Probabilistic Optimal Power Flow for Day-Ahead Dispatching of Power Systems with High-Proportion Renewable Power Sources

With the increasing proportion of uncertain power sources in the power grid; such as wind and solar power sources; the probabilistic optimal power flow (POPF) is more suitable for the steady state analysis (SSA) of power systems with high proportions of renewable power sources (PSHPRPSs). Moreover; PSHPRPSs have large uncertain power generation prediction error in day-ahead dispatching; which is accommodated by real-time dispatching and automatic generation control (AGC). In summary; this paper proposes a once-iterative probabilistic optimal power flow (OIPOPF) method for the SSA of day-ahead dispatching in PSHPRPSs. To verify the feasibility of the OIPOPF model and its solution algorithm; the OIPOPF was applied to a modified Institute of Electrical and Electronic Engineers (IEEE) 39-bus test system and modified IEEE 300-bus test system. Based on a comparison between the simulation results of the OIPOPF and AC power flow models; the OIPOPF model was found to ensure the accuracy of the power flow results and simplify the power flow model. The OIPOPF was solved using the point estimate method based on Gram–Charlier expansion; and the numerical characteristics of the line power were obtained. Compared with the simulation results of the Monte Carlo method; the point estimation method based on Gram–Charlier expansion can accurately solve the proposed OIPOPF model


Introduction
The steady state analysis (SSA) of power systems is the most fundamental method for the investigation of power system planning and operations. In particular, a calculation method for the steady state operation of power systems is based on power flow. The injection power and voltage of traditional power flow calculation are both determinate quantities, and the bus voltage and line power obtained are also determinate quantities [1,2]. With an increase in proportion of uncertain power sources such as wind and solar power generation in the power grid, there is a corresponding increase in the number of indeterminate variables in the power grid. A grid with a high proportion of uncertain power sources is referred to as a power system with high proportions of renewable power sources. Due to the indeterminate variables, in accordance with the law of probability and statistics, power flow model. The model was solved in this study using the point estimate method based on the Gram-Charlier expansion, and the numerical characteristics of the target variables were obtained. The OIPOPF mathematical model and its solution algorithm were validated on the modified Institute of Electrical and Electronic Engineers (IEEE) 39-bus test system and modified IEEE 300-bus test system. Moreover, the simulation results verify the accuracy of the model and the effectiveness of the method.
The remainder of this paper is structured as follows. Section 2 presents the mathematical model of OIPOPF and details the solution of the proposed OIPOPF using the point estimation method based on the Gram-Charlier expansion. Section 3 presents the simulation results for the OIPOPF, as applied to the IEEE 39-bus test system, to verify the accuracy of the OIPOPF mathematical model. Thereafter, the conclusions of this study are presented in Section 4.

POPF Mathematical Model
Renewable energy sources such as wind power and solar power have uncertainties based on their randomness and intermittent nature. For the PSHPRPSs whose bus variables obey probability and statistics law, it is suitable to use the POPF for grid SSA. In this paper, an OIPOPF mathematical model is used to formulate a day-head dispatching plan for the PSHPRPSs.
For a PSHPRPSs with n buses, the bus active power difference, ∆P, is defined as expressed by Equation (1): where ∆P is a 1 × n − 1 dimensional vector; P S is the bus active power; and P 0 e is the initial calculated value of the bus active power. P S = (P S,1 , P S,2 , · · · , P S,i , · · · , P S,n ), P S,i is the ith variable of P S vector. If P S,i is an uncertain renewable energy source, its mean value is µ S,i , and its standard deviation is σ S,i . Moreover, P 0 e is calculated using AC power flow equations, which can reduce the error in the calculation of ∆P.
The voltage phase angle error calculated by the modified active power error is more accurate. The grid bus active power difference, ∆P, and the bus voltage phase angle difference, ∆θ, satisfy the functional relationship expressed by Equation (2): where ∆θ is a 1 × n − 1 dimensional vector. The bus voltage phase angle difference ∆θ is defined as expressed by Equation (3): where θ S is the bus voltage phase angle, and θ e is the calculated value of bus voltage phase angle. In Equation (2), the coefficient matrix, B, is an n − 1 × n − 1 dimensional matrix with elements as expressed by Equation (4): Given that B is an invertible matrix, Equation (2) is transformed into Equation (5): In Equation (5), E = B −1 , and E is a n − 1 × n − 1 dimension matrix. For any branch, l, the from-bus is the i bus, and the to-bus is the j bus. The corresponding voltage phase angle of bus i is θ i , and that of bus j is θ j . Moreover, θ i and θ j are the ith and jth elements in the Sustainability 2020, 12, 518 4 of 19 θ e vector, respectively. The active power of the line between bus i and bus j is defined as expressed by Equation (6): where B ij represents the reactance between bus i and bus j; G ij represents the conductance between bus i and bus j; and y i0 represents the admittance between bus i and the zero-potential bus.
For a grid with L lines, the line power is P L = p l1 , p l2 , · · · , p ij , · · · , p L T . The day-ahead dispatching of PSHPRPSs considers the distribution of active power in the power grid. Therefore, according to Equation (6), the line power of the OIPOPF is defined as expressed by Equation (7): The day-ahead dispatching of PSHPRPSs not only needs to set generation plan and reserves accommodation reserve power for the grid, but also needs to reserve line active power margin for the grid to prevent line power congestion during real-time dispatching. Hence, the line power in the day-ahead dispatching plan of PSHPRPSs should be minimized, so as to reserve line power margin. The OIPOPF model should be defined as expressed by Equation (8), to constrain the minimization of line power.
where P i represents the power of power sources, and P l represents the line power. The OIPOPF mathematical model for the day-ahead dispatching of PSHPRPSs is constrained by power balance constraints, tie-line power constraints, and reserve capacity constraints.

Power Balance Constraints
Equation (9) is the power balance expression; where P d represents the total load power of the power grid, P u represents the total power of the uncertain power generation, and P c represents the total power generated by conventional power sources such as thermal power and hydropower sources.

Tie-Line Power Constraints
The maximum line power is P lmax , and the power of each line in Equation (7) is P l . Moreover, P l represents the lth line power. Equation (7) satisfies the line power constraint expression as shown in Equation (10).

Reserve Capacity Constraints
The number of power sources in the grid is assumed to be m. Moreover, P i represents the ith power source. The power source described in P i satisfies the reserve capacity constraints in Equation (11). For a continuous random variable X of any positive integer k, the x k function can be integrated on (−∞, +∞) with respect to f (x). The k order moment of X can be expressed by Equation (12) [39]: where α k represents the k order moment. Moreover, the mean value of the continuous random variable X is µ, and the k order central moment of X can then be defined as expressed by Equation (13): where β k represents the k order central moment.
The k order moment, α k , and the k order central moment, β k , of the continuous random variable X satisfy the relationship expressed by Equation (14) [40]: where the zero-order origin moment α 0 = 1. Moreover, the coefficient C j k satisfies Equation (15): According to Equation (14), the k order central moment, β k , can be obtained by the k order moment, α k . The relationship between the central moment and the moment satisfies the relationship expressed by Equation (16):

Point Estimation Method
For a grid with n buses, P i represents the ith bus active power; with a mean of µ i standard deviation of σ i . In the point estimation method, for n − 1 input variables (P 1 , P 2 , · · · , P i , · · · , P n−1 ), each variable contains j points. From the first variable to the n − 1th variable, the POPF is calculated when only one point is considered for one variable at a time and the other variables are mean. For example, for the ith input variable, the POPF calculated at the jth estimation point is expressed by Equation (17) [41]: P Li,j = F µ 1 , µ 2 , · · · , P i,j , · · · , µ n−1 .
The relationship between P i and µ i , σ i satisfies the relationship expressed by Equation (18). In Equation (18), η i,j represents the standard central moments, which are defined as expressed by Equation (19): In Equation (19), λ is calculated by the ratio of the central moment, M(P i ), to the standard deviation, σ i , which is defined as expressed by Equation (20).
where M(P i ) represents the central moment, which is defined as expressed by Equation (13). After calculating each point in turn, the grid line power, P L , and the line power, P Li,j , calculated by each point estimation satisfy the relationship expressed by Equation (21).
where w i,j represents the weight coefficient, which is defined as expressed by Equation (22).

Gram-Charlier Expansion
Based on the Gram-Charlier expansion, the cumulative distribution function (CDF) of a continuous random variable, X, is expressed by Equation (23), and the probability distribution function (PDF) is expressed by Equation (24) [42].
where Φ(x) and ϕ(x) are the CDF and the PDF of the standard normal distribution, respectively; and ξ i is a constant coefficient as shown in Equation (25).

Algorithm Calculation Steps
The steps for calculating the OIPOPF using the point estimation method based on the Gram-Charlier expansion are as follows: (1) The standard central moment, λ, of the probability variable is calculated according to Equation (20).
(2) The standard location, η, of the probability variable is calculated according to Equation (19).
A flow chart for the calculation of the OIPOPF based on the Gram-Charlier expansion point estimation method is presented in Figure 1.
The steps for calculating the OIPOPF using the point estimation method based on the Gram-Charlier expansion are as follows: (1) The standard central moment,  , of the probability variable is calculated according to Equation (20).
(2) The standard location,  , of the probability variable is calculated according to Equation (19).
(3) The weight, w , of the probability variable is calculated according to Equation (22). (4) The estimated point, , i j P , is calculated according to Equation (18).
A flow chart for the calculation of the OIPOPF based on the Gram-Charlier expansion point estimation method is presented in Figure 1. Figure 1. Flow chart for the calculation of the once-iterative probabilistic optimal power flow (OIPOPF) based on the Gram-Charlier expansion point estimation method. CDF-cumulative distribution function, PDF-probability distribution function.

Results
In this study, the OIPOPF mathematical model of PSHPRPSs was applied to the modified IEEE 39-bus test system and the modified IEEE 300-bus test system for verification. In this section, a comparison of the OIPOPF calculation results with those of the DA power flow model and AC power flow model is presented, to verify the accuracy of the OIPOPF mathematical model. The OIPOPF model proposed in this paper is solved by the point estimation method based on the Gram-Charlier expansion. The results were compared with those obtained using the Monte Carlo method. The simulations were conducted using MATLAB 8.3.0.532 (R2014a, TheMathWorks, Inc, Natick, MA, USA) on a personal computer with an i3-4170 CPU, 3.70 GHz processor, and 4.0 GB of random-access memory (RAM).

Modified IEEE 39-Bus Test System
The modified IEEE 39-bus test system diagram is presented in Figure 2. As shown in Figure 2, the modified IEEE 39-bus test system adds uncertain renewable energy sources such as wind and photovoltaic power.

Results
In this study, the OIPOPF mathematical model of PSHPRPSs was applied to the modified IEEE 39-bus test system and the modified IEEE 300-bus test system for verification. In this section, a comparison of the OIPOPF calculation results with those of the DA power flow model and AC power flow model is presented, to verify the accuracy of the OIPOPF mathematical model. The OIPOPF model proposed in this paper is solved by the point estimation method based on the Gram-Charlier expansion. The results were compared with those obtained using the Monte Carlo method. The simulations were conducted using MATLAB 8.3.0.532 (R2014a, TheMathWorks, Inc, Natick, MA, USA) on a personal computer with an i3-4170 CPU, 3.70 GHz processor, and 4.0 GB of random-access memory (RAM).

Modified Power Test System Raw Data
The modified IEEE 39-bus test system diagram is presented in Figure 2. As shown in Figure 2, the modified IEEE 39-bus test system adds uncertain renewable energy sources such as wind and photovoltaic power. The IEEE 39 bus test power system contains 10 power sources. In the modified test system, four of the power sources were replaced by uncertain renewable energy sources. The numerical The IEEE 39 bus test power system contains 10 power sources. In the modified test system, four of the power sources were replaced by uncertain renewable energy sources. The numerical characteristics such as the mean and standard deviation of each uncertain renewable energy sources prediction error are shown in Table 1. The IEEE 300 bus test power system contains 69 power sources. In the modified test system, 28 power sources replaced were by uncertain renewable energy sources. The numerical characteristics such as the mean and standard deviation of each uncertain renewable energy source prediction error are shown in Table 2.

Comparison of Simulation Results with DC Power Flow Model Results
The day-ahead dispatching of PSHPRPSs considers the distribution of active power in the power grid. In this study, the once-iteration power flow model, DC power flow model, and AC power flow model were used to determine the active power of power lines in the power grid, and the results were compared to verify the accuracy of the OIPOPF. In order to verify the accuracy of the calculation results of the power flow model proposed in this paper, the calculation results of the OIPOPF model and the DC power flow model are compared with those of the AC power flow model. The relative error of the simulation results of the two models relative to the AC power flow model is defined by Equation (26).
where error represents the percentage error of the simulation results; P AC L represents the simulation results of the AC power flow model; P 1 L represents the simulation results of the OIPOPF model or DC power flow model. Based on the percentage error, the average error was calculated. The average error error is defined as expressed by Equation (27), and N L represents the total number of percentage errors.
According to Table 3, the average relative error of the AC power flow and DC power flow calculation result for the modified IEEE 39-bus test system is 4.45%; the average relative error of the AC power flow and the OIPOPF model calculation result for the modified IEEE 39-bus test system is 0.64%. The average relative error of the AC power flow and DC power flow calculation result for the modified IEEE 300-bus test system is 11.91%; the average relative error of the AC power flow and the OIPOPF model calculation result for the modified IEEE 300-bus test system is 0.90%. Regardless of whether the modified IEEE 39-bus test system or the modified IEEE 300-bus test system was used, the simulation results show that the results of the optimal power flow model proposed in this paper are more accurate than those of the DC power flow model, and similar to those of the AC power flow model. Furthermore, in the PSHPRPSs, the uncertain power source accounts for a relatively high proportion, and its power prediction error is generally 25-40% [43]. The relative average error of the OIPOPF and AC power flow models for the modified IEEE 39-bus test system and the modified IEEE 300-bus test system are 0.64% and 0.90%, respectively, which is significantly less than the power prediction error of PSHPRPSs. Therefore, the simulation results prove the accuracy of the OIPOPF model proposed in this paper. In order to verify that the probabilistic power flow model proposed in this paper can improve computing efficiency, the running time of the OIPOPF mathematical model, the DC power flow model and the AC power flow model are compared. The comparison results are shown in Table 4. As shown in Table 4, regardless of whether the modified IEEE 39-bus test system or the modified IEEE 300-bus test system was used, the operation time of the OIPOPF model is similar to that of the DC power flow, and shorter than that of the AC power flow. Therefore, the simulation results show that the proposed model can maintain high operation efficiency.
To sum up, using the OIPOPF mathematical model to set the day-ahead dispatching plan of PSHPRPSs can not only maintain high calculation accuracy but also have high operation efficiency.

Simulation Results of OIPOPF Model
In the modified IEEE 39-bus test system and the modified IEEE 300-bus test system, point estimation methods based on Gram-Charlier expansion and Monte Carlo simulation method were employed for the calculation of the OIPOPF model proposed in this paper. The Monte Carlo simulation method was found to be more accurate, and it is generally used to verify the accuracy of the simulation results of other models or algorithms. In this study, the simulation results of the two abovementioned methods were compared to verify that the point estimation method based on Gram-Charlier expansion can be used to solve the proposed OIPOPF model. The mean and standard deviation of the line power calculated using these two simulation methods are shown in Table A3 (Appendix B).
The simulation results of the point estimation method based on Gram-Charlier expansion are represented by A, and the simulation results of the Monte Carlo simulation method are represented by A mcs . The relative error, ε, of the simulation result is expressed by Equation (28): The relative error, ε, can represent the mean relative error, ε µ , or the standard deviation relative error, ε σ .
To verify the accuracy of the point estimation method based on Gram-Charlier expansion, the cumulative distribution average root mean square error (CDARMSE) was defined as expressed by Equation (29).
where N represents the number of point estimates; A i represents the cumulative distribution value obtained using the point estimation method based on Gram-Charlier expansion for the ith point estimation; and A mcs represents the cumulative distribution value obtained using the Monte Carlo simulation method for the ith point estimation. Table A4 (Appendix B) presents the relative errors and CDARMSE values of the line power for the modified IEEE 39-bus test system calculated using the Gram-Charlier point estimation and Monte Carlo simulation methods. As seen in Table A4, the CDARMSE values of the two methods were generally less than 0.001. Hence, the point estimation method based on Gram-Charlier expansion can be used to solve the proposed OIPOPF model. Table A4 indicates that the CDARMSE of the line power between busses 31 and 6 was the highest. The PDF and CDF of the line power between busses 31 and 6, as obtained using the two abovementioned methods, are shown in Figures 3 and 4. As shown in Figures 3 and 4   This paper uses 2m + 1 point estimation method based on Gram-Charlier expansion to calculate the OIPOPF mathematical model proposed in this paper. In order to obtain the calculation result, 2m + 1 simulation calculations are required, where m is the number of indeterminate variables. The modified IEEE 39-bus test system has four indeterminate variables, and the modified IEEE 300-bus test system has 28 indeterminate variables.
According to Table 5, the number of simulation calculations of different sizes of test systems is related to the number of uncertain variables. Table 5. Number of once-iterative probabilistic optimal power flow OIPOPF mathematical model simulation calculations.

Scale of Test Power System Variables PECCS Simulation Times MC Simulation Times
Operation time of IEEE 39-bus test system 4 9 1000 Operation time of IEEE 300-bus test system 28 57 1000 This paper compares the simulation results of the point estimation method based on Gram-Charlier expansion and Monte Carlo simulation method in the modified IEEE 300-bus test system to verify that the point estimation method based on Gram-Charlier expansion can calculate the OIPOPF model in large test power systems. In the simulation results of the modified IEEE 300-bus test system, the CDARMSE of the line power between busses 123 and 124 was the highest. The PDF and CDF of the line power between busses 123 and 124, as obtained using the two abovementioned methods, are shown in Figures 5 and 6. As shown in Figures 5 and 6, the simulation results of the two methods were very similar. Therefore, it is verified that the point estimation method based on Gram-Charlier expansion can calculate the OIPOPF model in large test power system.

Discussion
Because the PSHPRPSs contain a high proportion of uncertain renewable energy sources, the prediction error of the day-ahead power is large, and the day-ahead dispatching plan cannot be accurate. Therefore, using the probabilistic AC power flow model not only cannot effectively improve the calculation accuracy, but will also obviously reduce the calculation efficiency. The use of probabilistic DC power flow model can improve the calculation efficiency but reduce the calculation accuracy. In order to verify that the probabilistic power flow model proposed in this paper can not only maintain high calculation accuracy but also maintain high calculation efficiency, this section compares the OIPOPF calculation results with those of the DC power flow model and AC power flow model.
According to the simulation results in Section 3.2, regardless of whether the modified IEEE 39-bus test system or the modified IEEE 300-bus test system was used, the calculation accuracy of OIPOPF model is similar to that of AC power flow model, and higher than that of DC power flow model; the calculation efficiency of OIPOPF model is similar to that of DC power flow model, and higher than that of AC power flow model. To sum up, the OIPOPF model proposed in this paper not only maintains high calculation accuracy, but also maintains high calculation efficiency when setting the day-ahead dispatching plan for the PSHPRPSs.
In this paper, the point estimation method based on Gram-Charlier expansion is used to solve the OIPOPF mathematical model. In order to verify that the point estimation method based on Gram-Charlier expansion can solve the mathematical model of probabilistic optimal power flow proposed in this paper, the simulation results of point estimation method based on Gram-Charlier expansion and Monte Carlo simulation method are compared. According to the simulation results in Section 3.3, regardless of whether the modified IEEE 39-bus small test system or the modified IEEE 300-bus large test system was used, the CDARMSE of the two calculation methods are generally less than 0.001. The figures were drawn by selecting the line power corresponding to the maximum error of the simulation results in the modified IEEE 39-bus small test system and the modified IEEE 300-bus large test system. As shown in the figures in Section 3.3, the simulation results of point estimation method based on Gram-Charlier expansion are similar to those of Monte Carlo simulation method. In summary, by comparing the simulation results of the two algorithms, it is verified that the point estimation method based on Gram-Charlier expansion can accurately solve the OIPOPF mathematical model.
Based on the comparison of the simulation results, the accuracy and efficiency of the proposed OIPOPF model of PSHPRPSs was verified. Moreover, based on the comparison of the calculation results obtained using the two methods, it was included that the point estimation method based on Gram-Charlier expansion can accurately calculate the proposed OIPOPF model.

Conclusions
Given that PSHPRPSs contain a large number of uncertain power sources, the day-ahead power prediction error is significantly high. During the dispatch cycle, it is difficult for the day-ahead generation plan to meet the actual active power demand of the power grid. To maintain the power balance of the power grid, the day-ahead power prediction error is accommodated by real-time dispatching and AGC. Due to the significant uncertainty of the power generated by the uncertain power sources, the conventional power flow calculation is not suitable for the SSA of PSHPRPSs. Based on the above reasons, this paper proposes the OIPOPF mathematical model as the power flow model for the SSA of PSHPRPSs. In this paper, the OIPOPF and AC power flow models were applied to a modified IEEE 39-bus test system and modified IEEE 300-bus test system. The simulation results verify that the OIPOPF model can not only ensure the accuracy of power flow results, but also simplify the power flow model. The OIPOPF mathematical model was solved using the point estimation method based on the Gram-Charlier expansion. Compared with the Monte Carlo simulation method, the results indicate that the point estimation method based on Gram-Charlier expansion can be employed to accurately calculate the proposed OIPOPF model. Furthermore, the simulation results verify the accuracy of the OIPOPF model and algorithm.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix B
Appendix B presents the simulation results of the point estimation method for the modified test power system (see Section 3.3 for details). Table A3 presents the numerical characteristics of the line power simulation  results for the modified IEEE 39-bus test system. Table A4 presents the relative error and CDARMSE of the line power for the modified IEEE 39-bus test system.