Power transmission network multi-stage stochastic programming method considering dynamic frequency security constraint

Abstract

The invention discloses a power transmission network multi-stage stochastic programming method and a power transmission network multi-stage stochastic programming system considering dynamic frequency security constraints, which are characterized by comprising a model decomposition method based on a piecewise linear fitting method in a maximum affine function form and a step-by-step hedging algorithm. The method comprises the following steps: S1, establishing a multi-machine system frequency response model of dynamic frequency response of a new energy power system, deducing a frequency safety index expression, and proposing a piecewise linear fitting method based on a maximum affine function form; s2, establishing a multi-stage unexpected constraint, and proposing a power transmission network multi-stage stochastic programming model considering a dynamic frequency security constraint; and S3, designing a model decomposition method based on a step-by-step hedging algorithm. The power transmission network multi-stage stochastic programming method and system provided by the invention have more remarkable effectiveness and superiority in the aspects of frequency safety and uncertainty response, and in addition, the multi-stage stochastic programming model can obtain a more economical and more flexible programming scheme.

Description

Technical field:

[0001] The invention relates to multi-stage random programming of a power transmission network, and in particular to a multi-stage random programming method of a power transmission network considering dynamic frequency safety constraints. Technical background:

[0002] Under the background of dual carbon, large-scale grid connection of renewable energy has become an inevitable trend. However, from a transient perspective, new energy units have the characteristics of no inertia or low inertia, resulting in a low overall inertia level of high-proportion new energy power systems and weakened frequency regulation capabilities, which poses a huge challenge to the frequency safety and stability of the power system. In addition, from a steady-state perspective, due to the intermittent and fluctuating characteristics of the output of new energy units, when a high proportion of new energy units are connected to the grid, the power system will face complex uncertainties. Transmission network planning is the basis of power system operation and control. Reasonable transmission network planning schemes can effectively deal with the frequency safety risks and uncertainty risks brought by new energy to ensure large-scale access to new energy units. Therefore, it is necessary to consider the uncertainty of new energy output and the dynamic frequency safety of the system during the planning process of a high-proportion new energy transmission network.

[0003] At present, the research on dynamic frequency security issues of renewable energy power grids mainly focuses on control and operation scheduling. However, most of the existing transmission network planning issues ignore the frequency security of the power grid, or only consider the steady-state frequency security issues, while the research on how to effectively deal with dynamic frequency security issues such as transient frequency over-limit in the planning stage is still insufficient.

[0004] In addition, in the existing research, the main methods for dealing with uncertainty in power grid planning include stochastic programming and robust optimization. However, the results of robust optimization tend to be conservative, the model transformation is complex, and the computational burden is heavy. Most of the existing methods are based on a two-stage framework of upper-level planning and lower-level operation evaluation, which cannot flexibly adjust the planning schemes for each stage in response to the uncertainty of new energy in multi-stage planning, which will inevitably have a certain impact on the economy and feasibility of the planning scheme.

[0005] It can be seen that in order to deal with the frequency safety problem of new energy grid connection and ensure the planning economy and feasibility in a multi-stage uncertain environment, there are three main challenges: (1) The description of the dynamic frequency problem consists of a series of differential equations, which are difficult to be compatible with the algebraic planning model; (2) As the multi-stage uncertainties are gradually realized, how to achieve flexible adjustment modeling of the planning scheme and ensure its planning economy and feasibility is an important challenge. (3) The multi-stage and multi-scenario power grid planning model has high dimensionality and large scale, and its high efficiency and high precision solution problem is difficult to solve. Summary of the invention:

[0006] Symbols and definitions:

[0007] A. Indices, Sets, and Matrices:

[0008]

[0009] B. Variables:

[0010]

[0011]

[0012] C. Constants:

[0013]

[0014] D.Abbreviation:

[0015]

[0016]

[0017] The purpose of the present invention is to provide a multi-stage stochastic programming method for a transmission network considering dynamic frequency security constraints, and propose a multi-stage stochastic programming method for a transmission network based on dynamic frequency security constraints. The frequency security problem is dealt with through analysis and linearization of the frequency response model of the new energy power system, and the multi-stage uncertainty problem is dealt with based on unexpected constraints. The expression of the dynamic frequency security index of the new energy power grid is derived, and considering its highly non-convex and nonlinear characteristics, a piecewise linear fitting method based on the maximum affine function form is proposed. A multi-stage unexpected constraint is established to deal with the multi-stage uncertainty problem, and on this basis, a multi-stage stochastic programming model for a transmission network considering dynamic frequency security constraints is proposed. In view of the high-dimensional and multi-scenario characteristics of the planning model, a model decomposition method based on the stepwise hedging algorithm is designed, which ensures the calculation accuracy while significantly improving the calculation efficiency.

[0018] To achieve the above object, the present invention provides a multi-stage stochastic programming method for a transmission network considering dynamic frequency security constraints, characterized in that the model decomposition method based on a piecewise linear fitting method in the form of a maximum affine function and a stepwise hedging algorithm comprises the following steps:

[0019] S1. Establish a multi-machine system frequency response model for the dynamic frequency response of the new energy power system, derive the frequency security index expression, and propose a piecewise linear fitting method based on the maximum affine function form;

[0020] S2. Establish multi-stage unexpected constraints and propose a multi-stage stochastic programming model for the transmission network considering dynamic frequency security constraints;

[0021] S3. Design a model decomposition method based on the stepwise hedging algorithm.

[0022] 2. According to a multi-stage stochastic programming method and system for a power transmission network considering dynamic frequency security constraints according to claim 1, it is characterized in that the dynamic frequency security index of the new energy power grid is constructed as follows:

[0023] (1) Frequency response of new energy units

[0024] New energy units are connected to the power system through power electronic equipment, and they do not inherently provide inertia for the power system. New energy units can use corresponding control strategies to simulate the inertial response and primary frequency regulation response process similar to synchronous units, and participate in the frequency regulation process of the power system, as shown in equations (1) and (2):

[0025]

[0026] Among them, ΔP w (t) and ΔP w (s) is the frequency response power in the time domain and frequency domain, Δf(t) and Δf(s) are the frequency deviations in the time domain and frequency domain, D w is the droop coefficient of the new energy unit, T w H is the speed regulator time response coefficient w is the inertia constant of the new energy unit.

[0027] (2) Frequency response model of multi-machine system

[0028] On the basis of the low-order SFR model, the frequency regulation system of renewable energy units is comprehensively considered and the MMSFR model is established. This model consists of three parts: inertia and load frequency control, synchronous generator frequency control and renewable energy unit frequency control.

[0029] F Hτ , T Rτ , R Gτ , K mτ are the reheat coefficient, reheat time constant, mechanical power gain coefficient and governor speed constant of the τth synchronous machine, respectively. E is the comprehensive inertia of the system, D E is the damping coefficient of the system. ΔP w and ΔP g are the primary frequency regulation powers of synchronous generator sets and new energy generator sets in response to frequency deviations. Vw and T Vw are the virtual regulation coefficient of the w-th new energy station and the time constant of the inverter response respectively.

[0030] (3) Equivalent Aggregate Multi-machine System Frequency Response Model

[0031] The MMSFR model is aggregated into the AMSFR model using the parameter equivalent aggregation method. This model is a low-order linear transfer function model. The equivalent inertia constant of the AMSFR system is shown in formula (3):

[0032]

[0033] in, and are the rated power of synchronous generator set and new energy generator set, H E , H g and H w They are the equivalent inertia constant, the inertia constant of the synchronous generator set and the new energy generator set respectively.

[0034] The equivalent parameters of the AMSFR model are shown in equations (4), (5) and (6):

[0035]

[0036] Where: Ω g and Ω w are synchronous generator sets and new energy generator sets respectively, R E , R g and R w They are the equivalent droop coefficient, the droop coefficient of the synchronous generator set, and the droop coefficient of the new energy unit; F E 、F Hg and F Hw are the system equivalent work ratio, the synchronous generator set and the new energy generator set work ratio, T RE , T Rg and T Rw They are the equivalent reheat time constant, the reheat time constant of the synchronous generator set and the new energy unit respectively.

[0037] According to the AMSFR model transfer function block diagram, the step disturbance ΔP is derived L The analytical expressions of the post-frequency dynamics are shown in equations (7) and (8):

[0038]

[0039]

[0040] The frequency response model is subjected to Laplace inverse transform, and the time domain analytical expression of the frequency deviation of the AMSFR model is derived, as shown in equations (9) and (10):

[0041]

[0042] According to the time domain expression of AMSFR model, the maximum frequency change rate RoCoF and the lowest frequency point Δf of frequency safety index are derived.naclir and steady-state frequency Δf ss The corresponding constraints are established as shown in equations (11), (12) and (13):

[0043]

[0044] (4) Frequency safety constraint linearization

[0045] The frequency minimum point expression is given by (H E ,R E ,F E ,T RE ,D), which cannot be directly processed by the piecewise linearization method. The system damping is usually strictly limited to a small range, so D can be determined as a constant; the frequency deviation is less sensitive to the reheating time constant, so T RE can be assumed to be the same value without loss of accuracy. The variables in the frequency safety expression are further simplified to w = (H E ,R E ,F E ).

[0046] The frequency minimum point function can be approximately regarded as point-wise convex within the variable range used in this paper. It is linearized using the MA-PWL method and constructed as a maximum affine function, as shown in formula (14):

[0047]

[0048] in, b i is the coefficient of the i-th fitting hyperplane; m is the number of hyperplanes used in the calculation.

[0049] The least squares form is used to express the fitting error between the true value and the fitted value. The optimal fitting coefficient of the point-by-point function evaluation of the fixed evaluation point is determined by optimizing the minimum fitting error, as shown in formula (15):

[0050]

[0051] Since the fitting error optimization problem contains the "min-max" operator, it is difficult for current commercial solvers to solve it. New binary variables and continuous variables are introduced to reconstruct the maximum affine function term into a series of linear inequalities to eliminate the "max" operator in the objective function and reformulate the complete optimization problem in equation (15) as shown in equations (16) and (17):

[0052]

[0053] After solving the above problem with a commercial solver to obtain the optimal parameters of the fitted hyperplane, the linearization of the frequency security constraint is completed, and the lowest point constraint is integrated into the transmission network planning model by adding a set of linear inequalities.

[0054] Among the other two constraints in the dynamic frequency safety model, the maximum frequency change rate constraint (11) can be transformed into a linear constraint. The quasi-steady-state constraint (12) contains a nonlinear term of the product of continuous variables. Based on the previous assumption that the equivalent damping constant of the system can be determined as a constant, constraint (12) can also be regarded as a linear constraint.

[0055] 3. According to the method and system for multi-stage stochastic programming of a power transmission network considering dynamic frequency security constraints of claim 1, it is characterized in that the multi-stage stochastic programming model of a power transmission network considering dynamic frequency security constraints:

[0056] (1) Model of renewable energy output scenario tree

[0057] This paper constructs a multi-stage scenario tree for renewable energy output. Each node in the scenario tree is the realization of its output uncertainty. The probability of each node refers to the conditional probability that all uncertainties before the current stage are revealed. The first node of the scenario tree is the initial state, and the parent node is connected to the child node through branches at each stage, indicating the realization from one stage to another. Each path from the first node to the last node of the scenario tree represents a multi-stage scenario, which includes the realization process of random variables in multiple stages.

[0058] First, it is assumed that the implementation of each stage maintains the independence between stages. In each stage, a multi-scenario set of new energy output is formed by the Latin hypercube sampling method. Then, the backward scenario reduction method is used to reduce the scenarios, and finally a typical daily scenario set of new energy output in each stage is obtained.

[0059] The three-stage scenario tree generation process is as follows: In the first stage, it is assumed that a certain new energy output information is used as the initial node of the scenario tree, so the probability of this scenario is 1, represented by node 0, and the new energy output information of this node is The new energy output information of the second stage is determined by the initial node 0, and three possible scenarios are generated, represented by nodes 1, 2 and 3. The new energy output information of these three scenarios is and The probabilities are θ1, θ2 and θ3 respectively; the new energy output information of the third stage is determined by the nodes of the second stage, and the three nodes each generate three uncertainty scenarios, represented by nodes 4-12. The wind and solar output information of these nine scenarios are The probabilities are θ4-θ 12In this way, there are 9 paths from the root node to the end node, that is, the scene tree has 9 random scenes s1-s9, and the probabilities of the 9 scenes are shown in formula (18):

[0060]

[0061] (2) Unexpected constraints

[0062] Due to the random scenarios of uncertainty in the new energy power system 1 …ξ y It is gradually realized as the time series progresses, not generated at the same time. Therefore, the corresponding planning decisions should also be realized step by step, that is, observation (ξ 1 )→Planning decision (X1)→Observation (ξ 2 )→Planning decision (X2)→…→Observation (ξ y )→Planning Decision (X v ) process.

[0063] From the above steps, it can be seen that planning decision variables cannot change with random factors that have not been realized in the future, otherwise the model will not be able to obtain feasible planning results to deal with future uncertainties. For this reason, "unexpectedness" needs to be considered in the multi-stage stochastic planning process.

[0064] by Figure 3 The scene tree shown in the figure is used as an example to illustrate the unexpected. In this three-stage scene tree, taking scenes s1, s2 and s3 as examples, the uncertainty information that can be obtained for these three scenes in the third stage is and In the second stage, the uncertainty information available for the three scenarios is That is to say, in the second stage, these three scenarios are indistinguishable, and they will not evolve into different scenarios until the third stage. Therefore, the decision-making of these three scenarios in the second stage is This is the unexpected nature of the scene tree decision process.

[0065] Next, the unexpected constraints of the three stages of the scenario tree are constructed. For the first stage, its decision variables do not depend on the decisions of the second and third stages. The conditions that all scenarios need to meet are shown in formula (19):

[0066]

[0067] For the second stage, its decision variables do not depend on the decision of the third stage. The conditions that all scenarios need to meet are shown in formula (20):

[0068]

[0069] The above analysis has been completed Figure 3 The unexpected constraints of the three-stage scene tree shown are constructed. Further, based on the unexpected conditions, for s scenes, the unexpected constraints of the scene tree of y stages are shown in formula (21):

[0070]

[0071] Among them, A y is a scene cluster, representing the unexpected subspace of the scene, which is defined as: at each stage y, the scene set Ω s can be divided into a finite number of disjoint subsets, in any of which the various scenarios are considered indistinguishable in terms of decision making. The set of all scenario clusters at stage y is denoted as A y , as shown in formula (22):

[0072]

[0073] (3) Multi-stage stochastic programming model for transmission network considering dynamic frequency security constraints

[0074] In the multi-stage model, planning decisions must be able to adjust in time with the increase in available information in the uncertain scenario, so the representation of the uncertain information structure must be an important part of the planning decision representation. Based on the traditional transmission network stochastic planning model, the planning variables are associated with random factors. This section establishes a transmission network stochastic planning model that considers dynamic frequency security. In addition to configuring traditional thermal power units, unit planning also considers new energy units.

[0075] a) Objective Function

[0076] The objective function is to minimize the total cost, including investment cost and operating cost, as shown in formula (23):

[0077]

[0078] Among them, in the formula: Ω i is the node set, Ω s is the scene set, Ω t is the time period set, Ω g is the set of built generators, Ω′ g is the set of generators to be built, Ω l is the set of transmission lines, Ω w A collection of new energy units to be built; I l,s,y ,I g,s,y and I w,s,y are the decision variables for the construction of transmission lines, generators and new energy units respectively; C l , C g and C ware the investment and construction cost coefficients of transmission lines, generator sets, and new energy units respectively; p g,s,y,t is the power generation capacity of conventional units; is the load shedding amount; It represents the operating cost coefficient of conventional units; The starting cost of the unit; is the load shedding cost coefficient; η s is the probability of scene s; T h is the typical number of days in the planning stage; yr is the planning period, and d is the discount rate.

[0079] b) Investment decision constraints

[0080] Once the equipment is put into construction at a certain stage, it will be considered as constructed in the following stages. The constraints are shown in equations (25)-(28):

[0081]

[0082] I l，s,y , I g，s,y , I w,s,y ∈{0,1} (28)

[0083] c) DC power flow constraints

[0084] The model uses DC power flow constraints and ignores voltage changes and line losses, where M is a sufficiently large integer, as shown in equations (29)-(32):

[0085]

[0086] Among them, f l,s,y,t is the power flow of line l in stage y during period t of scenario s; b l is the susceptance of line l; θ i,s,y,t and θ j,s,y,t is the node voltage phase angle value of nodes i and j at both ends of line l; and are the upper and lower limits of the power flow capacity of line l respectively.

[0087] d) Constraints on new energy output

[0088] The new energy output constraint is shown in formula (33):

[0089]

[0090] Among them, p w,s,t,y is the output power of the new energy unit in stage y during period t in scenario s; α w,s,t is the normalized new energy output power.

[0091] e) Power balance constraints

[0092] The power balance constraint is shown in formula (34):

[0093]

[0094] Among them, L(i) to and L(i) from are the line sets at the power transmission and receiving ends of node i respectively; D i,s,t is the load power demand of node i.

[0095] f) Generator output upper and lower limit constraints

[0096] The upper and lower limits of the generator output are shown in equations (35) and (36):

[0097]

[0098] in, and are the maximum and minimum output power of generator set g respectively.

[0099] g) Upper and lower limit constraints of unit climbing

[0100] The upper and lower limit constraints of the unit climbing are shown in equations (37) and (38):

[0101]

[0102] Among them, δ g,ip and δ g,down They are the ramp-up rate and ramp-down rate of generator set g respectively.

[0103] i) Frequency safety constraints

[0104] Write out the maximum frequency change rate constraint and steady-state frequency constraint expressions related to the frequency safety constraint. The equivalent frequency parameter expressions are shown in equations (39)-(42):

[0105]

[0106] The frequency minimum point constraint after linearization is shown in equations (43)-(45):

[0107]

[0108] in, are m-1 linear inequality constraints, and They are respectively the system inertia constant, equivalent droop coefficient and equivalent work ratio coefficient after equivalent stage y in scenario s.

[0109] j) Unexpected constraints

[0110] The unexpected constraints are shown in equations (46)-(48):

[0111]

[0112] 3. According to claim 1, a multi-stage stochastic programming method and system for a power transmission network considering dynamic frequency security constraints is characterized in that the solution of the multi-stage stochastic programming model is:

[0113] (1) Multi-stage stochastic programming model of transmission network with unexpected constraints

[0114] The matrix form of the multi-stage stochastic programming model of the transmission network considering unexpected constraints can be expressed as:

[0115]

[0116] Where: X is the decision variable for the construction status of power grid planning equipment; P s is the continuous variable of system operation, including unit output, voltage phase angle, etc.; X FR are frequency-related variables, such as inertia constant, regulation coefficient, etc.; f(·) is the economic operation of the power grid; A is the feasible domain of the constraints related to the operation of the power grid; T and B T is the coefficient matrix related to the equipment construction constraints between different stages, C T and D T is the constant coefficient matrix of the frequency-safe convex feasible region constraint, E T and F T It is the relationship coefficient matrix between frequency variables and equipment investment and construction status decision variables.

[0117] It can be seen from the model that due to the existence of unexpected constraints, the decision variables between different scenarios are coupled. Therefore, based on the PHA algorithm, the multi-stage model is reconstructed into an augmented Lagrangian form, and the unexpected constraints are converted into equality constraints, which relaxes the unexpectedness of the original problem and realizes the scenario-based decoupling of the model, as shown in formula (50):

[0118]

[0119] Where W is the Lagrange multiplier, r is the penalty factor, Defined as an aggregation operator, it is a conditional expectation operator based on the uncertain information structure of the scene tree and the probability weight of the scene. It aggregates the possible different responses that the decision may make to the scenes in any scene cluster into a single compromise response to the scene cluster, as shown in formula (51):

[0120]

[0121] (2) PHA algorithm process

[0122] By selecting appropriate multipliers and penalties, the deviation between the planning scheme of the scenario sub-problem at each stage and the aggregate planning scheme represented by the aggregation operator is continuously reduced iteratively to restore the unexpectedness of the problem after decoupling and gradually approach the optimal solution of the original model. The implementation process of the PHA algorithm is as follows:

[0123] PHA Algorithm

[0124] Description of the drawings:

[0125] Figure 1 It is a flow chart of a multi-stage stochastic programming method for a transmission network considering dynamic frequency security constraints;

[0126] Figure 2 It is a schematic diagram of the MMSFR model;

[0127] Figure 3 It is a schematic diagram of the AMSFR model;

[0128] Figure 4 It is a schematic diagram of the three-stage scenario tree;

[0129] FIG5 is a schematic diagram of network topologies of four scenarios after planning according to an embodiment of the present invention;

[0130] 6 is a surface diagram of the frequency minimum point expression of fixed variables according to an embodiment of the present invention and a surface diagram of a hyperplane fitted using MA-PWL;

[0131] Figure 7 is a fitting error probability distribution histogram of all data points in the embodiment of the present invention;

[0132] FIG8 is a comparison diagram of the frequency trajectory of the system after disturbance of two model planning results according to an embodiment of the present invention;

[0133] Fig. 9 is a comparison chart of the solution time of directly solving a multi-stage model and using the PHA algorithm to solve the multi-stage model in an embodiment of the present invention;

[0134] Fig.10 is a comparison chart of installed capacity of four scenarios of wo / FC and w / FC models in an embodiment of the present invention;

[0135] Fig.11 It is a convergence error diagram of solving the model with different iteration times using the PHA algorithm in an embodiment of the present invention. Specific implementation method:

[0136] The present invention is further described below in conjunction with the accompanying drawings and specific embodiments.

[0137] Unless otherwise defined, technical or scientific terms used in the present invention shall have the common meanings understood by one having ordinary skills in the field to which the present invention belongs.

[0138] The present invention aims to provide a multi-stage stochastic programming method for a transmission network considering dynamic frequency security constraints. A multi-stage stochastic programming method for a transmission network based on dynamic frequency security constraints is proposed. The frequency security problem is dealt with through analysis and linearization of the frequency response model of the new energy power system, and the multi-stage uncertainty problem is dealt with based on unexpected constraints. The dynamic frequency security index expression of the new energy power grid is derived, and considering its highly non-convex and nonlinear characteristics, a piecewise linear fitting method based on the maximum affine function form is proposed. A multi-stage unexpected constraint is established to deal with the multi-stage uncertainty problem, and on this basis, a multi-stage stochastic programming model for a transmission network considering dynamic frequency security constraints is proposed. In view of the high-dimensional and multi-scenario characteristics of the planning model, a model decomposition method based on the stepwise hedging algorithm is designed. The method is decomposed into problems S1, S2, and S3.

[0139] In the embodiment study, a 6-node system and a New England IEEE 39-node system design example were used for analysis. All example simulation programs were completed on a laptop with a 2.30GHz CPU and 16GB RAM using the GAMs platform to call the commercial solver Gurobi.

[0140] A. 6-node system

[0141] 1) Test system and parameter settings

[0142] The 6-node system consists of 6 nodes, 7 branches, and 3 fire-point units, with node 1 being a balancing node. The parameter information of the units to be built is shown in Tables 1 and 2. The upper limit of the number of new lines in the original transmission corridor is 2. The load shedding cost is 1000$ / MWh, and the capital recovery coefficient is 0.15. The planning period is expected to be divided into three stages, each of which is planned based on a typical day and the period is set to one year. The system connects to the load at nodes 3, 4, and 5, and the load increases by 20% in each stage. At the same time, a 3-stage 4-scenario tree is used to characterize the uncertainty of wind turbine output. Each parent node in the scenario tree has two child nodes. The schematic diagram of the network topology of the four scenarios after planning is shown in Figure 5.

[0143] Table 1 Parameters of thermal power units to be built

[0144]

[0145] Table 2 Parameters of the wind farm to be built

[0146]

[0147] 2) MA-PWL fitting validity analysis

[0148] In order to analyze the linearization effect of the MA-PWL method, one of the four frequency parameters is fixed as a constant, and the frequency minimum point function is converted from four dimensions to three dimensions, so that the fitting effect can be intuitively displayed using images. Taking into account the range of the frequency safety function variables, as well as the scale of the inequality and 0-1 variables after linearization, four hyperplanes are selected to fit the frequency minimum point function. Figure 6 is a surface plot of the frequency minimum point expression with fixed variables and a surface plot of the hyperplane fitted using MA-PWL. The relative average error is used to analyze the fitting effect. 1,000 data points are randomly selected within the variable value range of the frequency minimum point function to calculate the relative error between the fitting function and the original function. Figure 7 is the histogram of the fitting error probability distribution for all data points.

[0149] From Figure 6, we can see that different fitting hyperplanes are used to replace the lowest frequency point surface in different evaluation point areas. Figure 7 It can be clearly seen that the probability of relative error within 2% is the highest. Finally, the relative average fitting error of 1000 data points is calculated to be 1.37%. Combined with Figure 6, it can be concluded that the MA-PWL method can fit the frequency minimum point surface with high accuracy.

[0150] 3) Effectiveness analysis of multi-stage stochastic programming method

[0151] In order to verify the effectiveness of the multi-stage stochastic planning method proposed in this paper, the traditional two-stage stochastic planning model is used as the control group for comparison. The planning schemes obtained by the two models are shown in Tables 3 and 4, and the cost comparison of the planning schemes is shown in Table 5.

[0152] Table 3 Results of the traditional two-stage stochastic programming model

[0153]

[0154] Table 4 Results of multi-stage stochastic programming model

[0155]

[0156] As can be seen from Table 3, the planning result of the traditional two-stage random programming model is a fixed scheme, which is determined by comprehensively considering the four scenarios and the corresponding probability characteristics. In contrast, the planning scheme obtained by the multi-stage random programming model shown in Table 4 is a decision tree scheme, which takes into account the sequential realization process of wind output uncertainty based on the timing stage. It can be seen from the scenario tree that the four scenarios share a parent node in the first stage, s1 and s2 have the same parent node in the second stage, and s3 and s4 have the same parent node in the second stage. After the model performs the first stage planning, it observes the specific realization of the uncertainty information in the second stage, and then the information is planned in the second stage, and the same is true for the third stage. As can be seen from the table, the four scenarios in the first stage have the same planning scheme, and all plan the same two wind farms; in the second stage, the scenarios s1 and s2 have the same planning scheme, planning two lines and one thermal power unit, and s3 and s4 have the same planning scheme, planning one wind farm. The above planning schemes meet the "unexpectedness" of the multi-stage random programming model.

[0157] Table 5 Planning results cost

[0158]

[0159] Table 5 shows the planning costs of the two planning models. It can be seen that the total cost of the traditional two-stage random planning model is 391.6M$, of which the planning cost is 218.6M$ and the operating cost is 173M$; the total cost of the multi-stage random planning model is 448.7M$, of which the planning cost is 279.2M$ and the operating cost is 169.5M$; the total cost of the multi-stage random planning model is 12.7% lower than that of the traditional two-stage random planning model, of which the planning cost is reduced by 21.7%, and the operating cost is not much different. The analysis results of this paper are verified: the multi-stage random planning model has better economy than the traditional two-stage model.

[0160] 4) Analysis of the effectiveness of planning schemes considering dynamic frequency safety constraints

[0161] In order to verify the effectiveness of dynamic frequency security constraints in multi-stage models, this section sets up a multi-stage random programming model of the transmission network with dynamic frequency security constraints for comparison. In terms of frequency constraints, the model considers the failure of a 60Mw (G3) unit in the system as instantaneous unbalanced power, the maximum frequency deviation is limited to less than 0.4Hz, the maximum frequency change rate is 0.5Hz / s, and the reference frequency is set to 50Hz. The models without dynamic frequency security constraints and with frequency security constraints are denoted as wo / FC and w / FC, respectively. Table 6 shows the planning scheme with dynamic frequency security constraints, Table 7 gives the comprehensive cost comparison of the two planning schemes, and Table 8 lists the frequency security indicators of the system after disturbances in the two models.

[0162] Table 6 Results of w / FC model planning model

[0163]

[0164] Table 7 Comparison of planning results and costs

[0165]

[0166] By comparing Table 4 and Table 6, it can be seen that after adding the dynamic frequency security constraint, in order to maintain the stability of the system frequency after planning, more thermal power units need to be planned to provide inertial response and primary frequency regulation response. As can be seen from Table 7, the planning costs of the two models are 219.2M$ and 350M$ respectively. The model with the frequency security constraint has an increased planning cost due to the additional planning of units. The operating cost increased from 172.5M$ to 183M$. This result shows that after adding the frequency security constraint, in addition to adding units, the output of thermal power units should also be increased to provide inertial response and primary frequency regulation response reserves.

[0167] Table 8 Frequency safety index of the system after disturbance in two models

[0168]

[0169] The frequency trajectories of the two models in each scenario under the power imbalance of 60Mw are shown in Figure 8. It can be seen that the planned system has better frequency characteristics after adding the frequency constraint. As can be seen from Table 8, the dynamic frequency security indicators of the planning results without adding the frequency security constraint model in all scenarios do not meet the set requirements, while the dynamic frequency security indicators of the planning results with the frequency security constraint model are within the set limits. The above analysis shows that incorporating the dynamic frequency security constraint into the multi-stage transmission network stochastic programming model does limit the frequency regulation of the planned system, and verifies the effectiveness of the constraint in the multi-stage stochastic programming model of the transmission network.

[0170] 5) Analysis of the effectiveness of the PHA algorithm

[0171] In order to verify the accuracy of the PHA algorithm, this section will compare the direct use of commercial solvers to solve the multi-stage model with the use of the PHA algorithm. The cost results of the planning schemes obtained by the two solving methods are shown in Table 9.

[0172] Table 9 Planning cost results of two solution methods

[0173]

[0174] From the cost comparison in Table 9, it can be seen that the error between the total cost obtained by PHA algorithm and the total cost solved directly is 0.94%, which is within an acceptable range.

[0175] The scale of the scenario tree has a great impact on the solution rate of the multi-stage stochastic programming model. The expansion of the scenario tree scale means the increase of scenarios and planning cycles, and the complexity of scenario sub-problems is also increasing. These factors make the multi-stage model more difficult to solve than the traditional model. To this end, this section sets up four different scenarios based on the 6-node transmission network stochastic programming model to analyze the impact of the scenario tree scale. The four scenarios are as follows:

[0176] Case 1: The scene tree has 5 stages and 16 scenes, and each parent node in the scene tree has 2 child nodes;

[0177] Case 2: The scene tree scale is 4 stages and 27 scenes, and each parent node in the scene tree has 3 child nodes;

[0178] Case 3: The scene tree scale is 4 stages and 64 scenes, and each parent node in the scene tree has 4 child nodes;

[0179] Case 4: The scene tree scale is 5 stages and 81 scenes, and each parent node in the scene tree has 3 child nodes.

[0180] The multi-stage models of the above four situations are solved directly using commercial solvers and using the PHA algorithm. The solution time of the four situations is shown in Table 8:

[0181] Table 10 Solution time of multi-stage model of scene tree of different scales

[0182] Solution Scenario 1 Scenario 2 Scenario 3 Scenario 4 Direct solution 77s 225s 493s 934s PHA algorithm solution 44s 81s 232s 443s

[0183] Fig. 9 The solution time comparison between directly solving the multi-stage model and using the PHA algorithm to solve the multi-stage model shows that as the scale of the scene tree increases, the solution time of directly solving the multi-stage model increases significantly. When using the PHA algorithm to solve, when the scale of the scene tree is small, the effect of the PHA algorithm is not obvious. This is due to the time consumption caused by the communication of the solver when solving the scene sub-problem in each PHA iteration process. When the scale of the scene tree becomes larger, the total solution time increases, and the communication time caused by the solver is no longer important, so the effect of the PHA algorithm becomes more obvious.

[0184] B. 39 and 118 node systems

[0185] 1) Test system and parameter settings

[0186] In this section, the proposed model and solution method are further implemented in the modified New England IEEE 39-node transmission system, which consists of 39 nodes, 46 branches, 10 generators, node 31 is a balancing node, and the system base power is 100MW. There are 10 candidate thermal power units, 12 wind farms, and 12 candidate photovoltaic power stations. The planning period is expected to be divided into three stages, each stage is planned based on a typical day and the period is set to one year, and the load increases by 20% in each stage. At the same time, a 3-stage 4-scenario tree is used to characterize the uncertainty of wind and light output. The upper limit of new lines in the original transmission corridor is 2.

[0187] 2) Effectiveness Analysis

[0188] Table 11 compares the planning costs of the two models. The total cost of the multi-stage model is 4980.9M$, of which the planning cost and operating cost are 1772.1M$ and 3208.8M$ respectively. The total cost of the traditional model is 5590.9M$, of which the planning cost and operating cost are 2118.8M$ and 3472.1M$ respectively. It can be seen that the planning cost of the multi-stage model is reduced by 16.3% compared with the traditional model, and the operating cost is reduced by 7.5% compared with the traditional model. The conclusions obtained through the above analysis are basically consistent with the 6-node example in the previous section, proving that the multi-stage random programming model has better economy than the traditional two-stage random programming model.

[0189] Table 11 Comparison of planning results and costs

[0190]

[0191] Fig.10 The figure shows the installed capacity comparison of the four scenarios of the wo / FC and w / FC models. It can be seen from the figure that after adding the frequency security constraint, the planned installed capacity of thermal power units increases, and the planned installed capacity of new energy units decreases. This result shows that after adding the frequency security constraint, the output of thermal power units should be increased to provide inertial response and primary frequency regulation response reserve.

[0192] Table 12 Frequency safety index of the system after disturbance in two models

[0193]

[0194] Table 12 shows the frequency safety factor of the system after the disturbance of the two models when the failure of the largest capacity unit G10 (550MW) in the system is considered as the instantaneous unbalanced power. It can be seen that the dynamic frequency safety index of the planning results without adding the frequency safety constraint model in all scenarios did not meet the set requirements, while the dynamic frequency safety index of the planning results with the frequency safety constraint model was within the set limits. The above analysis shows that incorporating the dynamic frequency safety constraint into the multi-stage transmission network stochastic programming model does limit the frequency regulation of the planned system, verifying the effectiveness of the constraint in the multi-stage stochastic programming model of the transmission network.

[0195] Table 13 Comparison of results of two solution methods

[0196] Direct solution PHA Solution Solution time(s) 6537 1451 Planning cost (M$) 4988.2 5157.8

[0197] Table 13 compares the results of directly calling the commercial solver to solve the model and using the PHA algorithm to solve the model. It can be seen from the table that for the multi-stage model with dynamic frequency security constraints, the solution time using the PHA algorithm is 77.8% shorter than the direct solution, and the relative error of the total cost obtained by the two solution methods is 3.4%. Compared with the improvement in the solution rate, it can be considered that the error is within an acceptable range. In summary, it can be shown that the PHA algorithm can effectively solve the multi-stage stochastic programming model of the transmission network based on dynamic frequency security constraints proposed in this chapter.

[0198] Fig.11 The convergence error diagram of different iteration numbers of the model solved by the PHA algorithm shows that in the initial iteration, the convergence error is as high as 0.9. As the iteration continues, the convergence error gradually decreases. Finally, after 19 iterations, the preset convergence accuracy is reached. The algorithm jumps out of the iterative loop and solves the planning scheme of the model.

Claims

1. A multi-stage stochastic programming method and system for a transmission network considering dynamic frequency security constraints, characterized in that: The model decomposition method based on the piecewise linear fitting method of the maximum affine function form and the stepwise hedging algorithm is as follows: S1. Establish a multi-machine system frequency response model for the dynamic frequency response of the new energy power system, derive the frequency security index expression, and propose a piecewise linear fitting method based on the maximum affine function form; S2. Establish multi-stage unexpected constraints and propose a multi-stage stochastic programming model for the transmission network considering dynamic frequency security constraints; S3. Design a model decomposition method based on the stepwise hedging algorithm.

2. A multi-stage stochastic programming method and system for a power transmission network considering dynamic frequency security constraints according to claim 1, characterized in that: Construction of dynamic frequency security indicators for new energy power grids: (1) Frequency response of new energy units New energy units are connected to the power system through power electronic equipment, and they do not inherently provide inertia for the power system. New energy units can use corresponding control strategies to simulate the inertial response and primary frequency regulation response process similar to synchronous units, and participate in the frequency regulation process of the power system, as shown in equations (1) and (2): Among them, ΔP w (t) and ΔP w (s) is the frequency response power in the time domain and frequency domain, Δf(t) and Δf(s) are the frequency deviations in the time domain and frequency domain, D w is the droop coefficient of the new energy unit, T w H is the speed regulator time response coefficient w is the inertia constant of the new energy unit. (2) Frequency response model of multi-machine system On the basis of the low-order SFR model, the frequency regulation system of renewable energy units is comprehensively considered and the MMSFR model is established. This model consists of three parts: inertia and load frequency control, synchronous generator frequency control and renewable energy unit frequency control. F Hτ 、T Rτ , R Gτ , K mτ are the reheat coefficient, reheat time constant, mechanical power gain coefficient and governor speed constant of the τth synchronous machine, respectively. E is the comprehensive inertia of the system, D E is the damping coefficient of the system. ΔP w and ΔP g are the primary frequency regulation powers of synchronous generator sets and new energy generator sets in response to frequency deviations. Vw and T Vw are the virtual regulation coefficient of the w-th new energy station and the time constant of the inverter response respectively. (3) Equivalent Aggregate Multi-machine System Frequency Response Model The MMSFR model is aggregated into the AMSFR model using the parameter equivalent aggregation method. This model is a low-order linear transfer function model. The equivalent inertia constant of the AMSFR system is shown in formula (3): in, and are the rated power of synchronous generator set and new energy generator set, H E , H g and H w They are the equivalent inertia constant, the inertia constant of the synchronous generator set and the new energy generator set respectively. The equivalent parameters of the AMSFR model are shown in equations (4), (5) and (6): Where: Ω g and Ω w are synchronous generator sets and new energy generator sets respectively, R E , R g and R w They are the equivalent droop coefficient, the droop coefficient of the synchronous generator set, and the droop coefficient of the new energy unit; F E 、F Hg and F Hw are the system equivalent work ratio, the synchronous generator set and the new energy generator set work ratio, T RE , T Rg and T Rw They are the equivalent reheat time constant, the reheat time constant of the synchronous generator set and the new energy unit respectively. According to the AMSFR model transfer function block diagram, the step disturbance ΔP is derived L The analytical expressions of the post-frequency dynamics are shown in equations (7) and (8): The frequency response model is subjected to Laplace inverse transform, and the time domain analytical expression of the frequency deviation of the AMSFR model is derived, as shown in equations (9) and (10): According to the time domain expression of AMSFR model, the maximum frequency change rate RoCoF and the lowest frequency point Δf of frequency safety index are derived. naclir and steady-state frequency Δf ss The corresponding constraints are established as shown in equations (11), (12) and (13): (4) Frequency safety constraint linearization The frequency minimum point expression is given by (H E ,R E ,F E ,T RE ,D), which cannot be directly processed by the piecewise linearization method. The system damping is usually strictly limited to a small range, so D can be determined as a constant; the frequency deviation is less sensitive to the reheating time constant, so T RE can be assumed to be the same value without loss of accuracy. The variables in the frequency safety expression are further simplified to w = (H E ,R E ,F E ). The frequency minimum point function can be approximately regarded as point-wise convex within the variable range used in this paper. It is linearized using the MA-PWL method and constructed as a maximum affine function, as shown in formula (14): in, b i is the coefficient of the i-th fitting hyperplane; m is the number of hyperplanes used in the calculation. The least squares form is used to express the fitting error between the true value and the fitted value. The optimal fitting coefficient of the point-by-point function evaluation of the fixed evaluation point is determined by optimizing the minimum fitting error, as shown in formula (15): Since the fitting error optimization problem contains the "min-max" operator, it is difficult for current commercial solvers to solve it. New binary variables and continuous variables are introduced to reconstruct the maximum affine function term into a series of linear inequalities to eliminate the "max" operator in the objective function and reformulate the complete optimization problem in equation (15) as shown in equations (16) and (17): After solving the above problem with a commercial solver to obtain the optimal parameters of the fitted hyperplane, the linearization of the frequency security constraint is completed, and the lowest point constraint is integrated into the transmission network planning model by adding a set of linear inequalities. Among the other two constraints in the dynamic frequency safety model, the maximum frequency change rate constraint (11) can be transformed into a linear constraint. The quasi-steady-state constraint (12) contains a nonlinear term of the product of continuous variables. Based on the previous assumption that the equivalent damping constant of the system can be determined as a constant, constraint (12) can also be regarded as a linear constraint.

3. A multi-stage stochastic programming method and system for a power transmission network considering dynamic frequency security constraints according to claim 1, characterized in that: Multi-stage stochastic programming model of transmission network considering dynamic frequency security constraints: (1) Model of renewable energy output scenario tree This paper constructs a multi-stage scenario tree for renewable energy output. Each node in the scenario tree is the realization of its output uncertainty. The probability of each node refers to the conditional probability that all uncertainties before the current stage are revealed. The first node of the scenario tree is the initial state, and the parent node is connected to the child node through branches at each stage, indicating the realization from one stage to another. Each path from the first node to the last node of the scenario tree represents a multi-stage scenario, which includes the realization process of random variables in multiple stages. Assuming that the realization of each stage maintains the independence between stages, a multi-scenario set of new energy output is formed in each stage through the Latin hypercube sampling method, and the backward scenario reduction method is used to reduce the scenarios to obtain a typical daily scenario set of new energy output in each stage. The three-stage scenario tree generation process is as follows: In the first stage, it is assumed that a certain new energy output information is used as the initial node of the scenario tree, so the probability of this scenario is 1, represented by node 0, and the new energy output information of this node is The new energy output information of the second stage is determined by the initial node 0, and three possible scenarios are generated, represented by nodes 1, 2 and 3. The new energy output information of these three scenarios is and The probabilities are and The new energy output information of the third stage is determined by the nodes of the second stage. The three nodes each generate three uncertainty scenarios, represented by nodes 4-12. The wind and solar output information of these nine scenarios are The probabilities are In this way, there are 9 paths from the root node to the end node, that is, the scene tree has 9 random scenes s1-s9, and the probabilities of the 9 scenes are shown in formula (18): (2) Unexpected constraints Due to the random scenarios of uncertainty in the new energy power system 1 …ξ y It is gradually realized as the time series progresses, not generated at the same time. Therefore, the corresponding planning decisions should also be realized step by step, that is, observation (ξ 1 )→Planning decision (X1)→Observation (ξ 2 )→Planning decision (X2)→…→Observation (ξ y )→Planning Decision (X v ) process. From the above steps, it can be seen that planning decision variables cannot change with random factors that have not been realized in the future, otherwise the model will not be able to obtain feasible planning results to deal with future uncertainties. For this reason, "unexpectedness" needs to be considered in the multi-stage stochastic planning process. Take the scene tree shown in Figure 3 as an example to illustrate the unexpected. In this three-stage scene tree, taking scenes s1, s2 and s3 as examples, the uncertainty information that can be obtained for these three scenes in the third stage is and In the second stage, the uncertainty information available for the three scenarios is That is to say, in the second stage, these three scenarios are indistinguishable, and they will not evolve into different scenarios until the third stage. Therefore, the decision-making of these three scenarios in the second stage is This is the unexpected nature of the scene tree decision process. Next, the unexpected constraints of the three stages of the scenario tree are constructed. For the first stage, its decision variables do not depend on the decisions of the second and third stages. The conditions that all scenarios need to meet are shown in formula (19): For the second stage, its decision variables do not depend on the decision of the third stage. The conditions that all scenarios need to meet are shown in formula (20): Through the above analysis, the unexpected constraint construction of the three-stage scene tree shown in Figure 3 has been completed. Further, based on the unexpected conditions, for s scenes, the unexpected constraints of the scene tree of y stages are shown in formula (21): Among them, A y is a scene cluster, representing the unexpected subspace of the scene, which is defined as: at each stage y, the scene set Ω s can be divided into a finite number of disjoint subsets, in any of which the various scenarios are considered indistinguishable in terms of decision making. The set of all scenario clusters at stage y is denoted as A y , as shown in formula (22): (3) Multi-stage stochastic programming model for transmission network considering dynamic frequency security constraints In the multi-stage model, planning decisions must be able to adjust in time with the increase in available information in the uncertain scenario, so the representation of the uncertain information structure must be an important part of the planning decision representation. Based on the traditional transmission network stochastic planning model, the planning variables are associated with random factors. This section establishes a transmission network stochastic planning model that considers dynamic frequency security. In addition to configuring traditional thermal power units, unit planning also considers new energy units. a) Objective Function The objective function is to minimize the total cost, including investment cost and operating cost, as shown in formula (23): Among them, in the formula: Ω i is the node set, Ω s is the scene set, Ω t is the time period set, Ω g is the set of built generators, Ω ′ g is the set of generators to be built, Ω l is the set of transmission lines, Ω w A collection of new energy units to be built; I l,s,y ,I g,s,y and I w,s,y are the decision variables for the construction of transmission lines, generators and new energy units respectively; C l , C g and C w are the investment and construction cost coefficients of transmission lines, generator sets, and new energy units respectively; p g,s,y,t is the power generation capacity of conventional units; is the load shedding amount; It represents the operating cost coefficient of conventional units; The starting cost of the unit; is the load shedding cost coefficient; η s is the probability of scene s; T h is the typical number of days in the planning stage; yr is the planning period, and d is the discount rate. b) Investment decision constraints Once the equipment is put into construction at a certain stage, it will be considered as constructed in the following stages. The constraints are shown in equations (25)-(28): I l,s,y ,I g,s,y ,I w,s,y ∈{0,1} (28) c) DC power flow constraints The model uses DC power flow constraints and ignores voltage changes and line losses, where M is a sufficiently large integer, as shown in equations (29)-(32): Among them, f l,s,y,t is the power flow of line l in stage y during period t of scenario s; b l is the susceptance of line l; θ i,s,y,t and θ j,s,y,t is the node voltage phase angle value of nodes i and j at both ends of line l; and are the upper and lower limits of the power flow capacity of line l respectively. d) Constraints on new energy output The output constraint of new energy is shown in formula (33): Among them, p w,s,t,y is the output power of the new energy unit in stage y during period t in scenario s; α w,s,t is the normalized new energy output power. e) Power balance constraints The power balance constraint is shown in formula (34): Among them, L(i) to and L(i) from are the line sets at the power transmission and receiving ends of node i respectively; D i,s,t is the load power demand of node i. f) Generator output upper and lower limit constraints The upper and lower limits of the generator output are shown in equations (35) and (36): in, and are the maximum and minimum output power of generator set g respectively. g) Upper and lower limit constraints of unit climbing The upper and lower limit constraints of the unit climbing are shown in equations (37) and (38): Among them, δ g,up and δ g,down They are the ramp-up rate and ramp-down rate of generator set g respectively. i) Frequency safety constraints Write out the maximum frequency change rate constraint and steady-state frequency constraint expressions related to the frequency safety constraint. The equivalent frequency parameter expressions are shown in equations (39)-(42): The frequency minimum point constraint after linearization is shown in equations (43)-(45): in, are m-1 linear inequality constraints, and They are respectively the system inertia constant, equivalent droop coefficient and equivalent work ratio coefficient after equivalent stage y in scenario s. j) Unexpected constraints The unexpected constraints are shown in equations (46)-(48):

4. A multi-stage stochastic programming method and system for a power transmission network considering dynamic frequency security constraints according to claim 1, characterized in that: Solution of multi-stage stochastic programming model: (1) Multi-stage stochastic programming model of transmission network with unexpected constraints The matrix form of the multi-stage stochastic programming model of the transmission network considering unexpected constraints can be expressed as: Where: X is the decision variable for the construction status of power grid planning equipment; P s is the continuous variable of system operation, including unit output, voltage phase angle, etc.; X FR are frequency-related variables, such as inertia constant, regulation coefficient, etc.; f(·) is the economic operation of the power grid; A is the feasible domain of constraints related to power grid operation; T and B T is the coefficient matrix related to the equipment construction constraints between different stages, C T and D T is the constant coefficient matrix of the frequency-safe convex feasible region constraint, E T and F T It is the relationship coefficient matrix between frequency variables and equipment investment and construction status decision variables. It can be seen from the model that due to the existence of unexpected constraints, the decision variables between different scenarios are coupled. Therefore, based on the PHA algorithm, the multi-stage model is reconstructed into an augmented Lagrangian form, and the unexpected constraints are converted into equality constraints, which relaxes the unexpectedness of the original problem and realizes the scenario-based decoupling of the model, as shown in formula (50): Where W is the Lagrange multiplier, r is the penalty factor, Defined as an aggregation operator, it is a conditional expectation operator based on the uncertain information structure of the scene tree and the probability weight of the scene. It aggregates the possible different responses that the decision may make to the scenes in any scene cluster into a single compromise response to the scene cluster, as shown in formula (51): (2) PHA algorithm process By selecting appropriate multipliers and penalties, the deviation between the planning scheme of the scenario sub-problem at each stage and the aggregate planning scheme represented by the aggregation operator is continuously reduced iteratively to restore the unexpectedness of the problem after decoupling and gradually approach the optimal solution of the original model. The implementation process of the PHA algorithm is as follows: PHA Algorithm

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