0095] The traditional K-means algorithm uses Euclidean distance to measure, which only considers the similarity of the overall numerical value, but does not consider the difference in the curve shape. The improvement of the K-means algorithm in this paper is also reflected in determining the optimal number of clusters through the improved silhouette coefficient to measure the clustering quality.

[0096] That is, after each cluster is obtained through the K-means algorithm, a clustering effect evaluation index is calculated according to its classification result to evaluate the current clustering result and measure the effectiveness of the clustering.

[0097] The silhouette coefficient method combined with the comprehensive difference distance of the output curve is used to determine the optimal number of clusters and evaluate the clustering quality, as shown in the following formula:

[0098]

[0099] Where SIL(x) represents the silhouette coefficient of sample x; x represents the sample belonging to C k Class samples; |C k |、|C j | respectively represent C k , C j The number of samples of the class; CD a (x) represents x and C k The average CD distance of the remaining objects in the class; CD b (x) means x and non-C k The minimum average CD distance of the remaining objects in the class;

[0100] The range of the silhouette coefficient is [-1, 1]. The closer the silhouette coefficient is to 1, the smaller the C to which x belongs. k The closer the clusters are, the better the clustering effect is. If the silhouette coefficient is less than 0, the clustering fails.

[0101] The mean value of the silhouette coefficient of all data samples is used to evaluate the effectiveness of clustering. The formula is as follows:

[0102]

[0103] The larger the value, the better the overall clustering effect. The maximum number of clusters is the optimal number of clusters.

[0104] Finally, repeat the above steps, loop (c max -c min +1) times, and output the number of clusters that maximizes the clustering effect evaluation index and the corresponding clustering results.

[0105] Finally, it should be noted that the above embodiments are only used to illustrate the technical solution of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that the technical solution of the present invention can be modified or replaced by equivalents without departing from the purpose and scope of the technical solution of the present invention, which should be included in the scope of the claims of the present invention.

Claims

1. A wind power output analysis method based on WOA-K-means, characterized in that: The steps include: S1. Calculate the characteristic value of the output curve data of each wind power day to perform data dimension reduction processing; S2, after reducing the dimension of the daily output data, construct a feature data set of samples to be clustered as the basic data for clustering, and obtain c cluster centers of the current c number of clusters through the whale optimization algorithm WOA optimization iteration; S3, respectively calculating the power generation difference and shape dimension fluctuation difference between each sample and the current cluster center, and then determining the comprehensive difference between the sample and the cluster center, which is used as the distance division basis in the K-means algorithm; S4, taking c cluster centers as the initial cluster centers of the K-means algorithm and dividing the samples into clusters with the smallest distance; After all data points have been assigned, the center of each cluster is recalculated; Update the iterative cluster center; repeat the assignment step and the update step until the cluster center no longer changes significantly or reaches a preset number of iterations; S5. Calculate the clustering effect evaluation index to evaluate the current clustering results and measure the effectiveness of clustering; S6, repeat steps S1-S5, cycle (c max -c min +1) times, and output the number of clusters that maximizes the clustering effect evaluation index and the corresponding clustering results.

2. The analysis method according to claim 1, characterized in that The characteristic evaluation index in step S1 includes: coefficient of variation C V , extreme volatility K pfr , average volatility coefficient K fluc , peak-to-valley difference rate K diff , output kurtosis K s , total power generation E.

3. The analysis method according to claim 2, characterized in that Among the characteristic evaluation indicators: Coefficient of Variation: average value It represents the average level of output in a period of time. The standard deviation σ is used to indicate the deviation of output in a period of time from the average output. The coefficient of variation can be used to reflect the fluctuation of the output curve. Extreme Volatility: Extreme Volatility K pfr The ratio of the number of moments exceeding or falling below the limit fluctuation rate to the total number of moments within a day, thus reflecting whether there is a situation of drastic fluctuation; where t is the moment within the cycle, T is the total number of moments within the cycle; ν t is the fluctuation rate at time t, obtained by differentiating the wind power output during this period, ν min and ν max Indicates the minimum and maximum fluctuation rate of change; Average volatility: This characteristic value can reflect the number of fluctuation reversals within the force sample period, where N t Represents the output value at time t, (N t -N t-1 )(N t -N t+1 )>0 indicates that the output is reversed once and there is fluctuation. The magnitude of the fluctuation can be analyzed by combining the extreme fluctuation power and the coefficient of variation. Peak-to-valley rate: Where N max 、N min They are the peak and valley values ​​in the output curve respectively; Output kurtosis: Where E is the mathematical expectation. Normal distribution is usually used as a benchmark. When the statistical data is steeper than the normal distribution, K s >0, and K s The larger the value, the steeper the slope, and vice versa. Total power generation: The total power generation is the area enclosed by the daily power generation curve and the coordinate axis, where x t is the output corresponding to the time t of the output curve.

4. The analysis method according to claim 3, characterized in that: The wind power output curve is reduced in dimension, and a feature attribute array is formed through the six feature evaluation indicators: X=(C V ,K pfr ,K fluc ,K diff ,K s ,E) (9)。 5. The analysis method according to claim 4, characterized in that In step S2, the initial cluster center and the number of clusters c are determined by the WOA algorithm. The specific steps are as follows: (1) Whale position coding: Assume that the data set to be clustered is X = {x1, x2, ...x n }, where x j It is the jth (j=1, 2..., n) sample feature attribute vector obtained by dimensionality reduction of the daily force curve data, which can be expressed as: x j =(x j1 ,x j2 ......x jm )=(C V ,K pfr ,K fluc ,K diff ,K s ,E) (23) Where m is the number of eigenvalues ​​of each sample, that is, m = 6; x kj is the j-th dimension eigenvalue of the k-th sample; According to the sample feature vector, the c×m-dimensional whale position code is constructed, and the i-th whale individual is represented as: P i =(p1,p2......p c ) (24) Where p c ∈{x j }; (2) Determine the fitness function The fitness function needs to measure the quality of the selected central curve set by using the sum of the Euclidean distances of all curves to their nearest central curve as the objective function. The specific form is as follows: For a whale individual, the fitness f(P i ) value is smaller, indicating that its location is better and the selection of cluster center is more reasonable.

6. The analysis method according to claim 1, characterized in that In step S3, the difference in power generation between the sample and the current cluster center is calculated using the following formula: The difference in shape dimension fluctuation between the sample and the current cluster center is calculated using the following formula: Among them, t is the time node, x is the xth wind power output scenario, θ x，t is the shape dimension fluctuation of the x-th output scenario.

7. The analysis method according to claim 6, characterized in that Combining the power generation difference and the shape and dimension fluctuation difference, the comprehensive difference distance is CD(x, y), as shown in the following formula: CD(x,y)=ΔE(x,y)·Δθ(x,y) (13) The smaller the comprehensive difference distance between the x and y curves, the higher the similarity between them.

8. The analysis method according to claim 1, characterized in that In step S5, the optimal number of clusters is determined by using the contour coefficient method combined with the comprehensive difference distance of the output curve, and the clustering quality is evaluated, which is specifically as follows: Where SIL(x) represents the silhouette coefficient of sample x; x represents the sample belonging to C k Class samples; |C k |、|C j | respectively represent C k , C j The number of samples of the class; CD a (x) represents x and C k The average CD distance of the remaining objects in the class; CD b (x) means x and non-C k The minimum average CD distance of the remaining objects in the class; The range of the silhouette coefficient is [-1, 1]. The closer the silhouette coefficient is to 1, the smaller the C to which x belongs. k The closer the clusters are, the better the clustering effect is. If the silhouette coefficient is less than 0, the clustering fails. The mean value of the silhouette coefficient of all data samples is used to evaluate the effectiveness of clustering. The formula is as follows: The larger the value, the better the overall clustering effect. The maximum number of clusters is the optimal number of clusters.

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