CN110533304B - A Method for Uncertainty Analysis of Power System Load - Google Patents

Abstract

The invention discloses a method for analyzing load uncertainty of a power system, which decomposes a load sample into a typical scene (a central sample), an extreme scene (an extreme sample) and a common scene (a common sample). To ensure robustness, the extreme scenario (extreme sample) is used to determine the crew composition scheme; the corresponding scheduling problem is computed using the typical scenario (center sample) with higher probability of occurrence. In view of the characteristic that each load sample is high-dimensional data, the PCA technology is adopted to reduce the original load sample to a low-dimensional space so as to better reveal the relationship between the data. Therefore, a scheduling scheme with high robustness and economy can be obtained.

Description

A Method for Uncertainty Analysis of Power System Load

technical field

The invention relates to the field of machine learning, in particular to an analysis method for power system load uncertainty.

Background technique

Nowadays, with the development of economy and society, the demand for electricity has increased dramatically. The emergence of some new load sources, such as high-speed railways and electric vehicles, has brought great difficulties to load forecasting. These difficulties are mainly reflected in the highly variable nature of these loads. However, the traditional unit combination determines the on/off state of the generator through a certain load forecast value. Uncertainty caused by load forecast errors is not considered. However, the uncertainty of load has brought great challenges to the economy and reliability of the grid, such as frequency drop, overloading of transmission lines, and even cascading failures when the system is under heavy load conditions, etc. In order to effectively solve this problem, researchers have proposed a large number of solutions, among which stochastic optimization [1]-[11] and robust optimization [12]-[17] are commonly used to deal with this type of problem.

Stochastic optimization is an optimization method based on random scenarios, which generates a large number of scenarios to simulate the uncertainty of load. Monte Carlo (MC) is a typical imitation method. In [7], Wu et al. proposed a scenario-based model for computing long-term safety-constrained unit combinations. Among them, MC is used to generate a large number of scenarios to simulate the uncertain change of load. In the literature [8], Wang et al. proposed a scenario-based safety-constrained unit combination model to deal with the intermittency and volatility of wind power. Among them, MC is used to simulate the output of wind power. In the literature [9], Wu et al. proposed a scene simulation method under the condition of multiple time scales. And use this method to calculate the scheduling plan to solve the problem of load forecast uncertainty. Another stochastic optimization approach is chance-constrained optimization, which deals with load uncertainty by placing probabilistic constraints on possible load scenarios. Wu et al. [10] proposed a chance-constrained unit combination model to address the impact of load forecast uncertainty on transmission line overload and power imbalance. In the literature [11], Wang et al. combined chance constraints and goal programming, and proposed a new model to optimize the unit combination (UC) scheme by adjusting the system risk threshold. Random optimization has a fatal shortcoming, that is, because some possible load scenarios are not considered, especially some extreme load scenarios, resulting in poor robustness of its unit combination scheme. In addition, in order to ensure the accuracy of the unit combination scheme, a large number of scenarios need to be generated, which will bring a huge burden to the calculation.

Robust optimization is another way to deal with load randomness. The purpose of robust optimization is to calculate the unit combination scheme under the worst scenario. If the schedule can be satisfied in the worst case, then this scheme can handle any scenario. This type of problem can be formulated as a min-max-min linear optimization problem, where the worst case is determined at an intermediate level. The entire optimization problem can be solved by a decomposition algorithm. But the solution complexity is high. In the literature [14], Hu et al. proposed a multi-band robust optimization unit combination model to solve the problem of time correlation of load uncertainty. In [15], Hu et al. proposed a robust optimization model to minimize the scheduling cost, which is used to ensure that when the load fluctuates within the forecast interval, the scheduling plan can be safely adjusted in real time. In [16], Bertsimas et al. proposed a stage-adaptive robust optimization model, which considers the uncertainty of load in both power supply and demand after the uncertainty subset is determined. Due to the consideration of the worst case, the corresponding unit combination scheme will be very conservative, that is, the operating cost is too high. In order to reduce costs. Blanco et al. [17] proposed a worst-case scenario method for solving a scenario-based two-stage safety-constrained unit combination model. This method only considers the worst-case division, which is used to reduce the conservatism of the scheduling scheme. And this method calculates that the conservatism of the scheduling scheme depends on the number of scene divisions, but the number of scene divisions is set manually. As a result, this method is not scientifically rigorous enough for the division of scenes. And it only considers the worst case, and other scene information is not fully utilized.

In addition, some other optimization methods have been proposed in recent years. Among them, interval optimization [18]-[20] only considers the uncertainty of these variables in the prediction interval, ignoring the probability distribution of these variables. In the literature [21], Zhao et al. proposed weight components based on robust optimization and stochastic optimization to reduce conservatism. This method simply combines the objective functions of robust optimization and stochastic optimization through weight components, without in-depth study of the intrinsic relationship between the two methods. Xiong et al. [23] proposed a distributed robust two-stage unit combination model that handles load uncertainty by defining fuzzy subsets. The purpose of this method is to minimize the expected total cost in the worst case, which is very similar to the method in [17]. These methods do not consider the relationship between scenes, and there is no clear scene division method. And the information of the scene is not fully used. For example, robust optimization and some improved methods only consider the use of the worst scenario, ignoring the information of other scenarios. In addition, the scene division of all methods is carried out in high-dimensional space, ignoring the property that the relationship between scenes is easily revealed in low-dimensional space.

Contents of the invention

The invention aims to provide a load uncertainty analysis method of a power system to improve the robustness of a scheduling scheme. In order to solve the above-mentioned technical problems, the technical solution adopted in the present invention is: a method for analyzing the load uncertainty of a power system, which is characterized in that it includes the following steps:

1) Remove the redundant information in the historical load samples, and project the historical load samples with redundant information removed into a low-dimensional space;

2) dividing the historical load samples of each cluster in the low-dimensional space into extreme samples, common samples and central samples;

3) Return all samples in the low-dimensional space divided in step 2) to the original dimension, and reconstruct the historical load sample data;

4) If there is a unit combination scheme in the extreme sample in the reconstructed historical load sample data, keep the unit combination scheme; otherwise, discard the extreme sample;

5) Calculate whether there is a corresponding economic dispatch plan for each common sample and center sample in the reconstructed historical load sample data, and record the score of each unit combination plan retained in step 4); if the unit combination plan is here If there is an economic dispatch solution under the common sample or the central sample, the score of the corresponding unit combination plan will increase. Use SI/NJ to calculate the feasibility ratio of each unit combination plan. SI is the total score of each unit combination plan, and NJ is the total Common sample and central sample size;

6) Select the central sample with the largest local density in each cluster to replace all sample information in this cluster; and show the possibility of power system load fluctuations through the central sample of each cluster and the occurrence probability of the corresponding cluster; Select the unit combination plan with the largest SI score, and calculate the economic scheduling results of each center sample through this unit combination plan; finally, combine the calculation results of all center samples through the occurrence probability of the corresponding cluster to obtain the final scheduling plan .

In step 1), redundant information in historical load samples is removed by principal component analysis.

In step 2), the average density of historical load samples in each cluster is calculated. If the local density of a historical load sample is greater than the average density, the sample is identified as a core sample, otherwise it is an edge sample, that is, an extreme sample; the core sample The cluster center in is the central sample, and the others are common samples.

In step 3), use the following formula to reconstruct the historical load samples

S_s为第s个负荷场景，N是负荷场景数量，Λ_m＝[e₁,e₂,……,e_m]，e₁,e₂,……e_τ为协方差矩阵ΔSΔS^T的特征向量阵，

in,

S _s is the sth load scenario, N is the number of load scenarios, Λ _m = [e ₁ , e ₂ ,..., e _m ], e ₁ , e ₂ ,... e _τ are the characteristics of the covariance matrix ΔSΔS ^T vector array,

In step 4), judge whether there is a unit combination scheme through the following unit combination model:

Among them, F _i (p _its ) represents the coal-burning cost of the i-th unit at the t-th moment in the scenario s, SU _its represents the start-up cost of the i-th unit at the t-th time in the scenario s, and NG represents the number of coal-fired units, NT represents the planned time scale, p _its represents the output of the i-th coal-fired unit at the t-th moment in the scenario s, PD _ts represents the load value at the t-th time in the scenario s, R _i represents the upper limit of the i-th unit’s slope, p _{min, i} ,p _max,i represent the minimum and maximum output of unit i respectively, I _its represents the state of unit i at the tth moment in scene s, X _on,i(t-1)s ,X _{off,i( t-1)s} represent the accumulative turn-on and turn-off time of the i-th unit at the t-1th moment in the scenario s, T _on,i , T _off,i respectively represent the minimum start-up and shutdown time required by the i-th unit , SF represents the system transfer factor matrix, K _p , K _D represent the unit and load node connection matrix respectively, PL _max represents the upper limit of line power flow, and the above matrices are all constant matrices; P _ts , PD _ts represent the Coal-fired unit output and system load; the system transfer factor matrix SF is a constant matrix.

In step 6), use the following formula to carry out economic dispatch calculation:

where NS represents the number of typical scenes, and π _s represents the probability of the sth typical scene occurring.

Compared with prior art, the beneficial effect that the present invention has is:

1) The present invention proposes a load uncertainty processing method based on machine learning. The advantage of this method is that it can generate a more economical scheduling scheme while ensuring robustness. Through a large number of simulation experiments, it is proved that the robustness of the unit combination scheme can be guaranteed by a small number of extreme scenarios.

2) The load scenarios are effectively divided into typical scenarios, extreme scenarios and common scenarios, and the selected extreme scenarios ensure the robustness of the unit combination scheme.

3) Use principal component analysis to project the load samples into a low-dimensional space, so that the load scenes can be separated more easily. The results show that some extreme load samples can be quickly found in some cluster samples, and the extreme load samples contain Information for most common load scenarios.

4) The calculated unit combination scheme is more robust.

Description of drawings

Figure 1 High-dimensional data (left) and low-dimensional data (right).

Figure 2 Decision diagram (left) and cluster diagram (right).

Figure 3 Scene separation and screening.

Figure 4. Model solution flow chart.

Figure 5. Comparison of clustering results with and without PCA.

Fig. 6 The influence of PCA on the robustness and economy of scheduling plan.

Figure 7. Scores of unit combination scenarios corresponding to each extreme sample.

Figure 8 Load curves for various methods.

Figure 9 Robustness comparison of four methods.

detailed description

In order to overcome the shortcomings of the random scenario method, some extreme load scenarios and typical scenarios need to be considered to improve the robustness of the scheduling scheme under the condition of ensuring economy. In this chapter, we adopt the method of CSFDP to separate historical samples into marginal samples, normal samples and central samples. Edge samples represent extreme load scenarios, central samples represent typical load scenarios with a relatively high probability of occurrence, and normal samples represent common scenarios. Principal component analysis is introduced to project high-dimensional loading samples into a low-dimensional space to better reveal the inner relationship between samples.

The principal component analysis of the present invention is introduced as follows.

In general, the data dimension of raw load is high, especially in large power systems. Take the IEEE 118 node system as an example. The system has 91 load nodes, so the data dimension of its 24-hour load scene is 91*24=2184. High-dimensional data not only requires a lot of calculation, but also the internal relationship between data is difficult to find. Such as the IEEE 188 node system, its high-dimensional load sample data is shown on the left side of Figure 1 below. The data overlap completely, and the relationship between the data is difficult to be revealed. Regarding this, PCA [24]-[26] provides an effective tool to extract key information by projecting raw data into a low-dimensional space. As shown on the right side of Figure 1, the relationship between the data can be easily observed when the load samples of the 118 system are reduced to a two-dimensional space. As shown in the figure, the distance between sample points reflects the similarity between two scenes. The number of sample points in a certain area reflects the probability of scene occurrence in this area. If there are more sample points, the probability of occurrence is greater, and vice versa. Of course. When PCA reduces high-dimensional data to low-dimensional space, most of the information of its original data will also be preserved, such as 118-node system, 97% of the information of its original data is preserved in low-dimensional space. The basic principle of PCA is described as follows: Suppose there are n load nodes and t time points, so the data dimension of the sth load scenario S _s is n×t. Suppose there are N load scenarios, then the average value of the scenarios is

Next, the difference between each load scenario and the average value is calculated by formula (2).

On this basis, we can use (3) to calculate the covariance matrix.

And decompose this covariance matrix, then you can get its eigenvector matrix e=[e ₁ ,e ₂ ,...e _τ ] and its corresponding eigenvalue vector λ=[λ ₁ ,λ ₂ ,... λ _τ ]. In this way, the original data is projected into a new space with m (m≤n×t) dimensions by formula (4).

Where Λ _m consists of m feature vectors extracted from _e , Λ _m =[e ₁ ,e ₂ ,...,e _m ]. Principal component analysis maps high-dimensional data to a low-dimensional space through the eigenvectors of the data itself, so its corresponding eigenvalues reveal how much information about the original data is preserved in the new space. The eigenvector with the largest eigenvalue contains the richest information in the original data and is called the principal component (PC). The percentage of information preserved in the new space can be estimated by computing the corresponding eigenvalues.

Equation (5) calculates how the information of the original data is preserved in the new coordinate system. where m represents the dimension of the new coordinates and τ represents the dimension of the original coordinates. In order to solve the safety-constrained unit combination model, the data must be inversely transformed back to the original dimension by (6) for calculation.

This verifies that PCA not only reduces the computational complexity, but also helps to reveal the intrinsic relationship between data. In actual calculation, at least 85% of the information needs to be retained to ensure the accuracy of the calculation. It should be noted that PCA can help reduce the dimensionality of data and reveal connections within the data, but cannot separate loading samples into certain categories. This will be achieved by the clustering by fast search and find of dense typeaks algorithm.

The fast search and find density peak clustering (CSFDP) algorithm of the present invention is as follows.

After mapping the original load data into a new low-dimensional coordinate system, the next step is to separate the low-dimensional samples. Fast Search and Discovery Density Peak Clustering [27] can be used to separate scenes. Because this method has an important parameter, the local density. It represents how close the scene is to other surrounding scenes. This reflects the probability of such a scenario occurring. Therefore, the occurrence probability of each scene can be calculated by this parameter. According to the difference of local density (the probability of occurrence), the scene can be divided into typical scene, normal scene and extreme scene.

CSFDP is a method to cluster data with similar information by local density and distance. Compute the local density using a Gaussian kernel.

where dc represents the cutoff distance, _djk represents the distance between the jth data and the kth data, and _ρj is the local density of the _jth scene. To choose a suitable cut-off distance, the distance between any two points is calculated, and to arrange these calculated distances in ascending order, we have

Among them, θ∈[0.01 0.02], [Mθ] indicates that the value of Mθ is rounded to an integer. Local density is an important parameter reflecting the probability of scene occurrence. Another important parameter, δ _j , represents the minimum distance to point j among all points with a local density greater than j; if point j is the point with the largest local density, it represents the maximum distance to point j among all sample points . Multiplying ρ _j and δ _j , the formula (9) can be obtained

γ _j = ρ _j · δ _j j = 1,2,...,N (9)

where γ _j is an important parameter to generate a decision map for determining cluster centers. Arranging γ _j in descending order, the decision diagram in the new coordinate system is shown in Fig. 2.

It can be seen from the left figure of Figure 2 that all the points arranged from right to left reflect the change trend of the γ value of the point. It can be seen from the figure that there is an obvious jump when the γ value of all points transitions from below the dotted line to above the dotted line. It indicates that the points above the dotted line are cluster centers, and the points below the dotted line are ordinary points. These common points will be clustered by cluster centers. The cluster center is represented by a solid point in the right figure of Figure 2, and the non-cluster center is the corresponding hollow point.

Then, the average density of these samples in each cluster is calculated and used as a lower bound on the separation density of the loaded samples. In each cluster, if the local density of the sample is greater than the average density, the sample is identified as a core sample, otherwise it is a marginal sample. The cluster centers in the core samples are identified as central samples, and the others are normal samples. Through this clustering algorithm, all load samples are divided into multiple clusters, and in each cluster the samples are further divided into core samples and marginal samples. The core samples consist of central samples and common samples, and the central samples have the highest probability (largest local density) in each cluster, so they correspond to the typical scenarios of the load. Likewise, common samples with high density (high probability of occurrence) represent common scenarios of load; conversely, marginal samples represent extreme scenarios of load due to their extremely low local density (low probability of occurrence).

The sample separation and screening process of the present invention is introduced as follows.

In the general scenario method, a large number of scenarios will be used to compute the group combination scheme to improve the reliability of the scheduling plan. Each scenario has its own characteristics. If the scenario is used indiscriminately, it will affect the reliability of the scheduling scheme and increase the computational burden. Also, some scenarios have similar properties. If scenarios with similar information are recalculated in the calculation of the scheduling scheme. They can only increase the computational burden and do not contribute to the improvement of the scheduling plan.

However, these scenes can be classified according to their similar characteristics, and the typical features of each classification can be easily extracted. The typical characteristics of the scenarios are used to calculate the safety-constrained unit combinations, which not only improves the reliability of the scheduling scheme, but also reduces the computational burden. Therefore, scene selection is necessary. Most of the current methods only consider the reduction of scenes to reduce the computational burden, and do not consider the relationship between scenes, and lack a clear way to distinguish these scenes.

As shown in Fig. 3, in the present invention, we propose a clear scene separation and screening method. PCA is used to reveal the relationship between scenes, and CSFDP is used to separate the scenes. The high-dimensional original scene is projected into a two-dimensional space, while PCA preserves 97% of the information of the original data. The result after dimensionality reduction can be seen on the right side of Figure 1, where each point represents a scene. The distance between two points reflects the similarity between two scenes, and the number of points in an area reflects the probability of occurrence in this area. In this way, the relationship between samples has been clearly revealed. These scenes are divided into several categories in low-dimensional space by CSFDP, and the scenes of each cluster are further divided into typical scenes, common scenes and extreme scenes according to the local density (probability of occurrence). Finally, these scenarios are returned to their original dimensions by inverse PCA transformation for computing the safety-constrained unit combination model.

If the unit combination scheme can satisfy extreme scenarios, it can also satisfy most normal scenarios. In order to ensure the robustness of the scheduling plan, extreme scenarios must be selected for computation. At the same time, the typical scenario represents the maximum possibility of the scenario, in order to improve the economic level, it should be used to calculate the economic dispatch. The common use of these two types of scenarios can make the robustness and economy of the scheduling plan well balanced.

The problem of combination of safety-constrained units before the present invention is introduced as follows.

After the load scenario is determined, the day-ahead safety-constrained unit combination model can be described as the following formula:

subject to

Where (10) represents the objective function, F _i (p _its ) represents the coal-burning cost of the i-th unit at the t-th moment in the scenario s, SU _its represents the start-up cost of the i-th unit at the t-th moment in the scenario s , NS represents the number of typical scenarios, π _s represents the probability of occurrence of the s-th typical scenario, NG represents the number of coal-fired units, NT represents the planning time scale, p _its represents the output of the i-th coal-fired unit at the t-th time under the scenario s , PD _ts represents the load value at the t-th moment in scenario s, R _i represents the climbing limit of the i-th unit, p _min,i , p _max,i represent the minimum and maximum output of the i-th unit respectively, and I _its represents the unit i In the state of the tth moment in the scene s, X _on,i(t-1)s , X _off,i(t-1)s represent the accumulative on and off of the i unit at the t-1th moment in the scene s The shutdown time, T _{on, i} , T _{off, i} represent the minimum start-up and shutdown time required by the i-th unit, SF represents the system transfer factor matrix, K _p , K _D represent the unit and load node connection matrix, PL _max represents the upper limit of line power flow, and the above matrices are all constant matrices. P _ts and PD _ts respectively represent the coal-fired unit output and system load at time t in scenario s. (11) is the power balance constraint, (12) is the climbing constraint, (13) is the generator output boundary constraint, (14) is the minimum start-stop constraint, and (15) is the line power flow constraint. As mentioned earlier, the traditional unit combination model does not consider the uncertainty of load. A good unit combination scheme should satisfy both economy and robustness. To ensure robustness, some extreme scenarios need to be considered, although they occur with relatively low probability. At the same time, operating costs should be controlled if extreme cases are included in unit combination calculations. The day-ahead safety-constrained unit combination model is solved by a two-stage optimization problem. In the first stage, the switching state of the generator is determined according to the selected extreme load scenario; in the second stage, an economic dispatch subproblem with stochastic safety constraints is solved. Typical scenarios are selected for computing this subproblem, thus ensuring the economy of the scheduling scheme. A flowchart of the program is shown in 4.

The specific solution steps are shown in the figure below:

Step 1: Use principal component analysis to remove redundant information in historical data. The raw data is projected into a two-dimensional space, in which at least 97% of the information is retained for subsequent calculations of safety-constrained unit combinations.

The second step: using the method of CSFDP, the historical load samples of each cluster are divided into extreme samples, normal samples and central samples.

Step 3: Reconstruct the load sample through (6), and return the divided low-latitude scene data to the original dimension

Step 4: Each extreme sample is used to compute the group combination problem (16). If there is a unit combination plan for extreme samples, the unit combination plan will be retained; otherwise, the sample will be discarded.

s.t constraints (11)-(15)

The above optimization problem will be solved under each extreme scenario (extreme sample).

Step 5: Calculate whether there is a corresponding economic dispatch plan for each core sample (collectively referred to as central sample and common sample), and record the score of each unit combination plan retained in step 4. If the unit combination scheme has an economic dispatch (SCED) solution in this core scenario, the score of the corresponding unit combination scheme will increase (SI=SI+1). Use SI/NJ to calculate the feasibility ratio of each unit combination plan, SI is the total score of each unit combination plan, and NJ is the total core sample size.

Step 6: Select the unit combination scheme with the largest SI score, through which, the central sample of each cluster is used for economic dispatch calculation, and its model is (17).

s.t. Constraints (11)-(13), (15)

where π _s represents the occurrence probability of the central sample, that is, the probability of each cluster, and NS represents the number of central samples, that is, the number of all clusters. The on/off state of the unit is determined in the fourth step.

The scheduling plan calculated by this method has high robustness and low conservatism. The robustness of the scheme is enhanced by selecting extreme scenarios of loads and only considering typical high-probability load scenarios (central samples), ensuring the economy of the system. The effectiveness of this method will be verified in the case study in the next section.

We tested the proposed method using an IEEE 118-node system. The load data comes from PJM’s 24-hour historical load from 2017 to 2018 [28]. The simulation is performed on a 3.9GHz personal computer with 8GB of memory, and the model and algorithm are implemented in MATLAB.

As shown in the left panel of Figure 5, there are 241 sample scenes participating in the experiment. Common samples and extreme samples are represented by blue and green points, respectively, and red points are the central samples (cluster centers) of each cluster. In the top left panel of Fig. 5, it shows the sample separation results produced by our method. When only CSFDP is used to separate samples, the clustering results are obtained in a high-dimensional space, but for the convenience of direct observation, the classification results are projected into a two-dimensional space, as shown in the lower left figure of Figure 5. Their respective 24-hour core samples and extreme samples are shown on the right side of Figure 5. The green dotted line is the extreme load sample data, and the cyan solid line is the core load sample data. Among them, the extreme load sample (202) with the highest score using our method is represented by the black solid line in the upper right panel of Fig. 5. In the lower right panel of Fig. 5, the solid black line represents the loading sample data (190) that achieved the highest score using only CSFDP separation.

From the left half of Figure 5, we can see the distribution of load samples in the two-dimensional coordinate system. Experimental results show that extreme samples (green dots) can be easily separated using our method. In the upper left panel of Figure 5, it shows that the core samples are surrounded by extreme samples which are far away from the regions containing a large number of samples. In fact, the number of samples in an area reflects the probability of occurrence of samples in this area. In reality, the probability of occurrence of extreme samples is very low. This means that extreme samples can be easily separated from the data using our method. In the lower left panel of Figure 5, it can be found that some extreme samples are separated in some regions with a large number of samples, while some extreme samples are not separated. For example, in the lower left figure of Figure 5, in the lower left corner area of the figure, those points that are obviously far away from the concentration area are not separated. This means that some extreme samples cannot be accurately separated using only CSFDP clustering. In the upper right of Fig. 5, it can be found that the 39 extreme samples selected by our method can cover the core samples, that is, the cyan solid line area is covered by the green dotted line area and have similar shapes. This means that extreme scenes contain the information of normal and typical scenes. Therefore, if crew combination schemes can satisfy extreme scenarios, then they are likely to satisfy other common and typical scenarios as well. But in the lower right panel of Fig. 5, we can see that only the 51 extreme scenes separated by CSFDP cannot fully surround those ordinary scenes. This means that the extreme scenes separated only by CSFDP cannot fully contain the information of ordinary scenes. And we can see that the black solid lines selected by the two methods are not at the highest position, but near the top, and the two black solid lines are not the same. It means that the two black solid lines contain the most abundant information of the common scene in their respective methods, but they do not contain the same amount of information, which will affect the robustness of the crew combination scheme.

In the left diagram of Fig. 6, it represents the ratio of the number of scenarios that can be executed by the unit combination scheme to the number of scenarios in common scenarios and extreme scenarios. It can be seen from the figure that in the normal scenario and the extreme scenario, the ratio of scenarios where the unit combination scheme calculated by our method can be executed is 1.00 and 0.94, respectively. This means that the crew combination proposals generated by our method can cope with 100% of normal scenarios and 94% of extreme scenarios. However, the proportions at which the unit combination scheme calculated using only the CSFDP method can be executed are 1.00 and 0.84 in normal and extreme scenarios, respectively. This means that the robustness of the computer group combination scheme using only CSFDP is similar to the robustness of our method computer group combination scheme in common scenarios. But in extreme scenarios, the robustness of our method is better than that using only CSFDP. In addition, the running costs calculated by the two methods are similar, and the running cost of our method is higher than the cost of only using CSFDP method by 6649$/h (increased by 0.52%). This means that our method can generate more robust unit combination schemes under similar economic conditions. The separation of extreme scenarios affects the robustness of the calculated crew combination scheme, and our method can improve the accuracy of scene separation.

The unit combination scheme scores for each extreme sample are shown in Fig. 7. One of the dots has a size proportional to the value of the fraction and is distinguished by a different color. It can be seen that the largest black dot has the highest score of 202 points. The percentage of the number of scenarios that the corresponding unit combination scheme can be executed is 1.00. Those samples surrounded by black dotted circles are samples for which the unit combination scheme cannot be calculated. An important observation can be made. The scores in the graph increase gradually from left to right. This means that if we choose the sample on the right side of the figure, the unit combination scheme calculated by it can satisfy more load scenarios. This provides us with a quick way to identify load samples that yield highly robust unit combination scenarios. That is, we only need to select load samples on the right side of the plot.

To verify the effectiveness of the proposed method, we compare it with Monte Carlo (MC) methods, chance-constrained optimization methods, and worst-scenario based robust optimization methods. Using the typical scenario in Figure 5, the expected load of chance constraints is obtained by its probability-weighted average. Set the line overload probability to 10% and the deviation to 4% of the desired value. The load curve is shown in the lower right panel of Figure 8. The MC method will generate 3000 scenes by simulation, and then reduce them to 10 scenes in the lower left picture of Figure 8 through scene reduction technology. And the load curve obtained using our method is shown in the upper right panel of Fig. 8.

It can be seen from Fig. 8 that the load curves obtained by PCA and CSFDP are smooth, and their shapes are very close to those of historical loads. But the MC method generates load curves with irregular shapes. In addition, the scenarios simulated by Monte Carlo (MC) are mainly concentrated in areas with high probability. Extreme scenarios are difficult to simulate, and the chance-constrained optimization method mainly considers the expected value of the load, and does not consider some necessary extreme scenarios. On the contrary, our method fully considers some extreme scenarios to ensure the robustness of the unit combination scheme.

Table I The operating costs of the four methods

We will compare the economics and robustness of the unit combination schemes of the four methods. Table I gives the operating costs of the four methods. where the third and fourth columns are the operating cost deviation and the corresponding percentage increase/decrease, respectively. Compared with our method, the chance-constrained method has the lowest cost and the worst-scenario method has the highest cost. The cost of our method approaches that of MC simulations and chance-constrained methods. The cost of our method is 7113$/h (0.56%) higher than the cost of the Monte Carlo simulation method and 55996$/h (4.55%) higher than the running cost of the chance-constrained method. The cost of this method is 638451$/h lower than that based on the worst scenario method, which reduces the operating cost by 33.18%.

In terms of the robustness of the crew combination scheme, our method has advantages over other methods. It can be seen from Fig. 9 that the number of feasible scenarios of our method is much larger than that of Monte Carlo and chance-constrained methods in normal scenarios and extreme scenarios. In particular, our method is able to obtain a crew combination that is as robust as worst-scenario based methods. For example, the number of feasible scenarios for our method and the worst-case based method is 202 for common scenarios and 37 for extreme scenarios. Our method achieves a percentage of 1.00 in normal scenarios and 0.94 in extreme scenarios. This means that the schedules produced by our method can satisfy 100% of normal scenarios and 94% of extreme scenarios, very close to worst-case based methods. Among them, the number of feasible scenarios for the MC method and the chance-constrained method are 105 and 95, respectively, less than half of our method. This means that the crew combination scheme produced by our method is much more robust than the other two methods.

The list of references used in the present invention is as follows:

[1] Q.P.Zheng, J.Wang and A.L.Liu, "Stochastic Optimization for UnitCommitment—A Review," in IEEE Transactions on Power Systems, vol.30, no.4, pp.1913-1924, July 2015.

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Claims (6)

1. A method for analyzing load uncertainty of an electric power system is characterized by comprising the following steps:

1) Removing redundant information in the historical load samples, and projecting the historical load samples with the redundant information removed into a low-dimensional space;

2) Dividing the historical load samples of each cluster in the low-dimensional space into extreme samples, common samples and central samples;

3) Returning all samples in the low-dimensional space divided in the step 2) to the original dimension, and reconstructing historical load sample data;

4) If the extreme samples in the historical load sample data obtained by reconstruction have a unit combination scheme, the unit combination scheme is reserved; otherwise, discarding the extreme sample;

5) Calculating whether each common sample and the center sample in the reconstructed historical load sample data have a corresponding economic dispatching scheme or not, and recording the score of each unit combination scheme reserved in the step 4); if the unit combination scheme has an economic scheduling solution under the common sample or the central sample, the fraction of the corresponding unit combination scheme is increased, the feasibility proportion of each unit combination scheme is calculated by using SI/NJ, wherein SI is the total fraction of each unit combination scheme, and NJ is the total number of the common samples and the central sample;

6) Selecting the central sample with the maximum local density in each cluster to replace all sample information in the cluster; the probability of fluctuation of the load of the power system is shown through the occurrence probability of the center sample of each cluster and the corresponding cluster; selecting a unit combination scheme with the maximum SI score, and calculating an economic dispatching result of each center sample according to the unit combination scheme; and finally, combining the calculation results of all the central samples through the occurrence probability of the corresponding clusters to obtain a final scheduling plan.

2. The method for analyzing uncertainty of load of electric power system according to claim 1, characterized in that in step 1), redundant information in historical load samples is removed by using principal component analysis method.

3. The method according to claim 1, wherein in step 2), the average density of the historical load samples in each cluster is calculated, and if the local density of a certain historical load sample is greater than the average density, the sample is identified as a core sample, otherwise, the sample is an edge sample, i.e., an extreme sample; the cluster center in the core sample is the center sample, and the others are the common samples.

4. The method according to claim 1, wherein in step 3), the historical load samples are reconstructed using the following equation

Wherein,

S _s for the s-th load scenario, N is the number of load scenarios, Λ _m ＝[e ₁ ,e ₂ ,......,e _m ]，e ₁ ,e ₂ ,……e _τ Is a covariance matrix Δ S ^T The feature vector matrix of (a) is,

5. the method for analyzing the load uncertainty of the power system as claimed in claim 1, wherein in the step 4), whether a unit combination scheme exists is judged by using the following unit combination models:

wherein, F _i (p _its ) Represents the coal-fired cost, SU, of the ith unit at the tth moment in the scene s _its Representing the startup cost of the ith unit at the tth moment under the scene s, NG representing the number of coal-fired units, NT representing the planned time scale, and p _its Represents the output of the ith coal-fired unit at the t moment under the scene s, PD _ts Representing the load value, R, at time t under scene s _i Represents the climbing upper limit, p, of the ith unit _min,i ,p _max,i Respectively representing the minimum and maximum output of the ith unit, I _its Representing the state of the unit i at the t-th moment in the scene s, X _on,i(t-1)s ,X _off,i(t-1)s Respectively representing the cumulative opening and closing time of the ith unit at the T-1 th moment under the scene s, T _on,i ,T _off,i Respectively representing the minimum startup and shutdown time required by the ith unit, wherein SF represents a system transfer factor matrix, K _p ,K _D Respectively representing unit and load node connection matrices, PL _max Representing the upper limit of the line flow; p _ts ,PD _ts Respectively representing the output of the coal-fired unit and the system load at the time t of the scene s.

6. The method for analyzing uncertainty of load in electric power system according to claim 5, wherein in step 6), the economic dispatch calculation is performed using the following formula:

where NS denotes the number of typical scenes, π _s Representing the probability of the occurrence of the s-th typical scene.

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