Probability distribution of wind power volatility based on the moving average method and improved nonparametric kernel density estimation

1 Introduction

In view of the continuous growth of the wind power access capacity,the fluctuation and randomness of the generated output power will introduce new challenges to the dispatching, planning, and safe and stable operation of power systems.Therefore,approaches needed to reduce the negative impact of grid-connected wind power on the power system have become a research hotspot.One of the effective ways to solve the aforementioned adverse effects is to analyze the wind power fluctuation characteristics (WPFC) and accurately establish its probability distribution model.This approach may also provide more safe and economical decision-making means for dispatch and planning agencies [1–3].

At present, scholars have conducted extensive research on WPFC modeling [4–9].In [4], the author defined three quantitative indicators based on the measured wind power output data, and analyzed WPFC from the aspects of the differential time scale and installed wind farm capacity.As the research progressed, some researchers began to use a mixed distribution of multiple functions as the prior distribution of wind power fluctuations.In [5], the author proposed a method of fitting WPFC with a Gaussian mixture model instead of a single distribution function model based on the measured data of a wind farm group.In [6], the author analyzed a large number of measured data of wind farm output, and used various probability distribution models to simulate the WPFC of wind farms.In [7], the authors proposed a hybrid t location–scale distribution model to describe the probability distribution of WPFC.In [8], the authors proposed a probability distribution model of a finite mixed Laplace distribution based on a large number of measured wind farm data.In [9], the authors proposed a hybrid Logistic distribution model to quantitatively describe WPFC, and validated it with measured data from a wind farm.Overall, prior publications used the theoretical distributions of one or more functions to fit the distribution curve of the wind power fluctuation index as the prior distribution, and then used historical data to estimate its parameters.Nevertheless,the universal applicability of the constructed model has been proven difficult to guarantee.Wind farms in China are extensively distributed throughout the country.Additionally, the WPFC in different regions may follow different forms of probability densities, and the modeling method based on parameter estimation is difficult to guarantee its universal applicability.

In view of this,some studies have proposed the modeling of WPFC with nonparametric estimation [10–14].Compared with the parameter estimation method,this method directly builds a probabilistic model based on historical sample data,and does not need to make a prior judgment on the standard form of wind power fluctuation.Therefore,it is of great advantage to use nonparametric kernel density estimation (NPKDE) methods when it is not clear which standard parameter form the probability density of the wind power fluctuations will have.In [10], the authors first applied a statistical approach and the method of NPKDE to construct a distribution function of wind power prediction errors.Based on this, they established a benefit model based on wind farm wind energy loss and energy storage device planning considerations.In [11],the authors predicted the wind power based on the actual power curve, and then combined the distribution characteristics of the prediction errors to predict the error distribution characteristics using NPKDE.In [14], the authors first used the hybrid Copula function to characterize the correlations between multiple failure modes.Based on these, they constructed the corresponding probability density function and cumulative distribution function based on the NPKDE method.However, if NPKDE was used to model the WPFC, problems still persisted.Accordingly, the bandwidth is the only parameter that needs to be determined in the NPKDE model.Traditional NPKDE often uses the same bandwidth for all samples in a model.However,owing to the large wind power samples, different samples may be adapted to different bandwidths.Using only one bandwidth will lead to a low-degree of fitness at a certain point or in a certain interval of the sample, which is the problem of so-called low-local fitness.

In summary,this study first uses the moving average method to extract the wind power fluctuations.It then uses an improved NPKDE method to model the extracted results.Subsequently, a constraint-order optimization algorithm is used to solve the model.The main research and innovation aspects of this study are as follows: 1.an improved NPKDE strategy is proposed, whereby the geometric distance is used as an index to identify the sample interval with a lower fitting accuracy, and its bandwidth is modified; 2.a method is proposed to model WPFC based on NPKDE.The advantages are: 1.the fixed, single bandwidth becomes adaptively adjusted by the sample interval, and the variable bandwidth improves the modeling accuracy of NPKDE; 2.the probability characteristics of the wind power fluctuation are modeled directly based on sample data, and there is no need to judge which standard distribution form it obeys.Therefore, it has higher accuracy and applicability.Finally, simulation results based on actual data obtained from a wind farm in Hubei have verified the correctness and effectiveness of the proposed modeling method.

2 Extraction of wind power fluctuation based on the method of moving average

Wind power contains continuous and minute components [15–17].The continuous components undergo smaller fluctuations, exhibit longer fluctuation periods, and have minor influences on the stability of the power system and the accuracy of wind power forecasting.However,the minute component is characterized by a large fluctuation and a short-fluctuation period.When the installed capacity of wind power is large,the component will seriously affect the power quality of the power system.At present, the lowpass filters,wavelet decomposition,and empirical mode decomposition,are used to extract the momentum of the wind power wave.However,these methods cannot use the time scale of the data to decompose the signal.In addition,some problems are associated with these methods, such as the boundary effect [18–20].

Therefore, according to the basic concept of the algorithm for the separation of load components in [21],this study uses the method of moving average to eliminate the wind power fluctuation.First, a time window is selected with a specific length,and the arithmetic average of all the values within the window is estimated.The value of the center point of the window is then replaced with the estimated average value.Finally, the window is moved backward by the dot pitch.The aforementioned steps are repeated.Suppose the length of the moving average is K min.If K is an even number, then the minute component PHt and continuous component PLt of the wind power at time t are calculated using (1).

where PLt is the continuous component,PHt is the minute component,which is the amount of change superimposed on the continuous component,Pt is the measured average power at t min,t is the measurement point,and Jis the total number of measurement points.

The length K of the moving time window is randomly selected,and its selection directly affects the extracted fluctuation components.If it is too long,the change trend of wind power will be reflected on the minute component PHt, and the minute component will no longer be a random variable.If it is too short, the fluctuation of wind power will be reflected on the continuous component PLt.In general,this can be selected according to experience, that is, 20 min is more suitable for general loads, and 30 min is suitable for large impact loads.Combined with the actual situation, this study selected the period of 20 min as the moving average sampling period [22].

3 Modeling of wind power fluctuation probability density based on improved NPKDE

3.1 Wind power fluctuation probability density model based on NPKDE

It is assumed that x1,x2,...,xn is an n-sample sequence denoting the wind power fluctuation.If (,) x b is the probability density function of wind power fluctuation, then the NPKDE of this probability density function is estimated as,

where xi represents the sample data of wind power fluctuation at the ith sampling point,b is the bandwidth, and is the kernel function.

To ensure the continuity of the estimated probability density function, the kernel function needs to be a symmetric smooth nonnegative function to meet the following characteristics,

where c is a constant.

There are many kernel function choices.However,different kernel functions have little impact on accuracy [23–25].Therefore,the kernel function selected in this study is a commonly used Gaussian function formulated as,

Combination of (2) and (4) allows the formulation of the nonparametric kernel density estimation of the wind power probability density function as,

To evaluate the goodness-of-fit of the model, three evaluation indices are used as the valid evaluation indices of the model,namely the correlation coefficient r,root mean square error (RMSE), and mean absolute error (MAE).The smaller the RMSE and MAE are, the more accurate the model will be.The closer the correlation coefficient value is to unity, the better the model fit is [26, 27].

The model deviation (geometric distance) can be calculated as,

where i =1,2,..., m,where m is the number of groupings of the wind power fluctuation sample frequency histogram,is the ordinate of the ith histogram,and is the function value of the fitted probability density function.

The three evaluation indicators are defined as follows,

where n is the number of sample sequences, and is the average value of the ordinate values of the histogram.

3.2 Model improvement strategy

As indicated in (5), the existing NPKDE theory uses a fixed bandwidth b. Thus,only one b is obtained to estimate the minimum sum of fitness of all the data samples.This processing method may be associated with the situation in which the evaluation index is abnormally large for individual sample data.If the bandwidth b corresponding to the sample data is modified in a targeted manner,the adaptive bandwidth can be solved.Therefore, changing the originally fixed bandwidth yields a set of adaptive bandwidth vectors that can ensure the local adaptive characteristics of the constructed probability model, and further improves the model’s goodness-of-fit.Therefore, on the basis of the aforementioned NPKDE method, the following improvement strategies are added.After the use of the bandwidth optimization model to obtain the optimal bandwidth bZ, the applicability of sample spacing is discriminated.For any sample interval if the following inequality is satisfied, there will be local adaptability problems in the sample interval.

where Tl(bz) is the model deviation in any sample interval l, is the average model deviation of the entire sample space,and λ is the adjustment coefficient.The specific value of λ can be determined according to the actual test situation.

The mathematical expression of the mean geometrical distance is,

For the interval of the local adaptability problem, the bandwidth adjustment model is constructed and the bandwidth matrix is modified to

where bl is the bandwidth of the l sample interval,nlis the number of samples in the sample interval, T(bz)mid is the median of the geometric distance in the sample interval,and δ is the kernel function threshold.Therefore, (5) can be modified in the following form, and an adaptive NPKDE model for the model is thus proposed,

where

In (xx) the number of sampling intervals to be adjusted is k,and bI is the modified bandwidth matrix of the interval I.

3.3 Bandwidth optimization model

In the NPKDE model, the selection of bandwidth bis important for kernel density estimation [28].In this study,the test of fitness is used as a constraint condition,and is incorporated into the bandwidth optimization model.A constrained bandwidth optimization model is proposed as follows

where (•) Xise is the integrated mean-square-error, R(x,b) is the true probability density function that takes the ordinate value of the frequency histogram,2 bχ is the test statistics of the NPKDE 2χ,and is the 2χ distribution with m-1 degrees-of-freedom at the significant level α.

4 Solution of bandwidth optimization model based on constrained-order optimization

The sequential optimization method is an effective method used to solve complex optimization problems [29–30].Traditional sequential optimization is generally used for unconstrained optimization problems.In this study, there are constraints in the bandwidth optimization model.Therefore, this study uses constrained sequential optimization based on the following steps:

Step 1: Determine the solution space Ω of the bandwidth value

Step 2: According to (3) and bandwidth value, calculate the wind power wave of the nonparametric kernel density estimation function value for momentum

Step 3: According to (17), calculate the number of solutions in the observation solution set

where (•)Prob is the alignment probability, Θ is a sufficiently good solution set,g is the number of real solutions,p is the number of solutions in the observed solution set P.Additionally, sshows that there are at least s truths,η shows the probability that the observation solution set P contains s sufficiently good solutions,and qis the probability that an adjective solution is actually observed in the solution space

Step 4: Select the 2χ test as the approximate model,and find the p solutions that satisfy it in Ω to form the observation solution set P

Step 5: Select the objective function of (16) as the exact model,and use the exact model to compare the solutions in the solution set P in order; then select the first s solutions to be true solutions

Step 6: Use (7)–(9) as the criterion, selecting the optimal solution bz, and then use (10)–(12) to modify the bandwidth to obtain the final optimal bandwidth sequence.

5 Data analyses

According to the measured data of a wind power field in Hubei Province, the simulation experiment was programmed in MATLAB.

5.1 Extraction of wind power volatility

From March 17 to April 19, 2009, 500 active output data were selected for analysis.The detailed wind power output data are shown in Appendix A.The sampling period of the data was 10 min and the wind farm consisted of 16 wind turbines rated at 850 kW.The moving average method was used to process the data of the first fan in this study,and the output comparison before and after processing is shown in Fig.1.

Fig.1 Comparison of wind power sampled sequences

Fig.1 shows that the moving average method can set a time window for the sampled data and estimate the local average.Furthermore, the fluctuating random error is filtered out,and the random fluctuation of wind power output is effectively reduced so as to obtain a smoother output result.The comparison of wind power fluctuation is shown in Fig.2.

Fig.2 Comparison of wind power fluctuations

Fig.2 shows that the fluctuation of wind power output can be slowed down after the method of moving average is used.The wave momentum shows a smooth variational trend to effectively reduce the impact of wind power fluctuation on the accuracy of model construction.

5.2 Comparative analysis

5.2.1 Comparison of the results of NPKDE before and after improvement

The probabilistic density model of the wind power fluctuation component is established for the wind power volatility quantity extracted in this study.Compared with the simulation results of the traditional NPKDE method, the effectiveness of the improved modeling strategy proposed in this study is verified.By analyzing the MAE, RMSE, the three r indices,the fitting degree of the model was evaluated.The probability density curve is shown in Fig.3.The results of the error operation are listed in Table1.

Table1 Comparison of evaluation indices before and after the improvement of the probability density model

Distribution model Mean absolute error (MAE) Root-meansquare error (RMSE)r Traditional nonparametric kernel density estimation method (NPKDE)4.917×10-5 3.104×10-4 0.8665 Model proposed in this study 2.773×10-5 1.704×10-4 0.9774

Fig.3 Comparison of goodness-of-fit before and after the improvement of the NPKDE method

By analyzing the results of Fig.3 and Table1, it can be found that the method used in this study is better than the traditional NPKDE method.The MAE, RMSE, and r values of the model are reduced by 43.6%, 45.1%, and improved by 12.8%, compared with the traditional model, respectively.The reason is attributed to the fact that the traditional NPKDE method is designed to minimize the total error of the entire sample and obtain a fixed bandwidth value.

The method of this study not only focuses on the sum of errors, but also modifies the bandwidth of the local intervals where the error is large.Although it is possible to reduce the accuracy of some individual intervals,the accuracies of the sample intervals with the largest errors have been improved considerably.Thus, the overall modeling accuracy has been increased by this method (as shown in Table1, the improved nonparametric correlation coefficient is much better than the traditional nonparametric correlation coefficient,and is close to unity).It can be observed that the overall modeling accuracy of the multivariable NPKDE method can be improved effectively by the adaptive adjustment of the bandwidth of the sample intervals with local adaptive problems.

5.2.2 Precision and fitness comparison between the parameter estimation method of the mixed distribution function and the adaptive NPKDE

The precision and fitness of wind power volatility of the proposed method is verified by using the parameter estimation method and the modeling method based on mixed logistic,mixed t location–scale,and mixed Gaussian distributions, respectively.The sampled data represent the active power data of the entire wind farm,and the studied time period was from March 17, 2015, to April 19, 2015.The sampling period was 10 min.The comparison results of the precision and fitness of different model evaluation indicators are listed in Table2.The comparison curve of precision for probability density is shown in Fig.4, and the fitness comparison curve is shown in Fig.5.

Table2 Comparison of the precision and fitness of different model evaluation indicators

Distribution model MAE RMSE r Precision Fitness Precision Fitness Precision Fitness Mixed Gaussian 7.015×10-5 8.486×10-5 3.772×10-4 4.271×10-4 0.9631 0.9534 Mixed t location–scale 6.881×10-5 8.443×10-5 3.696×10-4 4.202×10-4 0.9769 0.9687 Mixed logistic 5.473×10-5 8.167×10-5 3.146×10-4 4.248×10-4 0.9332 0.9295 Model proposed in this paper 2.773×10-5 6.138×10-5 1.704×10-4 3.393×10-4 0.9774 0.9954

Fig.4 Comparison of the precisions of different model evaluation indicators

From Fig.4 and Table2, it can be observed that among the four probability distributions of wind farm power fluctuation, the model in this study has the best fit, and the three indices are all the best.The r values of the mixed Gaussian and mixed logistic distributions were only 0.9631 and 0.9332.It can be observed that if the prior distribution selection was erroneous, the parameter estimation method could not easily obtain a better modeling accuracy.Although the correlation coefficient of the mixed t-location distribution was close to the model in this study, its MAE increased by 59.7%, and the RMSE increased by 53.9% compared with the model proposed in this study.Based on the analysis of the results, it can be concluded that the improved NPKDE method has a higher modeling accuracy than the mixed parameter estimation method.The main reason is that the method is driven by sample data to directly model the probability distribution.It is not necessary to select the distribution function form of the sample distribution in advance,while the modeling precision does not depend on the selection result of the distribution function that is only related to the bandwidth selection.

Fig.5 Comparison of the fitness values of different model evaluation indicators

Fig.5 and Table2 show that the model in this study yielded the best fitting effect among the four probability distributions of wind power volatility, and its rvalue was 0.9954.However, the modeling accuracy of the mixed t location–scale,the mixed Gaussian,and the mixed logistic distributions, were much lower, and their r values were only 0.9534, 0.9687, and 0.9295, respectively.Therefore, the improved NPKDE method proposed in this study is more applicable than the mixed parameter estimation method.The main reason is that the parameter estimation method needs to determine the form of probability distribution function in advance, but the probability distribution of different wind farms may follow different forms.Thus,the parameter estimation of different wind farms with the same distribution function may reduce the goodness-of-fit of the model.Therefore,the method proposed in this study can ensure the precision and fitness of modeling.

5.2.3 Comparison of Operation Time in Constructing Different Models

To test the calculation efficiency of the model constructed in this study,the calculation times of different models were compared,and the specific results are listed in Table3.

Table3 Comparison of operation times in the construction of different models

Distribution model Operation time/s Mixed Gaussian 87.14 Mixed t location–scale 89.53 Mixed logistic 85.77 Traditional NPKDE 83.94 Model proposed in this study 86.98

Table3 shows that the operation time of the model constructed by the method in this study has not been increased considerably when the modeling accuracy and the overall goodness-of-fit was improved.Compared with the traditional NPKDE method with the shortest operation time, the operation time of this method was only increased by 3.04 s.It can be concluded that the method in this study improves the modeling accuracy and goodness-of-fit, but with the improvement of modeling accuracy and goodness-of-fit, the operation time of the model did not increase significantly.These findings conform to the actual requirements of the modeling process.

6 Conclusions

A wind power fluctuation modeling method was proposed in this study based on the method of moving average and adaptive NPKDE.To improve the precision of the constructed model of this study,an algorithm was used to solve it based on constraint ordinal optimization.The specific conclusions are as follows:

(1) The method of moving average can reduce the random wind power output fluctuations in this study so as to effectively reduce the impact of these fluctuations on the accuracy of the constructed model

(2) The improved NPKDE method proposed in this study can solve the problem of large interval errors of local samples.Based on the bandwidth correction strategy, multiple adaptive bandwidths can replace the single, fixed bandwidth of traditional NPKDE, and can improve the goodness-of-fit of the NPKDE method

(3) Compared with the traditional modeling method based on parameter estimation, the improved NPKDE based on modeling method is more accurate and applicable.

Appendix A

Table Wind power fluctuation at various sampling points

Sampling points Wind power output 1 339.6875 101 99.5 201 234.5625 301 -1.1875 401 321.5 2 365.1875 102 79.875 202 189.5 302 -1.125 402 342.5 3 408.8125 103 80.3125 203 240.9375 303 -2 403 355 4 488.4375 104 90.1875 204 220.8125 304 -1.5625 404 436 5 533.875 105 99 205 176.3125 305 0.125 405 414.9375 6 588.3125 106 139.375 206 214.25 306 5.1875 406 287.5 Wind power output Sampling points Wind power output Sampling points Wind power output Sampling points Wind power output Sampling points 7 540.5 107 112 207 160.4375 307 -1.875 407 97.6875 8 416.1875 108 135.3125 208 187.4375 308 -0.8125 408 272.9375 9 486.5 109 107.625 209 207.25 309 53.6875 409 296.4375 10 456.5625 110 150.0625 210 132.8125 310 37.9375 410 260.25 11 462.6875 111 100.5625 211 125.25 311 6.25 411 208.4375 12 480.0625 112 92.1875 212 114.4375 312 23.8125 412 211.9375 13 537.5625 113 114.375 213 223.1875 313 29 413 217.9375 14 517 114 140.5 214 180.8125 314 26.625 414 195.625 15 547.1875 115 126.5625 215 145.625 315 40.5625 415 255.375 16 612.9375 116 119.875 216 184.3125 316 39.9375 416 265.75 17 564.4375 117 132.4375 217 160.1875 317 35.0625 417 228.5 18 436.1875 118 131 218 177.625 318 36.5625 418 222.9375 19 512 119 136.0625 219 140.625 319 62.875 419 174.125 20 531.75 120 160.875 220 107.4375 320 21.4375 420 200.625 21 529.125 121 188.4375 221 94.625 321 45.875 421 188.9375 22 446.625 122 167.625 222 128.3125 322 53 422 267.5 23 466.3125 123 184.1875 223 95.5 323 52.625 423 251.75 24 447.9375 124 211.4375 224 68.75 324 13.25 424 312 25 522.6875 125 163.5 225 69.5 325 -2.125 425 433.625 26 554 126 142.25 226 49.1875 326 -2.8125 426 363.6875 27 672.125 127 222.6875 227 42.3125 327 -1.9375 427 347.8125 28 629.5 128 211.9375 228 47.3125 328 -2.375 428 181.75 29 695.375 129 242.125 229 51.6875 329 16 429 662.5625 30 738.5 130 173.6875 230 34.6875 330 76.8125 430 600 31 638.0625 131 220.875 231 18.1875 331 224.875 431 591.0625 32 644.3125 132 202.125 232 16.125 332 254.4375 432 725.1875 33 621.75 133 146.125 233 9.5 333 258.8125 433 810.125 34 714.4375 134 148.6875 234 -0.1875 334 282.75 434 837.6875 35 721.75 135 160.75 235 -2 335 253.6875 435 842.0625 36 626.5 136 135.875 236 -2.25 336 256.125 436 825.625

continue

Sampling points Wind power output 37 577.625 137 161.8125 237 -1.3125 337 282.25 437 833.5625 38 619.75 138 157.75 238 -2.3125 338 307.375 438 802.5 39 613.6875 139 124.375 239 -2 339 320 439 808 40 608.1875 140 139.625 240 -2.125 340 334.0625 440 301.5 41 488.8125 141 112.9375 241 -2 341 319.25 441 818.3125 42 590.875 142 150.125 242 -2.0625 342 300.75 442 800.1875 43 508.3125 143 119 243 -2.0625 343 322.5625 443 789.9375 44 665.1875 144 98.25 244 -2.25 344 295.8125 444 798.875 Wind power output Sampling points Wind power output Sampling points Wind power output Sampling points Wind power output Sampling points 45 582.75 145 130 245 -2.125 345 275.5 445 812.125 46 677.875 146 120.6875 246 -2 346 273.25 446 822.375 47 544.0625 147 119.5 247 -2.0625 347 323.0625 447 825 48 513.625 148 158.6875 248 -1.125 348 341.625 448 788.25 49 511.5625 149 221.625 249 -1.1875 349 351.6875 449 830.3125 50 533.3125 150 267.5 250 -1.1875 350 314.5 450 841.25 51 471.625 151 332.3125 251 -1.3125 351 335.75 451 819.0625 52 469.4375 152 370.8125 252 -1.0625 352 301.25 452 842 53 466.375 153 310.6875 253 -1 353 287.375 453 821.9375 54 542.1875 154 353.0625 254 -1.125 354 246.875 454 812.75 55 530.375 155 300 255 -1.125 355 245.3125 455 821.3125 56 452.3125 156 286.9375 256 -1.1875 356 219.1875 456 830.1875 57 434.4375 157 381.3125 257 -1 357 183.6875 457 822 58 367.375 158 310.4375 258 -1.3125 358 212.1875 458 844.1875 59 412 159 303.75 259 -1.1875 359 243.625 459 790.125 60 382.0625 160 311.6875 260 -1.125 360 301.875 460 827.8125 61 308.3125 161 383.875 261 -2.1875 361 278.8125 461 249.1875 62 259 162 389.375 262 -2.6875 362 226 462 547.6875 63 488.9375 163 380.875 263 -1.25 363 187.75 463 115.8125 64 357.1875 164 378.875 264 -1.3125 364 194.1875 464 162.1875 65 399.8125 165 341.375 265 -0.6875 365 188.3125 465 -3.5 66 385.8125 166 443 266 0.5 366 208.9375 466 219.5625 67 314.6875 167 386.9375 267 0.75 367 248.5 467 274.25 68 238 168 357.9375 268 -1.125 368 292.3125 468 222.4375 69 257.5625 169 423.5625 269 -1.125 369 286 469 46.375 70 211.3125 170 403.6875 270 -1 370 264.125 470 845.9375 71 248.9375 171 391.4375 271 -1.1875 371 183.75 471 461.8125 72 377.1875 172 402.6875 272 -1.125 372 191.75 472 166.6875 73 292.4375 173 301.5625 273 -1.1875 373 173.25 473 170.3125 74 282.1875 174 322.4375 274 1.125 374 163.125 474 -2.875

continue

Sampling points Wind power output Sampling points Wind power output Sampling points Wind power output Sampling points Wind power output Sampling points Wind power output 75 363.75 175 355.875 275 21.125 375 142.625 475 92.0625 76 300.5 176 298.0625 276 -0.6875 376 137.25 476 829.4375 77 381.75 177 283.5 277 -1 377 111.875 477 840.4375 78 266.1875 178 286.1875 278 -1.9375 378 100.1875 478 848.5625 79 279.3125 179 326.875 279 -1.375 379 74.6875 479 200.625 80 245.875 180 305.375 280 -1.0625 380 3.5 480 844.5625 81 303.625 181 253.6875 281 -1.4375 381 -3.0625 481 826.375 82 223.8125 182 318.9375 282 -1.375 382 -4.125 482 849.9375 83 284.875 183 341.0625 283 -1.3125 383 47.9375 483 847.5 84 262.8125 184 326.5625 284 -1.125 384 142.3125 484 850.25 85 194.5 185 310.875 285 -1.8125 385 246.3125 485 848.5 86 353.8125 186 253.25 286 -1.125 386 197.3125 486 850.25 87 293.6875 187 355.6875 287 -1 387 133.375 487 850.5625 88 245.875 188 351.8125 288 -1.0625 388 186.25 488 841.25 89 344.8125 189 308.875 289 -1.0625 389 177 489 823.1875 90 311.375 190 234.875 290 -1.0625 390 141.1875 490 848 91 315.5625 191 228.375 291 -1.0625 391 239.625 491 847.8125 92 230.1875 192 178.4375 292 -1 392 282.5 492 850 93 205.9375 193 205 293 -1.4375 393 366.625 493 849.875 94 197.1875 194 174 294 -1.3125 394 266 494 848.75 95 214.8125 195 135.9375 295 -1.125 395 230.5 495 800.375 96 277.5 196 143.125 296 -1.125 396 225.4375 496 836.875 97 221 197 234.4375 297 -1.1875 397 215.625 497 788.6875 98 162.8125 198 239.125 298 -1.1875 398 170.8125 498 719.25 99 115.75 199 229.75 299 -1.125 399 210.6875 499 743.5 100 125.25 200 241.125 300 -1.1875 400 292 500 735

Acknowledgements

This work was supported by Science and Technology project of the State Grid Corporation of China “Research on Active Development Planning Technology and Comprehensive Benefit Analysis Method for Regional Smart Grid Comprehensive Demonstration Zone,” and National Natural Science Foundation of China (51607104).