Probabilistic optimal flow calculation method for electric-traffic combined system with photovoltaic pavement

Abstract

The method comprises the following steps: 1) establishing a probability model of multiple uncertain factors; 2) based on the probability model of multiple uncertain factors, establishing an electric-traffic coupling probability joint flow model of the photovoltaic road surface grid connection, and quantifying the influence of the multiple uncertain factors on the power distribution network and the traffic network; 3) based on the polynomial chaos expansion, solving the electric-traffic coupling probability joint flow model of the photovoltaic road surface grid connection, and obtaining the probability density function and statistical information of the influence. The electric-traffic coupling probability joint flow model is suitable for power and traffic system analysis, and provides decision support for the reasonable and safe operation of the power distribution network and the traffic network.

Description

Technical Field

[0001] This invention relates to the field of modeling and solving probabilistic joint flows of coupled power systems and transportation systems, and specifically to a method for calculating the probabilistic optimal flow of a power-transportation joint system with photovoltaic pavement. Background Technology

[0002] Due to increasing environmental degradation and the shortage of traditional fossil fuels, energy transition is imperative. In recent years, renewable energy, as a substitute for fossil fuels, has shown great potential and has developed rapidly. Especially with the maturation of photovoltaic materials and the development of road transportation, photovoltaic road power generation technology is becoming increasingly sophisticated. It is roughly estimated that if photovoltaic roads were laid on all highways in my country, the annual power generation would exceed 999.2 TWh. With the large-scale promotion and application of photovoltaic pavements and electric vehicles, the effective connection between the power grid's clean power production and the transportation network's low-carbon energy consumption from the source and load sides makes the coupled operation characteristics of the transportation network and the power grid increasingly significant. Therefore, research on photovoltaic pavement-transportation coupled systems can greatly reduce reliance on traditional fossil fuels, promote supply-side energy transition, facilitate the construction of green and intelligent transportation systems, and achieve the goals of carbon peaking and carbon neutrality. However, affected by multiple uncertainties in the transportation network and the power grid, the operating states of the coupled system increase, which may threaten the healthy operation of the power grid and the transportation network.

[0003] Against this backdrop, quantitatively analyzing the impact of multiple uncertainties on coupled systems is of great significance for improving the reliability of transportation network operation and the safety of distribution network operation. Existing research has considered the impact of uncertainties such as solar radiation, temperature, traffic flow, and vehicle dynamic shading on the electric-traffic coupled system. Some scholars have established photovoltaic pavement power generation models considering electric-traffic coupling and used traffic assignment models to quantify the impact of vehicle dynamic shading on photovoltaic pavement power generation. However, this model calculates the vehicle dynamic shading area inaccurately, assumes a fixed traffic flow, and only analyzes the impact of uncertainties on the output power of the photovoltaic pavement, without analyzing the impact of photovoltaic pavement grid connection on the distribution network. Other scholars have considered electric-traffic coupling models involving solar radiation, ambient temperature, vehicle dynamic shading, and mixed traffic flow, and then accurately measured vehicle dynamic shading based on solar geometry and the shoelace theorem. Furthermore, they have analyzed the impact of photovoltaic pavement grid connection on the fluctuations of distribution network node voltage and branch power flow. However, this also has many limitations, ignoring the strong correlation between multiple uncertainties and failing to analyze the impact of the distribution network on the traffic side. In summary, existing research tends to utilize the adjustable capacity of transportation networks to assist the safe and efficient operation of distribution networks and improve the consumption level of renewable energy, while neglecting the operational needs of the transportation side. Therefore, as two distinct entities, distribution networks and transportation networks should complement and coexist in a collaborative manner. Electricity-transportation coupling has a positive impact on building green transportation, promoting renewable energy consumption, and alleviating the burden on low-voltage and transmission lines. However, due to the multiple uncertainties in electricity-transportation coupled systems, the safe operation of both distribution and transportation networks faces severe challenges.

[0004] Electricity-traffic probabilistic joint flow models require simultaneous consideration of optimization and uncertainty analysis, increasing the complexity and difficulty of solving them. Existing solution methods such as Monte Carlo simulation and approximation methods cannot meet current needs. Polynomial chaotic expansion is a relatively advanced solution method, but its solution speed and accuracy are highly dependent on the sample quality in the experimental design. Therefore, selecting a reasonable sampling method is a challenge that needs to be addressed when applying this method. Summary of the Invention

[0005] The purpose of this invention is to provide a probabilistic optimal flow calculation method for an electric-transportation combined system with photovoltaic pavement, comprising the following steps:

[0006] 1) Establish a probability model for multiple uncertain factors.

[0007] 2) Based on the probability model of multiple uncertain factors, establish the power-traffic coupling probability joint flow model of photovoltaic road grid connection to quantitatively characterize the impact of multiple uncertain factors on the distribution network and transportation network.

[0008] The objective function of the electric-traffic coupled probabilistic joint flow model for the photovoltaic pavement grid connection is shown below:

[0009] minF P-T =F PDN +F T (1)

[0010] In the formula, minF P-T The objective function is the electric-traffic coupled probabilistic joint flow model for photovoltaic pavement grid connection.

[0011] Among them, the objective function F of the distribution network model PDN As shown below:

[0012]

[0013] In the formula, i is the starting node of the distribution network link l. i b i c i All are coefficients of a quadratic function. ρ1 represents the contract energy price with the main power grid. Let E be the active power of the generator at node i at time t. N P represents the set of nodes in the distribution network. main This indicates the amount of electricity purchased from the mainnet.

[0014] The objective function F of the traffic assignment model T As shown below:

[0015]

[0016] In the formula, 'a' represents a road segment. It is freedom through time. a Let c be the traffic flow on road segment a. a It refers to the carrying capacity of road segment a. Let J be the set of regular arcs. Let J be the objective function F. T Shape parameters. E is a collection of charging arcs. B Rated capacity of electric vehicle batteries. This indicates the electricity price at which the charging pile on road segment a is connected to node j of the distribution network.

[0017] The constraints of the electric-traffic coupled probabilistic joint flow model for photovoltaic road grid connection include conventional arc constraints, charging arc constraints, bypass arc constraints, traffic flow constraints, distribution network model constraints, and coupling constraints between the traffic network and the distribution network.

[0018] 3) Based on polynomial chaotic expansion, the electric-traffic coupling probabilistic joint flow model of photovoltaic road grid connection is solved to obtain the probability density function and statistical information of the impact.

[0019] Furthermore, the probability models for the multiple uncertainties include probability models for solar irradiance, temperature, vehicle dynamic shading, nonlinear correlation, and photovoltaic pavement power generation.

[0020] Furthermore, the probabilistic model for the solar irradiance is as follows:

[0021]

[0022] In the formula, x is the solar irradiance value. f(x; α, β) is the probability of solar irradiance. Γ represents the gamma function. α and β represent the two parameters of the Beta function, respectively.

[0023] The probability model for the temperature is shown below:

[0024]

[0025] In the formula, μ represents the mean of the temperature sample. σ represents the standard deviation of the temperature sample. x0 is the temperature. f(x0) is the probability of the temperature.

[0026] The probability model for the vehicle's dynamic shadow is shown below:

[0027]

[0028] In the formula, A d,t This represents the area of ​​the shadow cast on the road by the vehicle due to sunlight at time t, when the driving direction is d. and Let x represent the coordinates of vertex i1 of the vehicle's shadow outline at time t, when the driving direction is d. d,t,n+1 =x1 and y d,t,n+1 =y1. n is the total number of vertices of the vehicle's shadow outline.

[0029] The probabilistic model for the nonlinear correlation is shown below:

[0030]

[0031] In the formula, s is the temperature variable. t0 is the solar irradiance variable. C(u1,u2) is the probability of the nonlinear correlation between the two univariate marginal distribution functions, C(u1,u2)=C(F1(x k ),F2(x k+1 C is the join function. F1(x) k ), F2(x k+1 All are univariate marginal distribution functions. k = 1, 2, ..., N-1, where N is the total number of univariate marginal distribution functions. ρ is the correlation coefficient between the variables. Parameter u1 = F1(x k). Parameter u2 = F2(x k+1 ). φ -1 (u1), φ -1 (u2) represents the inverse function of the standard normal distribution of parameters u1 and u2, respectively.

[0032] The probability model for the photovoltaic pavement power generation is shown below:

[0033]

[0034] V s,t =V OC -K v T C,t (9)

[0035] I s,t =s t [I SC +K i (T C,t -25)] (10)

[0036]

[0037] P s,t =N PV V s,t I s,t FF (12)

[0038]

[0039]

[0040] In the formula, T C,t T A,t T OT These represent the battery temperature, ambient temperature, and normal operating temperature of the photovoltaic pavement power generation system, respectively. V s,t I s,t These represent the output voltage and current under solar irradiance S, respectively. OC I SC These represent the open-circuit voltage and short-circuit current, respectively. K v K i These represent the temperature coefficients of voltage and current, respectively. FF represents the fill factor. V MPP I MPP These represent the voltage and current at the maximum power point, respectively. N PV Indicates the number of modules. P s,t This indicates the output power of the photovoltaic module. P road,t This indicates the output power of the photovoltaic pavement. N veh,t Indicates the number of motor vehicles. A veh,t This indicates the shadow area projected onto the photovoltaic road surface by the vehicle. Aroad d represents the total area of ​​the photovoltaic pavement. L represents the length of the photovoltaic pavement. t This indicates the density of traffic flow. Q t V t These represent traffic flow and average speed, respectively. t This represents the solar irradiance at time t.

[0041] Furthermore, the conventional arc constraint is as follows:

[0042]

[0043] In the formula, 'a' represents road segment 'x'. a For traffic flow on road segment a, It is freedom through time. a It refers to the carrying capacity of road segment a. Let t be the set of regular arcs. a1 (x a () represents the time for a regular arc to pass through.

[0044] The charging arc constraint is as follows:

[0045]

[0046] In the formula, t a2 (x a ) represents the duration of the charging arc. J is a parameter that determines the shape of the function. It is a set of charging arcs.

[0047] The bypass arc constraint is as follows:

[0048]

[0049] In the formula, t a3 (x a () represents the time for the bypass arc to travel. It is the set of bypass arcs.

[0050] The traffic flow constraints are as follows:

[0051]

[0052]

[0053]

[0054]

[0055]

[0056] In the formula, k represents a feasible route, and od represents the destination. These represent the travel demand from the origin to the destination for both electric vehicles and gasoline-powered vehicles. These are the sets of feasible paths for gasoline-powered cars and electric cars, respectively, from the starting point to the destination. Let $\mathbf$ be the flow rate of a gasoline-powered car and an electric car on the kth feasible route between the starting point and the destination, respectively. These are the path-arc correlation matrices for electric vehicles and gasoline vehicles, respectively. When the feasible path k... e When arc a is included, δ E =1. When feasible path k e When arc a is not included, δ E =0. When feasible path k g When arc a is included, δ G =1. When feasible path k g When arc a is not included, δ G =0. T A It is a set of arcs. These represent the travel times for a gasoline-powered car and an electric car, respectively, on the k-th route between the starting point and the destination. These represent the lowest costs for gasoline-powered vehicles and electric vehicles, respectively.

[0057] The constraints of the power distribution network model are as follows:

[0058]

[0059]

[0060]

[0061]

[0062]

[0063]

[0064]

[0065]

[0066] In the formula, i and j are the starting node and ending node of link l, respectively. Let be the active power of the generator at node j at time t. Let E be the reactive power of the generator at node j at time t. N Let be the set of nodes in the distribution network. π(j) represents the set of child nodes of node j, and k represents the set of child nodes of node j. E l This is a collection of distribution network branches. This represents the active power flow from node i to node j and from node j to node k. This represents the reactive power flow from node i to node j and from node j to node k. Let represent the minimum and maximum active power of the generator at node j at time t, respectively. Let U represent the minimum and maximum reactive power of the generator at node j at time t, respectively. i,t U j,t These represent the voltage values ​​at node i and node j at time t, respectively. These represent the active and reactive power requirements of node j at time t, respectively. These represent the minimum and maximum voltage values ​​at node j, respectively. This represents the branch current value from node i to node j. This represents the maximum value of the branch current l from node i to node j. The resistance of branch l between node i and node j. P SR To contribute to photovoltaic pavement. The resistance of branch l between node i and node j. The impedance of branch l between node i and node j.

[0067] The coupling constraints between the transportation network and the power distribution network are as follows:

[0068]

[0069]

[0070] In the formula, This represents traditional electrical loads. This indicates the charging load of electric vehicles. This represents the electric vehicle traffic through charging link a. E B Rated capacity of electric vehicle batteries.

[0071] Furthermore, the impact includes:

[0072] 1) Overvoltage at nodes and power flow exceeding limits in the distribution network.

[0073] 2) Traffic congestion caused by routes exceeding capacity in the transportation network.

[0074] Furthermore, the polynomial chaos solution method includes a coherent D-optimal polynomial chaos solution method.

[0075] Furthermore, the solution steps for the coherent D-optimal polynomial chaos are as follows:

[0076] 3.1) Calculate the minimum coherence parameter, and select the sample set that minimizes the coherence parameter μ from the random variable ξ using the coherence optimal sampling method.

[0077] The minimum coherence parameter is as follows:

[0078]

[0079] In the formula, μ(Y) is the minimum coherence parameter. sup is the supremum function. P is the maximum order of the polynomial expansion. j1 is the order of the multivariate orthogonal polynomial expansion. Let represent the j1-th order multivariate orthogonal polynomial corresponding to the multivariate variables.

[0080] The weighting function w(ξ) is shown below:

[0081]

[0082] Wherein, polynomial φ j1 The uniform least upper bound B of the sum of squares of (ξ) 2 (ξ) is shown below:

[0083]

[0084] 3.2) Calculate the information moments that maximize the formation, and select high-quality samples from the sample set with the smallest coherence parameter μ using the D-optimal sampling method.

[0085] The information moments are as follows:

[0086]

[0087] In the formula, M represents the information moment. N represents the total number of high-quality samples. Φ represents the set of multivariate orthogonal polynomials.

[0088] The determinant for maximizing the information moments is shown below:

[0089] φ D =max|M| 1 / P (37)

[0090] In the formula, φ D This represents the maximum determinant. `max` is the maximum value function.

[0091] 3.3) Expand the probability density function of high-quality samples into a multivariate orthogonal polynomial, as shown below:

[0092]

[0093] In the formula, f(ξ) represents the high-quality sample ξ = [ξ1, ξ2, ξ3, ..., ξ]. NThe corresponding probability density function is given by [ ], where N is the total number of high-quality samples. Y is the probability density function of the corresponding high-quality samples. k is the order of the multivariate orthogonal polynomial expanded from the response probability density function Y. k These are the coefficients of the polynomial chaotic expansion.

[0094] Among them, the multivariate orthogonal polynomial φ (k) (ξ) is shown below:

[0095]

[0096] In the formula, i2 is the order of the multivariate orthogonal polynomial expansion. Indicates the kth order of the i2th order. i2 An orthogonal polynomial.

[0097] 3.4) Based on the minimum coherence parameter and minimum information moment, the coefficients c of the polynomial chaos are expanded using the least squares method. k Solving for the polynomial chaotic expansion coefficients c yields the results. k The least squares solution.

[0098] 3.4.1) Calculate the residuals of a set of sample response pairs [ξ,Y] as follows:

[0099]

[0100] In the formula, Y is the set of probability density functions of random variables. ε(C) represents the residual of the sample response pair [ξ,Y]. For variables i The probability density function of the random variable corresponding to 2. Let be a multivariate orthogonal polynomial. Φ is the set of multivariate orthogonal polynomials. C is the set of expansion coefficients.

[0101] The number of terms K in the polynomial expansion of the response probability density function Y is shown below:

[0102]

[0103] In the formula, P is the maximum order of the polynomial expansion.

[0104] 3.4.2) Find the partial derivatives of formula (40) and solve them by the least squares method to obtain the polynomial chaotic expansion coefficients. c k The least squares solution.

[0105] The least squares solution is shown below:

[0106] C=(Φ T Φ) -1 Φ T Y (42).

[0107] The technical effectiveness of this invention is undeniable. This invention models the probabilistic joint flow of photovoltaic road grid-connected power-traffic coupling and solves it efficiently based on an improved polynomial chaotic expansion method. It evaluates the impact of multiple uncertainties in the coupled system on the distribution network and transportation network. This invention first considers the influence of the correlation between light intensity and ambient temperature, making the photovoltaic pavement output calculation more accurate. Then, it establishes a probabilistic joint flow model of power-traffic coupling based on the photovoltaic pavement output model, traffic assignment model, and Distflow's optimal power flow model. Finally, it uses an improved polynomial chaotic expansion method to quantitatively evaluate the impact of multiple uncertainties (traffic volume, light intensity, etc.) of the coupled system on the distribution network and transportation network system.

[0108] The proposed electric-traffic coupled probabilistic joint flow model is applicable to the analysis of power and transportation systems, providing decision support for the rational and safe operation of distribution networks and transportation networks. The correlation between variables has a significant impact on the power output of photovoltaic pavements. A photovoltaic pavement power output model considering the correlation between temperature and irradiance can accurately estimate the actual power generation of photovoltaic pavements, which is beneficial for the operation and scheduling of distribution networks. Furthermore, due to uncertainties in irradiance, ambient temperature, traffic demand between origin and destination, and vehicle dynamic shading, overvoltage at certain nodes and branch power flow exceeding limits may occur in the distribution network, and some paths in the transportation network may exceed their capacity, leading to traffic congestion. However, the electric-traffic coupled probabilistic joint flow model can quantitatively characterize the probabilistic features of the coupled operating system, providing more comprehensive information for the joint analysis, operational status perception, and early warning of the coupled system. The improved polynomial chaotic expansion significantly reduces the computational burden while maintaining high fidelity. This method measures the degree of influence of uncertainties on the operation of the electric-traffic coupled system through a small amount of power flow assessment. Simulation results show that this method has good performance in both accuracy and efficiency. Attached Figure Description

[0109] Figure 1 For both original and extended transportation networks with charging stations;

[0110] Figure 2 The interaction between the power system and the transportation system;

[0111] Figure 3 This is a topology diagram of the transportation network;

[0112] Figure 4 This is a topology diagram of the power distribution network;

[0113] Figure 5 This is a quantile chart of voltage magnitude at 18 nodes during the week. Figure 5 (a) is the quantile map without considering correlation. Figure 5 (b) is a quantile plot considering correlation;

[0114] Figure 6 This is a power distribution diagram for the 24 branches during the week. Figure 6 (a) is the quantile map without considering correlation. Figure 6 (b) is a quantile plot considering correlation;

[0115] Figure 7 Box plot of traffic flow for a 12-node ring road network (excluding reverse roads);

[0116] Figure 8 Box plots showing voltage amplitude at coupling nodes in a distribution network under different conditions;

[0117] Figure 9 Traffic flow diagrams for charging stations under different conditions;

[0118] Figure 10 The probability density function of the power of the 32nd branch is compared using different methods;

[0119] Figure 11 The probability density function of the power of the 18th branch is compared using different methods;

[0120] Figure 12 The probability density function for comparing the voltage magnitude at node 19 using different methods;

[0121] Figure 13 The probability density function for comparing the voltage amplitude at node 33 using different methods;

[0122] Figure 14 The probability density function of the total cost of electric-transportation coupling is used to compare different methods. Detailed Implementation

[0123] The present invention will be further described below with reference to embodiments, but it should not be construed that the scope of the present invention is limited to the following embodiments. Various substitutions and modifications made based on ordinary technical knowledge and common practices in the art without departing from the above-described technical concept of the present invention should be included within the scope of protection of the present invention.

[0124] Example 1:

[0125] See Figures 1 to 12 A probabilistic optimal flow calculation method for an electric-transportation combined system with photovoltaic pavement includes the following steps:

[0126] 1) Establish a probability model for multiple uncertain factors.

[0127] 2) Based on the probability model of multiple uncertain factors, establish the power-traffic coupling probability joint flow model of photovoltaic road grid connection to quantitatively characterize the impact of multiple uncertain factors on the distribution network and transportation network.

[0128] The objective function of the electric-traffic coupled probabilistic joint flow model for the photovoltaic pavement grid connection is shown below:

[0129] minF P-T =F PDN +F T (1)

[0130] In the formula, minF P-T The objective function is the electric-traffic coupled probabilistic joint flow model for photovoltaic pavement grid connection.

[0131] Among them, the objective function F of the distribution network model PDN As shown below:

[0132]

[0133] In the formula, i is the starting node of the distribution network link l. i b i c i All are coefficients of a quadratic function. ρ1 represents the contract energy price with the main power grid. Let E be the active power of the generator at node i at time t. N P represents the set of nodes in the distribution network. main This indicates the amount of electricity purchased from the mainnet.

[0134] The objective function F of the traffic assignment model T As shown below:

[0135]

[0136] In the formula, 'a' represents a road segment. It is freedom through time. a Let c be the traffic flow on road segment a. a It refers to the carrying capacity of road segment a. Let J be the set of regular arcs. Let J be the objective function F. T Shape parameters. E is a collection of charging arcs. B Rated capacity of electric vehicle batteries. This indicates the electricity price at which the charging pile on road segment a is connected to node j of the distribution network.

[0137] The constraints of the electric-traffic coupled probabilistic joint flow model for photovoltaic road grid connection include conventional arc constraints, charging arc constraints, bypass arc constraints, traffic flow constraints, distribution network model constraints, and coupling constraints between the traffic network and the distribution network.

[0138] 3) Based on polynomial chaotic expansion, the electric-traffic coupling probabilistic joint flow model of photovoltaic road grid connection is solved to obtain the probability density function and statistical information of the impact.

[0139] Example 2:

[0140] The method for calculating the probabilistic optimal flow of an electric-transit combined system with photovoltaic pavement is described in Example 1. Furthermore, the probabilistic models of the multiple uncertainties include probabilistic models of solar irradiance, temperature, vehicle dynamic shading, nonlinear correlation, and photovoltaic pavement power generation.

[0141] Example 3:

[0142] The probabilistic optimal flow calculation method for an electric-transportation combined system with photovoltaic pavement is described in any one of Examples 1 and 2. Furthermore, the probabilistic model for solar irradiance is as follows:

[0143]

[0144] In the formula, x is the solar irradiance value. f(x; α, β) is the probability of solar irradiance. Γ represents the gamma function. α and β represent the two parameters of the Beta function, respectively.

[0145] The probability model for the temperature is shown below:

[0146]

[0147] In the formula, μ represents the mean of the temperature sample. σ represents the standard deviation of the temperature sample. x0 is the temperature. f(x0) is the probability of the temperature.

[0148] The probability model for the vehicle's dynamic shadow is shown below:

[0149]

[0150] In the formula, A d,t This represents the area of ​​the shadow cast on the road by the vehicle due to sunlight at time t, when the driving direction is d. and Let x represent the coordinates of vertex i1 of the vehicle's shadow outline at time t, when the driving direction is d. d,t,n+1 =x1 and y d,t,n+1 =y1. n is the total number of vertices of the vehicle's shadow outline.

[0151] The probabilistic model for the nonlinear correlation is shown below:

[0152]

[0153] In the formula, s is the temperature variable. t0 is the solar irradiance variable. C(u1,u2) is the probability of the nonlinear correlation between the two univariate marginal distribution functions, C(u1,u2)=C(F1(x k ),F2(x k+1 C is the join function. F1(x) k ), F2(x k+1 All are univariate marginal distribution functions. k = 1, 2, ..., N-1, where N is the total number of univariate marginal distribution functions. ρ is the correlation coefficient between the variables. Parameter u1 = F1(x k ). Parameter u2 = F2(x k+1 ). φ -1 (u1), φ -1 (u2) represents the inverse function of the standard normal distribution of parameters u1 and u2, respectively.

[0154] The probability model for the photovoltaic pavement power generation is shown below:

[0155]

[0156] V s,t =V OC -K v T C,t (9)

[0157] I s,t =s t [I SC +K i (T C,t -25)] (10)

[0158]

[0159] P s,t =N PV V s,t I s,t FF (12)

[0160]

[0161]

[0162] In the formula, T C,t T A,t , ... represent the battery temperature, ambient temperature, and normal operating temperature of the photovoltaic pavement power generation system, respectively. V s,t I s,t These represent the output voltage and current under solar irradiance S, respectively. OC I SC These represent the open-circuit voltage and short-circuit current, respectively. K v Ki These represent the temperature coefficients of voltage and current, respectively. FF represents the fill factor. V MPP I MPP These represent the voltage and current at the maximum power point, respectively. N PV Indicates the number of modules. P s,t This indicates the output power of the photovoltaic module. P road,t This indicates the output power of the photovoltaic pavement. N veh,t Indicates the number of motor vehicles. A veh,t This indicates the shadow area projected onto the photovoltaic road surface by the vehicle. A road d represents the total area of ​​the photovoltaic pavement. L represents the length of the photovoltaic pavement. t This indicates the density of traffic flow. Q t V t These represent traffic flow and average speed, respectively. t This represents the solar irradiance at time t.

[0163] Example 4:

[0164] The probabilistic optimal flow calculation method for an electric-transportation combined system with photovoltaic pavement is described in any one of Examples 1 to 3. Further, the conventional arc constraint is as follows:

[0165]

[0166] In the formula, 'a' represents road segment 'x'. a For traffic flow on road segment a, It is freedom through time. a It refers to the carrying capacity of road segment a. Let t be the set of regular arcs. a1 (x a () represents the time for a regular arc to pass through.

[0167] The charging arc constraint is as follows:

[0168]

[0169] In the formula, t a2 (x a ) represents the duration of the charging arc. J is a parameter that determines the shape of the function. It is a set of charging arcs.

[0170] The bypass arc constraint is as follows:

[0171]

[0172] In the formula, t a3 (x a () represents the time for the bypass arc to travel. It is the set of bypass arcs.

[0173] The traffic flow constraints are as follows:

[0174]

[0175]

[0176]

[0177]

[0178]

[0179] In the formula, k represents a feasible route, and od represents the destination. These represent the travel demand from the origin to the destination for both electric vehicles and gasoline-powered vehicles. These are the sets of feasible paths for gasoline-powered cars and electric cars, respectively, from the starting point to the destination. Let $\mathbf$ be the flow rate of a gasoline-powered car and an electric car on the kth feasible route between the starting point and the destination, respectively. These are the path-arc correlation matrices for electric vehicles and gasoline vehicles, respectively. When the feasible path k... e When arc a is included, δ E =1. When feasible path k e When arc a is not included, δ E =0. When feasible path k g When arc a is included, δ G =1. When feasible path k g When arc a is not included, δ G =0. T A It is a set of arcs. These represent the travel times for a gasoline-powered car and an electric car, respectively, on the k-th route between the starting point and the destination. These represent the lowest costs for gasoline-powered vehicles and electric vehicles, respectively.

[0180] The constraints of the power distribution network model are as follows:

[0181]

[0182]

[0183]

[0184]

[0185]

[0186]

[0187]

[0188]

[0189] In the formula, i and j are the starting node and ending node of link l, respectively. Let be the active power of the generator at node j at time t. Let E be the reactive power of the generator at node j at time t. N Let be the set of nodes in the distribution network. π(j) represents the set of child nodes of node j, and k represents the set of child nodes of node j. E l This is a collection of distribution network branches. This represents the active power flow from node i to node j and from node j to node k. This represents the reactive power flow from node i to node j and from node j to node k. Let represent the minimum and maximum active power of the generator at node j at time t, respectively. Let U represent the minimum and maximum reactive power of the generator at node j at time t, respectively. i,t U j,t These represent the voltage values ​​at node i and node j at time t, respectively. These represent the active and reactive power requirements of node j at time t, respectively. These represent the minimum and maximum voltage values ​​at node j, respectively. This represents the branch current value from node i to node j. This represents the maximum value of the branch current l from node i to node j. The resistance of branch l between node i and node j. P SR To contribute to photovoltaic pavement. The resistance of branch l between node i and node j. The impedance of branch l between node i and node j.

[0190] The coupling constraints between the transportation network and the power distribution network are as follows:

[0191]

[0192]

[0193] In the formula, This represents traditional electrical loads. This indicates the charging load of electric vehicles. This represents the electric vehicle traffic through charging link a. E B Rated capacity of electric vehicle batteries.

[0194] Example 5:

[0195] The probabilistic optimal flow calculation method for an electric-transit combined system with photovoltaic pavement, the main technical contents of which are described in any one of Examples 1 to 4, further, the influencing factors include:

[0196] 1) Overvoltage at nodes and power flow exceeding limits in the distribution network.

[0197] 2) Traffic congestion caused by routes exceeding capacity in the transportation network.

[0198] Example 6:

[0199] The probabilistic optimal flow calculation method for an electric-transit combined system with photovoltaic pavement is described in any one of Examples 1 to 5. Furthermore, the polynomial chaos solution method includes a coherent D-optimal polynomial chaos solution method.

[0200] Example 7:

[0201] The probabilistic optimal flow calculation method for an electric-transportation combined system with photovoltaic pavement, the main technical contents of which are described in any one of Examples 1 to 6, further, the solution steps for the coherent D-optimal polynomial chaos are as follows:

[0202] 3.1) Calculate the minimum coherence parameter, and select the sample set that minimizes the coherence parameter μ from the random variable ξ using the coherence optimal sampling method.

[0203] The minimum coherence parameter is as follows:

[0204]

[0205] In the formula, μ(Y) is the minimum coherence parameter. sup is the supremum function. P is the maximum order of the polynomial expansion. j1 is the order of the multivariate orthogonal polynomial expansion. Let represent the j1-th order multivariate orthogonal polynomial corresponding to the multivariate variables.

[0206] The weighting function w(ξ) is shown below:

[0207]

[0208] Among them, polynomial The uniform least upper bound B of the sum of squares 2 (ξ) is shown below:

[0209]

[0210] 3.2) Calculate the information moments that maximize the formation, and select high-quality samples from the sample set with the smallest coherence parameter μ using the D-optimal sampling method.

[0211] The information moments are as follows:

[0212]

[0213] In the formula, M represents the information moment. N represents the total number of high-quality samples. Φ represents the set of multivariate orthogonal polynomials.

[0214] The determinant for maximizing the information moments is shown below:

[0215] φ D =max|M| 1 / P (37)

[0216] In the formula, φ D This represents the maximum determinant. `max` is the maximum value function.

[0217] 3.3) Expand the probability density function of high-quality samples into a multivariate orthogonal polynomial, as shown below:

[0218]

[0219] In the formula, f(ξ) represents the high-quality sample ξ = [ξ1, ξ2, ξ3, ..., ξ]. N The corresponding probability density function is given by [ ], where N is the total number of high-quality samples. Y is the probability density function of the corresponding high-quality samples. k is the order of the multivariate orthogonal polynomial expanded from the response probability density function Y. k These are the coefficients of the polynomial chaotic expansion.

[0220] Among them, the multivariate orthogonal polynomial φ (k) (ξ) is shown below:

[0221]

[0222] In the formula, i2 is the order of the multivariate orthogonal polynomial expansion. Indicates the i2th order An orthogonal polynomial.

[0223] 3.4) Based on the minimum coherence parameter and minimum information moment, the coefficients c of the polynomial chaos are expanded using the least squares method. k Solving for the polynomial chaotic expansion coefficients c yields the results. k The least squares solution.

[0224] 3.4.1) Calculate the residuals of a set of sample response pairs [ξ,Y] as follows:

[0225]

[0226] In the formula, Y is the set of probability density functions of random variables. ε(C) represents the residual of the sample response pair [ξ,Y]. Let i2 be the probability density function of the random variable corresponding to i2. Let be a multivariate orthogonal polynomial. Φ is the set of multivariate orthogonal polynomials. C is the set of expansion coefficients.

[0227] The number of terms K in the polynomial expansion of the response probability density function Y is shown below:

[0228]

[0229] In the formula, P is the maximum order of the polynomial expansion.

[0230] 3.4.2) Find the partial derivative of formula (40) and solve it by least squares to obtain the polynomial chaotic expansion coefficients c. k The least squares solution.

[0231] The least squares solution is shown below:

[0232] C=(Φ T Φ) -1 Φ T Y (42).

[0233] Example 8:

[0234] See Figures 1 to 12 A probabilistic optimal flow calculation method for an electric-transportation combined system with photovoltaic pavement includes the following steps:

[0235] 1) Establish a probability model for multiple uncertain factors.

[0236] 2) Based on the probability model of multiple uncertain factors, establish the power-traffic coupling probability joint flow model of photovoltaic road grid connection to quantitatively characterize the impact of multiple uncertain factors on the distribution network and transportation network.

[0237] From the appendix Figure 2 As shown in the diagram, the photovoltaic pavement and electric vehicles couple to the distribution network and transportation network from the source-load side, respectively, which exacerbates the interaction within the coupled system. The objective function of the electric-traffic coupling probabilistic joint flow model for the photovoltaic pavement grid connection is as follows:

[0238] minF P-T =F PDN +F T (1)

[0239] In the formula, minF P-T The objective function is the electric-traffic coupled probabilistic joint flow model for photovoltaic pavement grid connection.

[0240] The goal of the optimal power flow model is to track the minimum generation cost by adjusting control variables based on the power flow equations and other security constraints. Currently, the ideal optimal power flow model uses the Distflow model to implement non-convex quadratic constraint programming. This reconstructs the optimal power flow problem for grid-connected photovoltaic pavements and fast charging piles into a nonlinear optimization problem, with the objective function F of the distribution network model being... PDN As shown below:

[0241]

[0242] In the formula, i is the starting node of the distribution network link l. i b i c i All are coefficients of a quadratic function. ρ1 represents the contract energy price with the main power grid. Let E be the active power of the generator at node i at time t. N P represents the set of nodes in the distribution network. main This indicates the amount of electricity purchased from the mainnet.

[0243] With the rise of electric vehicles and fast charging stations, traffic operations have become more complex, posing challenges to traffic optimization. To fully utilize road resources and enhance traffic service capabilities, a traffic assignment model is adopted. The objective function of the traffic assignment model is F. T As shown below:

[0244]

[0245] In the formula, 'a' represents the road segment. It is freedom through time. a Let c be the traffic flow on road segment a. a It refers to the carrying capacity of road segment a. Let J be the set of regular arcs. Let J be the objective function F. T Shape parameters. E is a collection of charging arcs. B Rated capacity of electric vehicle batteries. This indicates the electricity price at which the charging pile on road segment a is connected to node j of the distribution network.

[0246] The constraints of the electric-traffic coupled probabilistic joint flow model for photovoltaic road grid connection include conventional arc constraints, charging arc constraints, bypass arc constraints, traffic flow constraints, distribution network model constraints, and coupling constraints between the traffic network and the distribution network.

[0247] 3) Based on polynomial chaotic expansion, the electric-traffic coupling probabilistic joint flow model of photovoltaic road grid connection is solved to obtain the probability density function and statistical information of the impact.

[0248] This invention models the probabilistic joint flow of photovoltaic road grid-connected power-traffic coupling and solves it efficiently based on an improved polynomial chaotic expansion method. It evaluates the impact of multiple uncertainties in the coupled system on the power distribution network and transportation network. This invention first considers the influence of the correlation between light intensity and ambient temperature, making the calculation of photovoltaic pavement output more accurate. Then, it establishes a probabilistic joint flow model of power-traffic coupling based on the photovoltaic pavement output model, traffic assignment model, and Distflow's optimal power flow model. Finally, it uses an improved polynomial chaotic expansion method to quantitatively evaluate the impact of multiple uncertainties (traffic volume, light intensity, etc.) of the coupled system on the power distribution network and transportation network system.

[0249] Example 9:

[0250] The probabilistic optimal flow calculation method for an electric-transit combined system with photovoltaic pavement is described in Example 8. Furthermore, the probabilistic models for the multiple uncertainties include probabilistic models for solar irradiance, temperature, vehicle dynamic shading, nonlinear correlation, and photovoltaic pavement power generation.

[0251] Example 10:

[0252] The probabilistic optimal flow calculation method for an electric-transportation combined system with photovoltaic pavement is described in any one of Examples 8 to 9. Furthermore, the probabilistic model for solar irradiance is as follows:

[0253]

[0254] In the formula, x is the solar irradiance value. f(x; α, β) is the probability of solar irradiance. Γ represents the gamma function. α and β represent the two parameters of the Beta function, respectively.

[0255] It is generally assumed that the probability distribution of temperature follows a Gaussian distribution, and the probability model of temperature is shown below:

[0256]

[0257] In the formula, μ represents the mean of the temperature sample. σ represents the standard deviation of the temperature sample. x0 is the temperature. f(x0) is the probability of the temperature.

[0258] The probability model for the vehicle's dynamic shadow is shown below:

[0259]

[0260] In the formula, A d,t This represents the area of ​​the shadow cast on the road by the vehicle due to sunlight at time t, when the driving direction is d. and Let x represent the coordinates of vertex i1 of the vehicle's shadow outline at time t, when the driving direction is d. d,t,n+1 =x1 and y d,t,n+1 =y1. n is the total number of vertices of the vehicle's shadow outline. Additionally, it should be noted that the n vertices are listed in either clockwise or counterclockwise order.

[0261] Correlation between variables is ubiquitous. A direct method to comprehensively consider the correlation between random variables is to select an appropriate joint distribution. Copula theory can effectively capture the complex correlation structure between random variables and construct the joint probability distribution between them. According to Sklar's theorem, the high-dimensional joint probability distribution F(x1,x2,...,x...) is... N The function is decomposed into N univariate marginal distribution functions F1(x1), F2(x2), ..., F... N (x N) and the connection function C(u1,u2,...,u N The Copula function is used to describe the correlation structure of variables. When the marginal distributions are continuous, the Copula function is uniquely determined and satisfies: F(x1,x2,…,x…). N )=C(F1(x1),F2(x2),…,F N (x N ))

[0262] Bivariate copula functions generally include two types: Gaussian and T. We use the Gaussian copula function to investigate the correlation between output temperature and temperature. The probabilistic model for this nonlinear correlation is shown below:

[0263]

[0264] In the formula, s is the temperature variable. t0 is the solar irradiance variable. C(u1,u2) is the probabilistic model of the nonlinear correlation between the marginal distribution functions of two univariate variables, C(u1,u2)=C(F1(x k ),F2(x k+1 C is the join function. F1(x) k ), F2(x k+1 All are univariate marginal distribution functions. k = 1, 2, ..., N-1, where N is the total number of univariate marginal distribution functions. ρ is the correlation coefficient between the variables, which can be inferred from historical data. Parameter u1 = F1(x k ). Parameter u2 = F2(x k+1 ). φ -1 ( u1 ), φ -1 ( u2 ) represent the inverse functions of the standard normal distributions of parameters u1 and u2, respectively.

[0265] The principle of photovoltaic (PV) pavement power generation is basically the same as that of traditional PV power generation. The power generation of a PV pavement depends on the characteristics of the PV pavement modules and the actual solar irradiance available to the surface. However, the available solar irradiance is affected not only by temperature and climate, but also by traffic flow and vehicle dynamic shading. Considering the correlation between traffic flow, vehicle dynamic shading, temperature, and solar irradiance on the power output of the PV pavement, the probabilistic model for PV pavement power generation is as follows:

[0266]

[0267] V s,t =V OC -K v T C,t (9)

[0268] I s,t =s t [I SC +K i (T C,t -25)] (10)

[0269]

[0270] P s,t =N PV V s,t I s,t FF (12)

[0271]

[0272]

[0273] In the formula, T C,t T A,t T OT These represent the battery temperature, ambient temperature, and normal operating temperature of the photovoltaic pavement power generation system, respectively. V s,t I s,t These represent the output voltage and current under solar irradiance S, respectively. OC I SC These represent the open-circuit voltage and short-circuit current, respectively. K v K i These represent the temperature coefficients of voltage and current, respectively. FF represents the fill factor. V MPP I MPP These represent the voltage and current at the maximum power point, respectively. N PV Indicates the number of modules. P s,t This indicates the output power of the photovoltaic module. P road,t This indicates the output power of the photovoltaic pavement. N veh,t Indicates the number of motor vehicles. Aveh,t This indicates the shadow area projected onto the photovoltaic road surface by the vehicle. A road d represents the total area of ​​the photovoltaic pavement. L represents the length of the photovoltaic pavement. t This indicates the density of traffic flow. Q t V t These represent traffic flow and average speed, respectively. t This represents the solar irradiance at time t.

[0274] Example 11:

[0275] The probabilistic optimal flow calculation method for an electric-transit combined system with photovoltaic pavement is described in any one of Examples 8 to 10. Further, to simulate the decision of electric vehicles to stop for charging or continue driving, the fast-charging stations in the directed graph are extended by adding virtual arcs. The extended equivalent model is attached. Figure 1 As shown. Compared with the original model, the difference lies in the addition of an inlet node I, an outlet node E, a charging arc (I→FCS→E), a regular arc (T1→I, E→T2, T1→T2), and a bypass arc with zero travel time (I→E).

[0276] conventional arc t a (x a The function t is strictly monotonically increasing. The Public Roads Authority (BPR) function better matches the latent function t. a (x a The properties of ). The regular arc constraint is as follows:

[0277]

[0278] In the formula, 'a' represents road segment 'a'. 'xa' represents the traffic flow on road segment 'a', which can be calculated from the road segment length and the maximum speed limit of the road segment. It is freedom through time. a It refers to the carrying capacity of road segment a. Let t be the set of regular arcs. a1 (x a () represents the time for a regular arc to pass through.

[0279] Charging arc The time spent on this type of arc includes charging time and queuing time. The former is derived from the linear relationship between the rated charging power of the fast charging station and the charging demand of the electric vehicle battery. For the latter, if x a >c a If the queuing time exceeds the charging time, then the queuing time may exceed the charging time. Therefore, a Davidson function based on queuing theory is used to evaluate the travel time, and the charging arc constraint is as follows:

[0280]

[0281] In the formula, t a2 (x a ) represents the duration of the charging arc. J is a parameter that determines the shape of the function. It is a set of charging arcs.

[0282] 3) Bypass arc The transit time of gas-powered vehicles (GVs) and electric vehicles (EVs) is so short as to be negligible.

[0283] The bypass arc constraint is as follows:

[0284]

[0285] In the formula, t a3 (x a () represents the time for the bypass arc to travel. It is the set of bypass arcs.

[0286] Links associated with feasible paths are constrained by arc flow and path flow, a relationship known as traffic flow conservation. This formula represents a fundamental traffic flow constraint. Common sense dictates that traffic users tend to choose the path with the lowest travel cost. Assuming all drivers are rational and have access to detailed traffic information through various navigation applications, a user equilibrium is reached. The travel cost (kg / ke) for each path used between each pair of outer diameters is uniform and minimized, while the time spent on unselected paths is greater than that of the aforementioned paths. The equilibrium is explained using complementary constraints, as shown below:

[0287]

[0288]

[0289]

[0290]

[0291]

[0292] In the formula, k represents a feasible route, and od represents the destination. These represent the travel demand from the origin to the destination for both electric vehicles and gasoline-powered vehicles. These are the sets of feasible paths for gasoline-powered cars and electric cars, respectively, from the starting point to the destination. Let $\mathbf$ be the flow rate of a gasoline-powered car and an electric car on the kth feasible route between the starting point and the destination, respectively. These are the path-arc correlation matrices for electric vehicles and gasoline vehicles, respectively. When the feasible path k... e When arc a is included, δE =1. When feasible path k e When arc a is not included, δ E =0. When feasible path k g When arc a is included, δ G =1. When feasible path k g When arc a is not included, δ G =0. TA A It is a set of arcs. These represent the travel times for a gasoline-powered car and an electric car, respectively, on the k-th route between the starting point and the destination. These represent the lowest costs for gasoline-powered vehicles and electric vehicles, respectively.

[0293] Due to branch thermal constraints, the current amplitude cannot exceed the threshold. The constraints of the distribution network model are as follows:

[0294]

[0295]

[0296]

[0297]

[0298]

[0299]

[0300]

[0301]

[0302] In the formula, i and j are the starting node and ending node of link l, respectively. Let be the active power of the generator at node j at time t. Let E be the reactive power of the generator at node j at time t. N Let be the set of nodes in the distribution network. π(j) represents the set of child nodes of node j, and k represents the set of child nodes of node j. E l This is a collection of distribution network branches. This represents the active power flow from node i to node j and from node j to node k. This represents the reactive power flow from node i to node j and from node j to node k. Let represent the minimum and maximum active power of the generator at node j at time t, respectively. Let U represent the minimum and maximum reactive power of the generator at node j at time t, respectively. i,t U j,tThese represent the voltage values ​​at node i and node j at time t, respectively. These represent the active and reactive power requirements of node j at time t, respectively. These represent the minimum and maximum voltage values ​​at node j, respectively. This represents the branch current value from node i to node j. This represents the maximum value of the branch current l from node i to node j. The resistance of branch l between node i and node j. P SR Equivalent to the output force P of the photovoltaic pavement calculated in formula (10) road,t . The resistance of branch l between node i and node j. The impedance of branch l between node i and node j.

[0303] Electric vehicles charging at fast-charging stations generate a charging load. Therefore, the power distribution network and the transportation network are coupled on the load side. The total load of the power distribution network consists of two parts: the first is the traditional electrical load, and the second is the electric vehicle charging load, which is proportional to the traffic flow on the road served by node j.

[0304] The coupling constraints between the transportation network and the power distribution network are as follows:

[0305]

[0306]

[0307] In the formula, This represents traditional electrical loads. This indicates the charging load of electric vehicles. This represents the electric vehicle traffic through charging link a. E B Rated capacity of electric vehicle batteries.

[0308] Example 12:

[0309] The probabilistic optimal flow calculation method for an electric-transit combined system with photovoltaic pavement, the main technical contents of which are described in any one of Examples 8 to 11, further, the influencing factors include:

[0310] 1) Overvoltage at nodes and power flow exceeding limits in the distribution network.

[0311] 2) Traffic congestion caused by routes exceeding capacity in the transportation network.

[0312] Example 13:

[0313] The probabilistic optimal flow calculation method for an electric-transit combined system with photovoltaic pavement is described in any one of Examples 8 to 12. Furthermore, the polynomial chaos solution method includes a coherent D-optimal polynomial chaos solution method.

[0314] Example 14:

[0315] The probabilistic optimal flow calculation method for an electric-transportation combined system with photovoltaic pavement, the main technical contents of which are described in any one of Examples 8 to 13, further, the solution steps for the coherent D-optimal polynomial chaos are as follows:

[0316] Solving for the coefficients of a polynomial chaotic expansion depends on the experimental design. The experimental design requires selecting an appropriate sampling strategy to ensure the accuracy and efficiency of the approximate output quality. Generally, it is beneficial to consider all information about the given mathematical model in order to choose the correct method.

[0317] 3.1) Calculate the minimum coherence parameter, and select the sample set that minimizes the coherence parameter μ from the random variable ξ using the coherence optimal sampling method.

[0318] The principle of coherent optimal sampling is to use a Markov chain Monte Carlo sampler to obtain initial samples and then select the sample set that minimizes the coherence parameter μ. The coherence parameter μ plays a crucial role in the stability and convergence of the coefficients of the ordinary least squares polynomial solution. The smaller μ is, the more stable and accurate the polynomial coefficients are obtained by least squares. The minimum coherence parameter is as follows:

[0319]

[0320] In the formula, μ(Y) is the minimum coherence parameter. sup is the supremum function. P is the maximum order of the polynomial expansion. j1 is the order of the multivariate orthogonal polynomial expansion. Represents the first multivariate corresponding to the j1 A multivariate orthogonal polynomial of order 1.

[0321] The weighting function w(ξ) is shown below:

[0322]

[0323] There is generally no analytical expression for B(ξ), but for any value of ξ, it can be found that the polynomial... The uniform least upper bound B of the sum of squares 2 (ξ) is shown below:

[0324]

[0325] 3.2) Calculate the information moments that maximize the formation, and select high-quality samples from the sample set with the smallest coherence parameter μ using the D-optimal sampling method.

[0326] Optimal experimental design is a common practice in least squares regression. In experimental optimization design, the design sample ξ is selected according to the optimal letter criterion, which is a scalar function and serves as the information moment M, as shown below:

[0327]

[0328] In the formula, M represents the information moment. N represents the total number of high-quality samples. Φ represents the set of multivariate orthogonal polynomials.

[0329] The information matrix M is crucial for the stability of least squares, and its deviation from the identity matrix describes the stability of least squares. The information matrix M is not determined by the value of any response. Therefore, this means that different designs can be compared using the basis polynomial, determining their relative optimality for the representation of the variables of interest before any simulation. The D-optimal design principle is to maximize the determinant of the information matrix. The determinant for maximizing the information matrix is ​​shown below:

[0330] φ D =max|M| 1 / P (37)

[0331] In the formula, φ D This represents the maximum determinant. `max` is the maximum value function.

[0332] 3.3) In the domain of probability, we introduce a finite-dimensional probability space L. 2 The input variables and stochastic process are explained using the formula (Ω, F, P), where Ω is the sample set w, F is the delta operator on Ω, and P represents the probability measure, indicating the likelihood of a given event occurring. Therefore, it is necessary to establish a mathematical model f(X(ω)) and quantify the uncertainty using the spectral method. The response can be represented as a finite orthogonal polynomial corresponding to the random variable. Using polynomial chaotic expansion, the relationship between the input variables and the response can be analytically represented.

[0333] The probability density function of high-quality samples is expanded into a multivariate orthogonal polynomial, as shown below:

[0334]

[0335] In the formula, f(ξ) represents the high-quality sample ξ = [ξ1, ξ2, ξ3, ..., ξ]. N The corresponding probability density function is given by [ ], where N is the total number of high-quality samples. Y is the probability density function of the corresponding high-quality samples. k is the order of the multivariate orthogonal polynomial expanded from the response probability density function Y. kThese are the coefficients of the polynomial chaotic expansion.

[0336] Among them, multivariate orthogonal polynomials can be constructed from the tensor product of univariate orthogonal polynomials, and the multivariate orthogonal polynomial φ (k) (ξ) is shown below:

[0337]

[0338] In the formula, i2 is the order of the multivariate orthogonal polynomial expansion. Indicates the i2th order An orthogonal polynomial.

[0339] 3.4) Among numerous non-invasive methods, the least squares regression principle is applied to the fitting and solution of polynomial coefficients. Based on the minimum coherence parameter and the minimum information moment, the least squares method is used to expand the polynomial chaotic coefficients c. k Solving for the polynomial chaotic expansion coefficients c yields the results. k The least squares solution.

[0340] 3.4.1) Calculate the residuals of a set of sample response pairs [ξ,Y] as follows:

[0341]

[0342] In the formula, Y is the set of probability density functions of random variables. ε(C) represents the residual of the sample response pair [ξ,Y]. Let i2 be the probability density function of the random variable corresponding to i2. Let be a multivariate orthogonal polynomial. Φ is the set of multivariate orthogonal polynomials. C is the set of expansion coefficients.

[0343] The number of terms K in the polynomial expansion of the response probability density function Y is shown below:

[0344]

[0345] In the formula, P is the maximum order of the polynomial expansion.

[0346] 3.4.2) Find the partial derivative of formula (40) and solve it by least squares to obtain the polynomial chaotic expansion coefficients c. k The least squares solution.

[0347] The least squares solution is shown below:

[0348] C=(Φ T Φ) -1 Φ T Y (42).

[0349] Example 15:

[0350] See Figures 1 to 12 A probabilistic optimal flow calculation method for an electric-transportation combined system with photovoltaic pavement includes the following steps:

[0351] 1. Probabilistic Models with Multiple Uncertainties

[0352] 1) Establish a probabilistic model for solar irradiance.

[0353]

[0354] Where Γ represents the gamma function; α and β represent the two parameters of the Beta function, respectively.

[0355] 2) Establish a probabilistic model for temperature.

[0356] Temperature is generally considered to follow a Gaussian distribution, and the temperature model can be expressed as:

[0357]

[0358] μ and σ represent the sample mean and standard deviation, respectively.

[0359] 3) Establish a probabilistic model for vehicle dynamic shadows.

[0360]

[0361] In the formula A d,t Given a driving direction of d, at time t, the area of ​​the shadow cast by the vehicle on the road due to sunlight; x d,t,i and y d,t,i The x-coordinate represents the coordinates of vertex i; d,t,n+1 =x1 and y d,t,n+1 =y1. Furthermore, it should be noted that the n vertices are listed in either clockwise or counterclockwise order.

[0362] 4) Establish a probabilistic model for nonlinear correlation.

[0363] Correlation between variables is ubiquitous. A direct method to comprehensively consider the correlation between random variables is to select the appropriate joint distribution.

[0364] The theory can effectively capture the complex correlation structure between random variables and construct the joint probability distribution among them. According to Sklar's theorem, the high-dimensional joint probability distribution F(x1,x2,...,x...) is... N The function is decomposed into N univariate marginal distribution functions F1(x1), F2(x2), ..., F... N (x N) and the connection function C(u1,u2,...,u NThe Copula function is used to describe the correlation structure of variables. When the marginal distributions are continuous, the Copula function is uniquely determined and satisfies:

[0365] F(x1,x2,…,x N )=C(F1(x1),F2(x2),…,F N (x N (4)

[0366] Where C is the join function.

[0367] Bivariate copula functions generally include two types: Gaussian and T. We use the Gaussian copula function to examine the correlation between output temperature and temperature. Its probability density function can be expressed as follows:

[0368]

[0369] The parameter ρ is the correlation coefficient between the variables, which can be inferred from historical data.

[0370] 5) Probabilistic model of photovoltaic pavement power generation

[0371] The principle of photovoltaic (PV) pavement power generation is basically the same as that of traditional PV power generation. The power generation of a PV pavement depends on the characteristics of the PV pavement modules and the actual solar irradiance available to the surface. However, the available solar irradiance is affected not only by temperature and climate, but also by traffic flow and vehicle dynamic shading. Considering the correlation between traffic flow, vehicle dynamic shading, temperature, and solar irradiance on the power output of the PV pavement, a PV power generation model is established:

[0372]

[0373] V s,t =V OC -K v T C,t (7)

[0374] I s,t =s t [I SC +K i (T C,t -25)] (8)

[0375]

[0376] P s,t =N PV V s,t I s,t FF (10)

[0377]

[0378]

[0379] Where T C ,T A and T OT These represent battery temperature, ambient temperature, and normal operating temperature, respectively; V s and I s V represents the output voltage and current under solar irradiance S. OC and I SC K represents the open-circuit voltage and short-circuit current, respectively. v and K i Indicates the temperature coefficients of voltage and current; FF represents the fill factor; V MPP and I MPP P represents the voltage and current at the maximum power point; s N represents the output power of the photovoltaic module. PV Number of modules; P road

[0380] Output power of photovoltaic pavement; N veh Indicates the number of motor vehicles; A veh Indicates the shadow area of ​​the vehicle projected onto the photovoltaic road surface; A road d represents the area of ​​the entire photovoltaic pavement; L represents the length of the photovoltaic pavement; d represents the traffic flow density; Q t and V t These represent traffic flow and average speed, respectively.

[0381] 2. Electricity-Traffic Coupling Probabilistic Joint Flow Model for Grid-Connected Photovoltaic Pavements

[0382] (1) Transportation Network Model

[0383] This model simulates the decision of whether an electric vehicle should park to charge or continue driving by extending the fast-charging stations in the directed graph with virtual arcs. The extended equivalent model is attached. Figure 1 As shown. Compared with the original model, the difference lies in the addition of an inlet node I, an outlet node E, a charging arc (I→FCS→E), a regular arc (T1→I, E→T2, T1→T2), and a bypass arc with zero travel time (I→E).

[0384] 1) Conventional arc t a (x a The function t is strictly monotonically increasing. The Public Roads Authority (BPR) function better matches the latent function t. a (x a ) attributes.

[0385]

[0386] x a The traffic flow on arc a can be calculated using the arc length and the maximum speed limit of the road segment. It is freedom through time; c a It is the carrying capacity of arc a; It is a set of regular arcs.

[0387] 2) Charging arc The time spent on this type of arc includes charging time and queuing time. The former is derived from the linear relationship between the rated charging power of the fast charging station and the charging demand of the electric vehicle battery. For the latter, if x a >c a If the queuing time exceeds the charging time, then the waiting time may be longer. Therefore, the Davidson function based on queuing theory is used to evaluate the travel time, which is expressed as:

[0388]

[0389] The parameter J determines the shape of the function; It is a set of charging arcs.

[0390] 3) Bypass arc The transit time of gas-powered vehicles (GVs) and electric vehicles (EVs) is so short as to be negligible.

[0391]

[0392] in The set of charging arcs for bypass arcs.

[0393] 4) Traffic flow constraints

[0394] The links associated with feasible paths have constraints on both arc flow and path flow, a relationship known as traffic flow conservation. This formula is a fundamental traffic flow constraint, defined as:

[0395]

[0396]

[0397]

[0398] in These represent the travel demand of electric vehicles from their origin to their destination (OD). These are the sets of feasible paths between OD (Original Design Vehicle) and OD (Operational Design Vehicle) for both gasoline-powered and electric vehicles. δ represents the flow of GVs and EVs on the k-th feasible route between OD pairs; E δ GThese are the path-arc correlation matrices, where k is the feasible path. e k g When arc a is included, δ E δ G It equals 1, otherwise it equals 0.

[0399] Common sense dictates that traffic users tend to choose the path with the lowest travel cost. Assuming all drivers are rational and have access to detailed traffic information through various navigation applications, a user equilibrium will be reached. The travel cost (kg / ke) for each path between each pair of outer diameters is uniform and minimized, while the time spent not choosing a path is longer than for the aforementioned paths. This equilibrium can be explained using complementary constraints:

[0400]

[0401]

[0402] In the formula, The required GVs (EVs) travel time for the k-th path between OD pairs. The minimum cost required for GVs (EVs).

[0403] (2) Traffic Assignment Model

[0404] With the rise of electric vehicles and fast charging stations, traffic operations have become more complex, posing challenges to traffic optimization. To fully leverage road resources and enhance traffic service capabilities, a traffic assignment model was adopted.

[0405]

[0406] Constraint = {(13) - (20)}

[0407] Where EB represents the rated capacity of the electric vehicle battery.

[0408] (3) Distribution network model

[0409] The goal of the optimal power flow model is to track the minimum generation cost by adjusting control variables based on the power flow equations and other security constraints. Currently, the ideal optimal power flow model uses the Distflow model to implement non-convex quadratic constraint programming. This reconstructs the optimal power flow problem for grid-connected photovoltaic pavements and fast charging piles into a nonlinear optimization problem:

[0410] Objective function:

[0411]

[0412] constraint:

[0413]

[0414]

[0415]

[0416]

[0417]

[0418]

[0419]

[0420]

[0421] In the formula a i ,b i c i ρ represents the coefficients of the quadratic function; ρ represents the contract energy price with the main power grid. and Let E represent the active and reactive power of the generator at node j at time t; π(j) represents the set of child nodes of node j; N and E l is the set of distribution network nodes and branches; i,j are the starting and ending nodes of link l; Equation (30) indicates that due to branch thermal constraints, the current amplitude cannot exceed the threshold.

[0422] Electric vehicles charging at fast-charging stations generate a charging load. Therefore, the power distribution network and the transportation network are coupled on the load side. The coupling equation between the transportation network and the power distribution network can be expressed as follows.

[0423]

[0424]

[0425] In the formula Indicated via charging link The electric vehicle traffic flow, and the fast charging pile connected to node j of the distribution network; in equation (31)

[0426] In this context, the total load of the distribution network consists of two parts. The first part is the traditional electrical load, and the second part is the electric vehicle charging load, which is proportional to the traffic flow on the road served by node j.

[0427] From the appendix Figure 2 As shown in the diagram, the photovoltaic pavement and electric vehicles couple to the distribution network and transportation network from the source and load sides, respectively, which exacerbates the interaction within the coupled system. Based on this, a joint optimization model for the electric-transport coupling system incorporating photovoltaic pavement is established.

[0428] minF P-T =F PDN +F T (33)

[0429] Constraint={(13)-(20),(22)-(32)}

[0430] 3. Arbitrarily sparse polynomial chaotic expansion and coherent D-optimal sampling method

[0431] (1) Arbitrarily sparse polynomial chaotic expansion

[0432] In the field of probability, a finite-dimensional probability space L is introduced. 2 The input variables and stochastic process are explained using the formula (Ω, F, P), where Ω is the sample set w, F is the delta operator on Ω, and P represents the probability measure, indicating the likelihood of a given event occurring. Therefore, it is necessary to establish a mathematical model f(X(ω)) and quantify the uncertainty using the spectral method. The response can be represented as a finite orthogonal polynomial corresponding to the random variable. Using polynomial chaotic expansion, the relationship between the input variables and the response can be analytically represented.

[0433] Consider ξ=[ξ1,ξ2,ξ3,…,ξ N [The input variables are random variables, and each variable has a corresponding probability density function.] Then the corresponding random Y is defined as:

[0434]

[0435] Where c k These are the coefficients of the polynomial. Multivariable orthogonal polynomials can be constructed from the tensor product of univariate orthogonal polynomials:

[0436]

[0437] They satisfy the orthogonality condition:

[0438] <φ (m) (ξ),φ (n) (ξ)>:=∫φ (m) (ξ),φ (n) (ξ)dξ=δ mn γ n (36)

[0439] in k represents the i-th variable iOrthogonal polynomials. For variables following Gaussian, gamma, Beta, or uniform distributions, the basis functions can be directly obtained from the correlation polynomial. For variables with unknown distributions or those not conforming to the above distributions, recursive formulas and Stieltjes' quadrature are used to process the initial data. δ mn It is the Kronecker operator, when m = n, δ mn =1, otherwise δ mn =0. K is the number of terms in the response Y expansion, which is usually obtained through equation (37) in practical applications.

[0440]

[0441] Where N is the number of input variables, and P is the maximum order of the polynomial expansion.

[0442] (2) Solving for polynomial chaotic expansion coefficients

[0443] Among numerous non-invasive methods, the principle of least squares regression is applied to fitting and solving polynomial coefficients. Specifically, for a set of sample response pairs [ξ, Y], the expansion coefficients c are obtained by minimizing the sum of the squared residuals. k :

[0444]

[0445] Taking the partial derivative of the above equation, the general least squares solution is:

[0446] C=(Φ T Φ) -1 Φ T Y (39)

[0447] (3) Coherent D-optimal sampling method

[0448] Solving for the coefficients of a polynomial chaotic expansion depends on the experimental design. The experimental design requires selecting an appropriate sampling strategy to ensure the accuracy and efficiency of the approximate output quality. Generally, it is beneficial to consider all information about the given mathematical model in order to choose the correct method.

[0449] 1) Coherent optimal sampling

[0450] The principle of coherent optimal sampling is to use a Markov chain Monte Carlo sampler to obtain initial samples and then select the sample set that minimizes the coherence parameter μ. The coherence parameter μ plays a crucial role in the stability and convergence of the coefficients of the ordinary least squares polynomial solution. Coherence parameter μ(Y)

[0451] It can be obtained from the following formula:

[0452]

[0453] The smaller μ is, the more stable and accurate the polynomial coefficients obtained by least squares solution are. The weighting function w(ξ) can be expressed as:

[0454]

[0455] In the formula B 2 B(ξ) denotes the uniform least upper bound of the sum of squares of the base polynomials under consideration. There is generally no analytical expression for B(ξ), but for any value of ξ, it can be found that:

[0456]

[0457] 2) D-optimal sampling

[0458] Optimal experimental design is a common practice in least squares regression. In experimental optimization design, the design sample ξ is selected according to the letter optimality criterion, which is a scalar function, represented as the information moment M, and can be expressed as:

[0459]

[0460] The information matrix M is crucial for the stability of least squares, and its deviation from the identity matrix describes the stability of least squares. The information matrix M is not determined by the value of any response. Therefore, this means that different designs can be compared using the basis polynomial, determining their relative optimality for the representation of the variables of interest before any simulation. The D-optimal design principle is to maximize the determinant of the information matrix.

[0461] φ D =max|M| 1 / P (44)

[0462] Where P is the maximum expansion order of the polynomial basis. The D-optimal design is achieved using column rotation and QR decomposition techniques.

[0463] Example 16:

[0464] See Figures 1 to 12 A probabilistic optimal flow calculation method for an electric-transportation combined system with photovoltaic pavement includes the following steps:

[0465] like Figure 4The diagram shows the topology of the IEEE-33 node system coupled with a ring traffic system. The ring traffic network contains 64 road segments, all of which can be divided into four parts: SR1, SR2, SR3, and SR4. The lengths and widths of T1-T3, T2-T6, T10-T11, and T7-T12 are 1.5√2 km and 7 m, respectively. The design length of the other photovoltaic pavements is 1.5 km, and the width is 7 m. Each part is equipped with 3000 photovoltaic pavement power generation devices. The power output of the four photovoltaic pavement parts is connected to distribution network nodes 7, 19, 25, and 33, respectively. To more clearly analyze the impact of the electric-transportation coupled system on the independent distribution network and transportation network, and the performance of the coherent D-optimal sampling polynomial chaotic expansion method, the following three cases and four methods are compared:

[0466] Case 1: Considering the grid connection of photovoltaic pavement, the probabilistic joint flow calculation of electricity-traffic decoupling.

[0467] Case 2: Considering grid connection of photovoltaic pavement, the probabilistic joint flow calculation of power-traffic coupling.

[0468] Case 3: Probabilistic joint flow calculation of electricity-traffic coupling considering photovoltaic pavement not connected to the grid.

[0469] M1: Monte Carlo Simulation (MCS)

[0470] M2: Arbitrary Sparse Polynomial Chaotic Expansion Based on Monte Carlo Sampling (MC-ASPCE)

[0471] M3: Arbitrary Sparse Polynomial Chaotic Expansion Based on Latin Hypercube Sampling (LHS-ASPCE)

[0472] M4: Arbitrary sparse polynomial chaotic expansion based on coherent D-optimal sampling (proposed method)

[0473] The specific steps of the probabilistic joint flow modeling of electro-traffic coupling considering solar irradiance, ambient temperature, OD on traffic demand, and vehicle dynamic shading, and the probabilistic joint flow evaluation calculation method using polynomial chaotic expansion based on coherent D-optimal sampling to solve the model, are as follows:

[0474] (1) Input system parameters (topology of distribution network, topology of transportation network and hourly load curve), historical data (hourly solar irradiance data of the four road locations, ambient temperature, and hourly traffic flow data of the four roads) and photovoltaic pavement related parameters.

[0475] (2) First, 1000 samples of each variable are generated through Markov chain Monte Carlo simulation, and then Gaussian Copula is used as the formula (1).

[0476] The function constructs a joint probability distribution between solar irradiance and temperature, samples from the joint probability distribution, and updates the temperature and temperature samples; then, these samples are screened by minimizing the coherence parameter and the information moment, as in formulas (2) and (3), and 500 high-quality samples are selected as the final initial samples.

[0477]

[0478]

[0479] φ D =max|M| 1 / P (3)

[0480] (3) The historical data is classified using the K-Means clustering method to obtain the centroid of each traffic flow cluster; then the kernel density of the traffic flow volatility is estimated, and after obtaining the probability density function of the traffic flow volatility, the acceptance-rejection sampling method is used to generate samples representing the traffic flow volatility. Then, by adding the random samples of traffic flow volatility to the randomly selected cluster centroids, OD to traffic demand samples can be generated; finally, the dynamic shadow of the vehicle is calculated using formula (4).

[0481]

[0482] (4) Use kernel density estimation to fit the hourly solar irradiance data of the four road locations, as well as the ambient temperature and hourly traffic flow data of the four roads, to their respective probability distribution functions. Based on the probability distribution functions of each variable, construct multivariate polynomial basis functions using the three-term recursive formula, Stieltjes product, and formulas (5)-(6).

[0483]

[0484]

[0485] (5) Using the 500 samples of temperature and solar irradiance generated in step (1) as input to the photovoltaic power output model that considers the correlation between solar irradiance and temperature, the power output of individual photovoltaic panels of SR1, SR2, SR3 and SR4 without considering vehicle dynamic shadows is calculated by formula (7)-(13).

[0486]

[0487] V s,t =V OC -K v T C,t (8)

[0488] I s,t =st [I SC +K i (T C,t -25)] (9)

[0489]

[0490] P s,t =N PV V s,t I s,t FF (11)

[0491]

[0492]

[0493] (6) The 10,000 traffic demand samples calculated in step (2) and the output of a single photovoltaic road panel in the four regions calculated in step (4) are used as inputs to the electric-traffic coupling probability joint flow model. The traffic flow, branch power and node voltage of the distribution network, distribution network and traffic network costs and total cost of the coupling system in each road segment can be calculated by formulas (14)-(34) under the condition of 500 samples.

[0494] minF P-T =F PDN +F T (14)

[0495]

[0496]

[0497]

[0498]

[0499]

[0500]

[0501]

[0502]

[0503]

[0504]

[0505]

[0506]

[0507]

[0508]

[0509]

[0510]

[0511]

[0512]

[0513]

[0514]

[0515] (7) Substitute the 500 samples obtained in step (1) and the corresponding responses of interest obtained in step (6) into formula (35), and obtain the coefficients of the polynomial chaotic expansion through formula (36).

[0516]

[0517] C=(Φ T Φ) -1 Φ T Y (36)

[0518] (8) After obtaining the equivalent polynomial analytic function, 10,000 samples are randomly selected to obtain the probability characteristics of each output value. The impact of multiple uncertainties on the power distribution network and transportation network is analyzed to provide support and reference for the operation decision of the coupled system. Moreover, by comparing the results obtained by this method with those obtained by Monte Carlo simulation, the efficiency and accuracy of the coherent D-optimal sampling method are verified.

[0519] Example 17:

[0520] See Figures 1 to 12 A probabilistic optimal flow calculation method for electric-transportation combined systems with photovoltaic pavements, the main technical contents of which are as follows:

[0521] This invention provides a more accurate quantification of the probabilistic characteristics of multiple uncertainties affecting the joint operation of an electric-transportation coupled system, and offers precise and efficient solutions based on coherent D-optimal sampling and polynomial chaotic expansion. By considering the impact of solar irradiance, ambient temperature, traffic demand variations, vehicle dynamic shading, and the correlation between temperature and solar irradiance on photovoltaic pavement output, this invention establishes a probabilistic joint flow model of electric-transportation coupling, quantifying the impact of multiple uncertainties on the power distribution network and traffic. The model is then efficiently and accurately solved using coherent D-optimal polynomial chaotic expansion, and the probabilistic characteristics of the output variables are analyzed, providing a reference for the coordinated development of the power grid and transportation network. Experimental examples 1-3 are compared to each other, considering the impact of different situations on the electric-transportation coupled system, and M1-M4 are used for verification. The improved polynomial chaotic expansion demonstrates its performance advantages.

[0522] 1) The impact of correlation on the distribution network: Figure 6 and Figure 7 These represent the quantiles of the voltage amplitude at node 18 and the power flow in branches 24-25, respectively. Considering the correlation between temperature and solar irradiance, the voltage amplitude and branch power flow exhibit greater volatility. The impact is primarily concentrated between 10 AM and 6 PM. We can intuitively see... Figure 6 The peak voltage shifted from 11 AM to noon. Considering the interaction of multiple variables, Figure 7 The actual power status of branches 24-25 poses a greater challenge to the safe operation of the distribution network. However, positive aspects remain. In particular, the branch transmission power decreases significantly at 5 PM and 6 PM, reducing the probability of thermal constraint violations and indirectly improving the reliability of the power system.

[0523] 2) Impact on the transportation network: Figure 7 The data reveals the diversity of traffic flow distribution across the links, with links 1, 3, and 4 exhibiting significantly higher instability compared to others. Furthermore, the maximum traffic volume on routes 1, 3, 4, 45, 47, 48, 55, 57, and 58 all exceeds 4500 vehicles per hour. Therefore, when traffic accidents occur on these sections, the road's service capacity will deteriorate significantly, increasing the likelihood of traffic congestion. This data can provide decision support for future road expansion plans.

[0524] 3) Analysis of the differences between the three operating conditions of the power system and the transportation system: Figure 8 and Figure 9 This paper compares the changes in the number of electric vehicles at fast-charging stations and the voltage amplitude at coupled nodes of the distribution network under three case studies. Numerous scenarios arise in the operating states of the coupled system, which may increase the risk of unhealthy operation of the distribution network. From... Figure 8It can be seen that photovoltaic power grids have a greater impact on voltage amplitude than electric vehicles. However, this does not mean that the impact of electric vehicles on voltage amplitude can be ignored. (Combined with...) Figure 8 and Figure 9 It can be seen that the wider the vehicle flow distribution, the more drastic the voltage amplitude fluctuation. Therefore, we believe that as the penetration rate of electric vehicles (10% in this paper) continues to rise, its impact on voltage amplitude will further increase. Figure 9 It can be seen that Case 2 has a significant impact on the travel route planning and decision-making of electric vehicle drivers, while reducing traffic congestion and promoting the construction of green transportation.

[0525] 4) Verify the effectiveness of the proposed method: Simulation results are as follows Figures 10-14 As shown in the figure, intuitively, this method is significantly superior to the MC and LHS methods, and the proposed method can better capture the multimodal distribution of the output variable. Most importantly, the peak value of the voltage amplitude distribution obtained by this method matches the reference value well and can estimate branch power and total cost with reasonable accuracy. Tables 1 and 2 show the mean and standard deviation of voltage amplitude, branch power, and social optimal cost for the four methods, respectively. We can observe that both the mean and standard deviation of M4 are closer to M1 in terms of social optimal cost. The results show that the proposed method can simultaneously handle problems considering both optimization and uncertainty.

[0526] Table 1. Comparison of average values ​​of bus voltage, branch power flow, and total cost estimated by different methods.

[0527]

[0528] Table 2 Comparison of the standard deviations of bus voltage, branch power flow, and total cost estimated by different methods.

[0529]

[0530] 5) Verifying the efficiency of the proposed method: As shown in Table 3, M1 has the longest solution time because it requires 10,000 power flow calculations on the highly nonlinear electric-traffic coupled probabilistic joint flow model to ensure the accuracy of the results. M4, while maintaining high fidelity, significantly reduces computational costs, confirming the feasibility of the method in practical engineering. Furthermore, it is worth noting that the experimental design time for M4 is longer than that for M2 and M3. The main reason is that the coherent D-optimal sampling method requires additional computation time to generate high-quality initial samples. However, we believe that sacrificing some time to ensure accuracy is worthwhile.

[0531] Table 3 Comparison of calculation time using different methods

[0532]

[0533] The experimental results show that:

[0534] This invention provides a more accurate quantification of the probabilistic characteristics of multiple uncertainties affecting the joint operation of an electric-transportation coupled system, and offers precise and efficient solutions based on coherent D-optimal sampling and polynomial chaotic expansion. By considering the impact of solar irradiance, ambient temperature, traffic demand variations, vehicle dynamic shading, and the correlation between temperature and solar irradiance on photovoltaic pavement output, this invention establishes a probabilistic joint flow model of electric-transportation coupling, quantifying the impact of multiple uncertainties on the power distribution network and transportation. The model is then solved efficiently and accurately using coherent D-optimal polynomial chaotic expansion, and the probabilistic characteristics of the output variables are analyzed, thus providing a reference for the coordinated development of the power grid and transportation network.

[0535] Example 18:

[0536] See Figures 1 to 12 A probabilistic optimal flow calculation method for an electric-transportation combined system with photovoltaic pavement includes the following steps:

[0537] 1) Establish a probability model for multiple uncertain factors.

[0538] 2) Based on the probability model of multiple uncertain factors, establish the power-traffic coupling probability joint flow model of photovoltaic road grid connection to quantitatively characterize the impact of multiple uncertain factors on the distribution network and transportation network.

[0539] 3) Based on the coherent D-optimal polynomial chaotic expansion, the electric-traffic coupling probabilistic joint flow model of photovoltaic road grid connection is solved to obtain the probability density function and statistical information of the impact.

[0540] The probabilistic models for the multiple uncertainties include probabilistic models for solar irradiance, temperature, vehicle dynamic shading, nonlinear correlation, and photovoltaic pavement power generation.

[0541] The probabilistic model for solar irradiance is shown below:

[0542]

[0543] In the formula, x is the solar irradiance value. f(x; α, β) is the probability of solar irradiance. Γ represents the gamma function. α and β represent the two parameters of the Beta function, respectively.

[0544] The probability model for the temperature is shown below:

[0545]

[0546] In the formula, μ represents the mean of the temperature sample. σ represents the standard deviation of the temperature sample. x0 is the temperature. f(x0) is the probability of the temperature.

[0547] The probability model for the vehicle's dynamic shadow is shown below:

[0548]

[0549] In the formula, A d,t This represents the area of ​​the shadow cast on the road by the vehicle due to sunlight at time t, when the driving direction is d. and Let x represent the coordinates of vertex i1 of the vehicle's shadow outline at time t, when the driving direction is d. d,t,n+1 =x1 and y d,t,n+1 =y1. n is the total number of vertices of the vehicle's shadow outline.

[0550] The probabilistic model for the nonlinear correlation is shown below:

[0551]

[0552] In the formula, s is the temperature variable. t0 is the solar irradiance variable. C(u1,u2) is the probability of the nonlinear correlation between the two univariate marginal distribution functions, C(u1,u2)=C(F1(x k ),F2(x k+1 C is the join function. F1(x) k ), F2(x k+1 All are univariate marginal distribution functions. k = 1, 2, ..., N-1, where N is the total number of univariate marginal distribution functions. ρ is the correlation coefficient between the variables. Parameter u1 = F1(x k ). Parameter u2 = F2(x k+1 ). φ -1 ( u1 ), φ -1 ( u2 ) represent the inverse functions of the standard normal distributions of parameters u1 and u2, respectively.

[0553] The probability model for the photovoltaic pavement power generation is shown below:

[0554]

[0555] V s,t =V OC -K v T C,t (6)

[0556] I s,t =s t [I SC +Ki (T C,t -25)] (7)

[0557]

[0558] P s,t =N PV V s,t I s,t FF (9)

[0559]

[0560]

[0561] In the formula, T C,t T A,t T OT These represent the battery temperature, ambient temperature, and normal operating temperature of the photovoltaic pavement power generation system, respectively. V s,t I s,t These represent the output voltage and current under solar irradiance S, respectively. OC I SC These represent the open-circuit voltage and short-circuit current, respectively. K v K i These represent the temperature coefficients of voltage and current, respectively. FF represents the fill factor. V MPP I MPP These represent the voltage and current at the maximum power point, respectively. N PV Indicates the number of modules. P s,t This indicates the output power of the photovoltaic module. P road,t This indicates the output power of the photovoltaic pavement. N veh,t Indicates the number of motor vehicles. A veh,t This indicates the shadow area projected onto the photovoltaic road surface by the vehicle. A road d represents the total area of ​​the photovoltaic pavement. L represents the length of the photovoltaic pavement. t This indicates the density of traffic flow. Q t V t These represent traffic flow and average speed, respectively. t This represents the solar irradiance at time t.

[0562] The objective function of the electric-traffic coupled probabilistic joint flow model for the photovoltaic pavement grid connection is shown below:

[0563] minF P-T =F PDN +F T (12)

[0564] In the formula, minF P-T The objective function is the electric-traffic coupled probabilistic joint flow model for photovoltaic pavement grid connection.

[0565] Among them, the objective function F of the distribution network model PDN As shown below:

[0566]

[0567] In the formula, i is the starting node of the distribution network link l. i b i c i All are coefficients of a quadratic function. ρ1 represents the contract energy price with the main power grid. Let E be the active power of the generator at node i at time t. N P represents the set of nodes in the distribution network. main This indicates the amount of electricity purchased from the mainnet.

[0568] The objective function F of the traffic assignment model T As shown below:

[0569]

[0570] In the formula, 'a' represents the road segment. It is freedom through time. a Let c be the traffic flow on road segment a. a It refers to the carrying capacity of road segment a. Let J be the set of regular arcs. Let J be the objective function F. T Shape parameters. E is a collection of charging arcs. B Rated capacity of electric vehicle batteries. This indicates the electricity price at which the charging pile on road segment a is connected to node j of the distribution network.

[0571] The constraints of the electric-traffic coupled probabilistic joint flow model for photovoltaic road grid connection include conventional arc constraints, charging arc constraints, bypass arc constraints, traffic flow constraints, distribution network model constraints, and coupling constraints between the traffic network and the distribution network.

[0572] The conventional arc constraint is as follows:

[0573]

[0574] In the formula, 'a' represents road segment 'x'. a For traffic flow on road segment a, It is freedom through time. a It refers to the carrying capacity of road segment a. Let t be the set of regular arcs. a1 (x a () represents the time for a regular arc to pass through.

[0575] The charging arc constraint is as follows:

[0576]

[0577] In the formula, t a2 (x a ) represents the duration of the charging arc. J is a parameter that determines the shape of the function. It is a set of charging arcs.

[0578] The bypass arc constraint is as follows:

[0579]

[0580] In the formula, t a3 (x a () represents the time for the bypass arc to travel. It is the set of bypass arcs.

[0581] The traffic flow constraints are as follows:

[0582]

[0583]

[0584]

[0585]

[0586]

[0587] In the formula, k represents a feasible route, and od represents the destination. These represent the travel demand from the origin to the destination for both electric vehicles and gasoline-powered vehicles. These are the sets of feasible paths for gasoline-powered cars and electric cars, respectively, from the starting point to the destination. Let $\mathbf$ be the flow rate of a gasoline-powered car and an electric car on the kth feasible route between the starting point and the destination, respectively. These are the path-arc correlation matrices for electric vehicles and gasoline vehicles, respectively. When the feasible path k... e When arc a is included, δ E =1. When feasible path k e When arc a is not included, δ E =0. When feasible path k g When arc a is included, δ G =1. When feasible path k g When arc a is not included, δ G =0. T A It is a set of arcs. These represent the travel times for a gasoline-powered car and an electric car, respectively, on the k-th route between the starting point and the destination. These represent the lowest costs for gasoline-powered vehicles and electric vehicles, respectively.

[0588] The constraints of the power distribution network model are as follows:

[0589]

[0590]

[0591]

[0592]

[0593]

[0594]

[0595]

[0596]

[0597] In the formula, i and j are the starting node and ending node of link l, respectively. Let be the active power of the generator at node j at time t. Let E be the reactive power of the generator at node j at time t. N Let be the set of nodes in the distribution network. π(j) represents the set of child nodes of node j, and k represents the set of child nodes of node j. E l This is a collection of distribution network branches. This represents the active power flow from node i to node j and from node j to node k. This represents the reactive power flow from node i to node j and from node j to node k. Let represent the minimum and maximum active power of the generator at node j at time t, respectively. Let U represent the minimum and maximum reactive power of the generator at node j at time t, respectively. i,t U j,t These represent the voltage values ​​at node i and node j at time t, respectively. These represent the active and reactive power requirements of node j at time t, respectively. These represent the minimum and maximum voltage values ​​at node j, respectively. This represents the branch current value from node i to node j. This represents the maximum value of the branch current l from node i to node j. The resistance of branch l between node i and node j. P SR To contribute to photovoltaic pavement. The resistance of branch l between node i and node j. The impedance of branch l between node i and node j.

[0598] The coupling constraints between the transportation network and the power distribution network are as follows:

[0599]

[0600]

[0601] In the formula, This represents traditional electrical loads. This indicates the charging load of electric vehicles. This represents the electric vehicle traffic through charging link a. E B Rated capacity of electric vehicle batteries.

[0602] The impacts include:

[0603] 1) Overvoltage at nodes and power flow exceeding limits in the distribution network.

[0604] 2) Traffic congestion caused by routes exceeding capacity in the transportation network.

[0605] The solution steps for the coherent D-optimal polynomial chaos are as follows:

[0606] 3.1) Calculate the minimum coherence parameter, and select the sample set that minimizes the coherence parameter μ from the random variable ξ using the coherence optimal sampling method.

[0607] The minimum coherence parameter is as follows:

[0608]

[0609] In the formula, μ(Y) is the minimum coherence parameter. sup is the supremum function. P is the maximum order of the polynomial expansion. j1 is the order of the multivariate orthogonal polynomial expansion. Let represent the j1-th order multivariate orthogonal polynomial corresponding to the multivariate variables.

[0610] The weighting function w(ξ) is shown below:

[0611]

[0612] Among them, polynomial The uniform least upper bound B of the sum of squares 2 (ξ) is shown below:

[0613]

[0614] 3.2) Calculate the information moments that maximize the formation, and select high-quality samples from the sample set with the smallest coherence parameter μ using the D-optimal sampling method.

[0615] The information moments are as follows:

[0616]

[0617] In the formula, M represents the information moment. N represents the total number of high-quality samples. Φ represents the set of multivariate orthogonal polynomials.

[0618] The determinant for maximizing the information moments is shown below:

[0619] φ D =max|M| 1 / P (37)

[0620] In the formula, φ D This represents the maximum determinant. `max` is the maximum value function.

[0621] 3.3) Expand the probability density function of high-quality samples into a multivariate orthogonal polynomial, as shown below:

[0622]

[0623] In the formula, f(ξ) represents the high-quality sample ξ = [ξ1, ξ2, ξ3, ..., ξ]. N The corresponding probability density function is given by [ ], where N is the total number of high-quality samples. Y is the probability density function of the corresponding high-quality samples. k is the order of the multivariate orthogonal polynomial expanded from the response probability density function Y. k These are the coefficients of the polynomial chaotic expansion.

[0624] Among them, the multivariate orthogonal polynomial φ (k) (ξ) is shown below:

[0625]

[0626] In the formula, i2 is the order of the multivariate orthogonal polynomial expansion. Indicates the i2th order An orthogonal polynomial.

[0627] 3.4) Based on the minimum coherence parameter and minimum information moment, the coefficients c of the polynomial chaos are expanded using the least squares method. k Solving for the polynomial chaotic expansion coefficients c yields the results. k The least squares solution.

[0628] 3.4.1) Calculate the residuals of a set of sample response pairs [ξ,Y] as follows:

[0629]

[0630] In the formula, Y is the set of probability density functions of random variables. ε(C) represents the residual of the sample response pair [ξ,Y]. Let i2 be the probability density function of the random variable corresponding to i2. Let be a multivariate orthogonal polynomial. Φ is the set of multivariate orthogonal polynomials. C is the set of expansion coefficients.

[0631] The number of terms K in the polynomial expansion of the response probability density function Y is shown below:

[0632]

[0633] In the formula, P is the maximum order of the polynomial expansion.

[0634] 3.4.2) Find the partial derivative of formula (40) and solve it by least squares to obtain the polynomial chaotic expansion coefficients c. k The least squares solution.

[0635] The least squares solution is shown below:

[0636] C=(Φ T Φ) -1 Φ T Y (42).

[0637] The orthogonal polynomials meet the orthogonality condition as follows:

[0638] <φ (m) (ξ),φ (n) (ξ)>:=∫φ (m) (ξ),φ (n) (ξ)dξ=δ mn γ n (43)

[0639] In the formula, m and n are the ordinal numbers of the expanded polynomial. δ mn It is the Kronecker operator, when m = n, δ mn =1. When m≠n, δ mn =0. γ n This represents the integral value of the expanded m and n-order multivariate orthogonal polynomials.

[0640] The D-optimal sampling method is implemented using column rotation and orthogonal triangular decomposition.

Claims

1. A method for calculating the probabilistic optimal flow of an electric-transportation combined system with photovoltaic pavement, characterized in that, Includes the following steps: 1) Establish a probabilistic model with multiple uncertainties; 2) Based on a probabilistic model with multiple uncertainties, a power-traffic coupling probabilistic joint flow model for photovoltaic road grid connection is established to quantitatively characterize the impact of multiple uncertainties on the distribution network and transportation network. The objective function of the electric-traffic coupled probabilistic joint flow model for the photovoltaic pavement grid connection is shown below: minF P-T =F PDN +F T (1) In the formula, minF P-T The objective function is the electric-traffic coupling probabilistic joint flow model for photovoltaic pavement grid connection. Among them, the objective function F of the distribution network model PDN As shown below: In the formula, i is the starting node of the distribution network link l; a i b i c i All are quadratic function coefficients; ρ1 is the contract energy price with the main power grid; E represents the active power of the generator at node i at time t. N P represents the set of nodes in the distribution network. main This represents the electrical energy purchased from the mainnet; The objective function F of the traffic assignment model T As shown below: In the formula, 'a' represents a road segment; It is freedom through time; x a Let c be the traffic flow on road segment a. a It refers to the carrying capacity of road segment a; Let J be the set of regular arcs; J is the objective function F. T Shape parameters; A collection of charging arcs; E B Rated capacity of electric vehicle batteries; This indicates the electricity price at which the charging pile on road segment a is connected to node j of the distribution network; The constraints of the electric-traffic coupling probabilistic joint flow model for photovoltaic road grid connection include conventional arc constraints, charging arc constraints, bypass arc constraints, traffic flow constraints, distribution network model constraints, and coupling constraints between the traffic network and the distribution network. 3) Based on polynomial chaotic expansion, the electric-traffic coupling probabilistic joint flow model of photovoltaic road grid connection is solved to obtain the probability density function and statistical information of the impact.

2. The probabilistic optimal flow calculation method for an electric-transit combined system with photovoltaic pavement according to claim 1, characterized in that, The probabilistic models for the multiple uncertainties include probabilistic models for solar irradiance, temperature, vehicle dynamic shading, nonlinear correlation, and photovoltaic pavement power generation.

3. The probabilistic optimal flow calculation method for an electric-transportation combined system with photovoltaic pavement according to claim 2, characterized in that, The probabilistic model for solar irradiance is shown below: In the formula, x is the solar irradiance value; f(x; α, β) is the probability of solar irradiance; Γ represents the gamma function; α and β represent the two parameters of the Beta function, respectively; The probability model for the temperature is shown below: In the formula, μ represents the mean of the temperature sample; σ represents the standard deviation of the temperature sample; x0 is the temperature; and f(x0) is the probability of the temperature. The probability model for the vehicle's dynamic shadow is shown below: In the formula, A d,t This represents the area of ​​the shadow cast on the road by the vehicle due to sunlight at time t, when the driving direction is d; and This represents the coordinates of vertex i1 of the vehicle's shadow outline at time t, when the driving direction is d; x d,t,n+1 =x1 and y d,t,n+1 =y1; n is the total number of vertices of the vehicle's shadow outline; The probabilistic model for the nonlinear correlation is shown below: In the formula, s is the temperature variable; t0 is the solar irradiance variable; C(u1,u2) is the probability of the nonlinear correlation between the two univariate marginal distribution functions, C(u1,u2)=C(F1(x k ),F2(x k+1 C is the join function; F1(x) k ), F2(x k+1 All are univariate marginal distribution functions; k = 1, 2, ..., N-1, where N is the total number of univariate marginal distribution functions; ρ is the correlation coefficient between variables; parameter u1 = F1(x k ); parameter u2 = F2(x k+1 );φ -1 (u1), φ -1 (u2) represents the inverse function of the standard normal distribution with parameters u1 and u2, respectively; The probability model for the photovoltaic pavement power generation is shown below: V s,t =V OC -K v T C,t (9) I s,t =s t [I SC +K i (T C,t -25)] (10) P s,t =N PV V s,t I s,t FF (12) In the formula, T C,t T A,t T OT These represent the battery temperature, ambient temperature, and normal operating temperature of the photovoltaic pavement power generation system, respectively; V s,t I s,t V represents the output voltage and current under solar irradiance S, respectively. OC I SC K represents open-circuit voltage and short-circuit current, respectively. v K i These represent the temperature coefficients of voltage and current, respectively; FF represents the fill factor; V MPP I MPP N represents the voltage and current at the maximum power point, respectively; PV Indicates the number of modules; P s,t P represents the output power of the photovoltaic module. road,t N represents the output power of the photovoltaic pavement; veh,t Indicates the number of motor vehicles; A veh,t Indicates the shadow area of ​​the vehicle projected onto the photovoltaic road surface; A road The area of ​​the entire photovoltaic pavement is represented by L; the length of the photovoltaic pavement is represented by d. t Q represents the density of traffic flow; t V t These represent traffic flow and average speed, respectively; s t This represents the solar irradiance at time t.

4. The probabilistic optimal flow calculation method for an electric-transit combined system with photovoltaic pavement according to claim 1, characterized in that, The conventional arc constraint is as follows: In the formula, a represents the road segment; x a For traffic flow on road segment a, It is freedom through time; c a It refers to the carrying capacity of road segment a; It is a set of regular arcs; t a1 (x a () represents the time for a regular arc to pass through; The charging arc constraint is as follows: In the formula, t a2 (x a ) represents the time during which the charging arc travels; J is a parameter that determines the shape of the function; A collection of charging arcs; The bypass arc constraint is as follows: In the formula, t a3 (x a () represents the time for the bypass arc to travel; It is the set of bypass arcs; The traffic flow constraints are as follows: In the formula, k represents a feasible route; od is the destination; These represent the travel demand from the origin to the destination for both electric vehicles and gasoline-powered vehicles. These are the sets of feasible paths for gasoline-powered cars and electric cars, respectively, from the starting point to the destination; These represent the flow rates of gasoline-powered vehicles and electric vehicles on the k-th feasible route between the starting point and the destination, respectively. These are the path-arc correlation matrices for electric vehicles and gasoline vehicles, respectively. When the feasible path k... e When arc a is included, δ E =1; when feasible path k e When arc a is not included, δ E =0; when feasible path k g When arc a is included, δ G =1; when feasible path k g When arc a is not included, δ G =0;T A A set of arcs; These represent the travel times for a gasoline-powered car and an electric car, respectively, on the k-th route between the starting point and the destination. These are the lowest costs for gasoline-powered vehicles and electric vehicles, respectively. The constraints of the power distribution network model are as follows: In the formula, i and j are the start node and end node of link l, respectively; Let be the active power of the generator at node j at time t; E represents the reactive power of the generator at node j at time t. N Let be the set of nodes in the distribution network; π(j) represents the set of child nodes of node j, and k represents the child nodes of node j; E l For distribution network branch collection; This represents the active power flow from node i to node j and from node j to node k; This represents the reactive power flow from node i to node j and from node j to node k; Let represent the minimum and maximum active power of the generator at node j at time t, respectively; U represents the minimum and maximum reactive power of the generator at node j at time t, respectively; i,t U j,t These represent the voltage values ​​at node i and node j at time t, respectively. These represent the active and reactive power requirements of node j at time t, respectively. Let represent the minimum and maximum voltage values ​​at node j, respectively; This represents the branch current value from node i to node j; This represents the maximum value of the branch current l from node i to node j; The resistance of branch l between node i and node j; P SR To contribute to photovoltaic pavement; The resistance of branch l between node i and node j; The impedance of branch l between node i and node j; The coupling constraints between the transportation network and the power distribution network are as follows: In the formula, Indicates traditional electrical load; Indicates the charging load of electric vehicles; This represents the electric vehicle traffic through charging link a. E B Rated capacity of electric vehicle batteries.

5. The probabilistic optimal flow calculation method for an electric-transit combined system with photovoltaic pavement according to claim 1, characterized in that, The impacts include: 1) Overvoltage at nodes and power flow exceeding limits in branch circuits in the distribution network; 2) Traffic congestion caused by routes exceeding capacity in the transportation network.

6. The probabilistic optimal flow calculation method for an electric-transit combined system with photovoltaic pavement according to claim 1, characterized in that, The polynomial chaos solution method includes the coherent D-optimal polynomial chaos solution method.

7. The probabilistic optimal flow calculation method for an electric-transit combined system with photovoltaic pavement according to claim 6, characterized in that, The solution steps for the coherent D-optimal polynomial chaos are as follows: 3.1) Calculate the minimum coherence parameter, and select the sample set that minimizes the coherence parameter μ from the random variable ξ using the coherence optimal sampling method; The minimum coherence parameter is as follows: In the formula, μ(Y) is the minimum coherence parameter; sup is the supremum function; P is the maximum order of the polynomial expansion; j1 is the order of the multivariate orthogonal polynomial expansion; Let represent the j1-th order multivariate orthogonal polynomial corresponding to the multivariate variables; The weighting function w(ξ) is shown below: Among them, polynomial The uniform least upper bound B of the sum of squares 2 (ξ) is shown below: 3.2) Calculate the information moments that maximize the formation, and select high-quality samples from the sample set with the smallest coherence parameter μ using the D-optimal sampling method; The information moments are as follows: In the formula, M is the information moment; N is the total number of high-quality samples; Φ is the set of multivariate orthogonal polynomials; The determinant for maximizing the information moments is shown below: f D =max|M| 1 / P (37) In the formula, φ D The maximum determinant is given by `max`, which is the maximum value function. 3.3) Expand the probability density function of high-quality samples into a multivariate orthogonal polynomial, as shown below: In the formula, f(ξ) represents the high-quality sample ξ = [ξ1, ξ2, ξ3, ..., ξ]. N The corresponding probability density function, where N is the total number of high-quality samples; Y is the probability density function of the corresponding high-quality samples; k is the order of the multivariate orthogonal polynomial expanded from the response probability density function Y; c k These are the coefficients of the polynomial chaotic expansion; Among them, the multivariate orthogonal polynomial φ (k) (ξ) is shown below: In the formula, i2 is the order of the multivariate orthogonal polynomial expansion; Indicates the i2th order A number of orthogonal polynomials; 3.4) Based on the minimum coherence parameter and minimum information moment, the coefficients c of the polynomial chaos are expanded using the least squares method. k Solving for the polynomial chaotic expansion coefficients c yields the results. k The least squares solution; 3.4.1) Calculate the residuals of a set of sample response pairs [ξ,Y] as follows: In the formula, Y is the set of probability density functions of random variables; ε(C) represents the residual of the sample response pair [ξ,Y]. Let i2 be the probability density function of the random variable corresponding to i2; Let Φ be a multivariate orthogonal polynomial; Φ be the set of multivariate orthogonal polynomials; and C be the set of expansion coefficients. The number of terms K in the polynomial expansion of the response probability density function Y is shown below: In the formula, P is the maximum order of the polynomial expansion; 3.4.2) Find the partial derivative of formula (40) and solve it by least squares to obtain the polynomial chaotic expansion coefficients c. k The least squares solution; The least squares solution is shown below: C=(Φ T F) -1 F T Y (42).

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