Water supply pump station non-convex optimization scheduling method based on group contract endpoint method

Abstract

The invention discloses a water supply pump station non-convex optimization scheduling method based on a group contract endpoint method, and the method comprises the steps: building an urban water supply pump station optimization scheduling model, enabling the model input to be a decision variable for guaranteeing an annual water distribution mode and a water demand and a water supply pump station optimization scheduling constraint condition, and enabling the model output to be a water supply system pump station energy-saving objective function; according to the optimal scheduling model of the urban water supply pump station, establishing a new objective function convex optimization model considering the pipeline head loss, and establishing a new objective function homotopy equation; and according to the new objective function homotopy equation and the water supply pump station optimal scheduling constraint condition, obtaining an optimal solution of the urban water supply pump station optimal scheduling model on the homotopy path. Compared with the prior art, the constraint form is adjusted through homotopy parameters, the method is suitable for processing complex constraint scenes, and meanwhile the solution stability is improved through the interior point method.

Description

Technical Field

[0001] The present invention relates to the technical field of optimization of water supply pump stations in urban water supply systems, and in particular to a non-convex optimization scheduling method and system for urban daily water supply pump stations. Background Art

[0002] Currently, existing technologies have constructed urban water supply pump station optimization scheduling models based on meta-heuristic methods, including particle swarm optimization algorithm, ant colony optimization algorithm, genetic algorithm, NSGA-II algorithm and other methods, to provide support for urban water supply optimization scheduling; however, meta-heuristic methods may reduce the robustness of water supply optimization problems. In certain circumstances, small perturbations in model parameters may lead to different local optimal solutions, resulting in different solution plans, and reducing the reliability of the optimization model.

[0003] In view of the above problems, how to realize a water supply scheduling optimization algorithm with a deterministic convergence criterion and a fast solution speed is a technical problem to be solved urgently in the present invention.

[0004] Non-convex optimization models refer to optimization problems in which the objective function or constraints do not satisfy convexity conditions. Convexity is a key concept in mathematical optimization, as convex optimization problems possess many desirable properties, such as the fact that local optimal solutions are also global optimal solutions. However, not all practical problems can be described using convex optimization models, and therefore non-convex optimization models are widely used in many fields.

[0005] Solving non-convex optimization problems is often more difficult than solving convex ones. This is because non-convex problems may have multiple local optimal solutions, and these local optimal solutions are not necessarily equal to the global optimal solution. Therefore, solving non-convex optimization problems requires the use of specialized algorithms and techniques. For example, metaheuristic algorithms, such as particle swarm optimization and swarm optimization, mimic natural phenomena to find solutions.

[0006] In machine learning, many deep learning model training problems are non-convex problems. In signal processing, many optimal control problems are also non-convex. In operations research, many production planning and resource allocation problems are also non-convex. Summary of the Invention

[0007] The present invention proposes a non-convex optimization scheduling method for water supply pumping stations based on the combined homotopy interior point method. The nonlinear characteristics of the optimization scheduling of urban water supply pumping stations are taken into account by using the homotopy method to transform the non-convex optimization problem of water supply pumping stations into an equivalent convex optimization problem, and then the problem is solved by combining the primal-dual interior point algorithm.

[0008] The present invention proposes a non-convex optimization scheduling method for a water supply pump station based on a combined homotopy interior point method, comprising the following steps:

[0009] Establish an optimal scheduling model for urban water supply pumping stations. The inputs of the model are decision variables for ensuring the annual water volume allocation pattern and water demand, as well as the optimal scheduling constraints of the water supply pumping stations. The output of the model is the energy-saving objective function of the pumping stations in the water supply system.

[0010] Based on the optimal scheduling model for urban water supply pumping stations, establish a convex optimization model and a homotopy equation for the energy-saving objective function of the new water supply system pumping stations considering pipeline head loss.

[0011] According to the homotopy equation of the energy-saving objective function of the new water supply system pumping stations and the optimal scheduling constraints of the water supply pumping stations, obtain the optimal solution of the optimal scheduling model for urban water supply pumping stations on the homotopy path.

[0012] In some embodiments, the method further includes the following steps:

[0013] Conduct modeling of the applicable condition model, where the applicable condition model includes a water transfer and water use plan model, a time-of-use electricity price policy model, a pump operation parameter and characteristic curve model, an engineering water conveyance capacity model, and a regulating pond and water treatment plant purification capacity model.

[0014] Combine one or more of the applicable condition models to analyze the optimal solution of the optimal scheduling model for urban water supply pumping stations.

[0015] In some embodiments, the energy-saving objective function of the water supply system pumping stations is shown as the following formula:

[0016] <(

[0017] The pipeline head loss equation is shown as the following formula:

[0018]

[0019] In the formula, h f，hw represents the pipeline head loss, C pipe represents the Hazen-Williams coefficient, D pipe represents the pipeline diameter, L pipe represents the pipeline length, F price represents the operating cost of the pumping stations in the water supply system, T represents the total number of time periods within the scheduling cycle; M represents the total number of pumps in the water supply system, PR t represents the power supply price in the t-th time period, CV t represents the conversion coefficient between energy consumption and electricity price in the t-th time period, Q m,t 、H m,t respectively represent the flow rate and head of the m-th pump in the t-th time period, η m,t represents the operating efficiency of the m-th pump in the t-th time period.

[0020] In some embodiments, the convex optimization model of the energy-saving objective function of the new water supply system pumping stations is shown as the following formula:

[0021]

[0022] Wherein, H d，m，t and H u，m，t respectively represent the downstream water level and the upstream water level corresponding to the m-th water pump in the t-th time period, corresponding to the relevant regulating pond or reservoir respectively, and h f，hw represents the pipeline head loss;

[0023] The homotopy equation of the energy-saving objective function of the new water supply system pumping station is shown as follows:

[0024]

[0025] Wherein, θ is the homotopy parameter, and H m,0 represents the linear approximation term of the head difference between the upstream and downstream of the m-th water pump, and H m,t represents the head difference between the upstream and downstream in the t-th time period, and η m,0 represents the linear approximation term of the water pump operation efficiency, and η m,t represents the water pump operation efficiency in the t-th time period. Within the normal operation range, the water pump operation efficiency is a function of the water supply flow rate and the head, and h f,hw,m,0 represents the linear approximation term of the pipeline head loss homotopy equation corresponding to the m-th water pump, and h f,hw,m,t represents the non-linear term of the pipeline head loss homotopy equation corresponding to the m-th water pump. According to formulas (9), (10), and (11), h in formula (12) is obtained f,hw,m,t .

[0026] In some embodiments, the optimization scheduling constraint conditions of the water supply pumping station include linear approximation constraint conditions and non-linear term constraint conditions.

[0027] In some embodiments, the linear approximation constraint conditions further include the water volume balance constraint of the regulating pond or reservoir, the engineering scale constraints of the reservoir, the regulating pond and the water pipeline, and the water pump operation constraints.

[0028] In some embodiments, establishing the non-linear term further includes the water level-storage capacity relationship constraint, the water pump operation power constraint, and the flow rate constraint, performing convex optimization processing on the non-linear term, and establishing a non-linear term homotopy equation, including:

[0029] i. The convex optimization processing result of the water level-storage capacity relationship constraint is shown as follows:

[0030] HR n,t (t, θ) = (1 - θ)(α Γ VR n,t + β Γ ) + θVR n,t

[0031] where α Γ and β Γ represent approximate parameters of the water level - storage capacity relationship;

[0032] ii. The convex optimization result of the pump operation power constraint is shown as follows:

[0033] P pump (t - θ)=(1 - θ)ρgQ m,t (H m,0 +h f,hw,m,0 ) / η m,0

[0034] +θρgQ m,t (H m,t +h f,hw,m,t ) / η m,t

[0035] where P pump (t - θ) represents the homotopy equation expression of the pump operation efficiency, ρ represents the density of water; g represents the acceleration of gravity, Q m,t (H m,0 +h f,hw,m,0 ) represents the output power of the pump, η m,t represents the input efficiency of the m - th pump in the t - th time period;

[0036] iii. The convex optimization result of the flow rate constraint is shown as follows:

[0037]

[0038] where Q n,i (t) represents the inflow of the n - th reservoir in the t - th time period, and Q n,o (t) represents the outflow of the n - th reservoir in the t - th time period.

[0039] In some embodiments, the optimization solution of the urban water supply pump station optimization scheduling model on the homotopy path is obtained according to the homotopy equation of the new water supply system pump station energy - saving objective function and the water supply pump station optimization scheduling constraint conditions. The water supply pump station optimization scheduling constraint conditions further include: starting from the initial point, using the interior - point method to search for the corresponding solution within the feasible region formed by the homotopy equation of the new water supply system pump station energy - saving objective function and the linear - term homotopy equation in an interior - point manner, and approaching the optimal solution by adjusting the relaxation factor, and iteratively completing the solution of the optimization problem on the homotopy path.

[0040] Compared with the prior art, the beneficial effects and advantages of the present invention are as follows:

[0041] 1. Introduce homotopy parameters for the model and non - linear term constraints, and gradually transform the non - convex problem into a series of sub - problems with simpler structures (such as convex ones); the homotopy method introduces a "smoothing effect" through parameters, temporarily eliminating non - convexity, making the objective function convex or quasi - convex in the intermediate steps, thereby reducing the risk that the optimization scheduling problem to be solved falls into a local optimum;

[0042] 2. Continue to use the interior - point method to efficiently solve the interior points of the homotopy equation at each parameter point, track the changes in the solution path, which may avoid local optima, has good path - tracking and global convergence potential, and is conducive to guiding out the global solution of the original non - convex problem when the path is continuous; after combining the interior - point method with the homotopy method, it can gradually approach the feasible solution under non - convex constraints, avoid boundary conflicts, the efficiency of the interior - point method supports multi - step rapid solution, can reduce the difficulty of single - step solution, and can be adaptively adjusted;

[0043] 3) By adjusting the constraint form through homotopy parameters, it is possible to transform non - convex constraints into convex or near - convex forms. This method has strong ability to handle complex constraints, and at the same time improves the solution stability through the interior - point method. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] Figure 1 is the overall flowchart of a non - convex optimization scheduling method for urban water supply pumping stations based on the combined homotopy interior - point method (CHIPM) of the present invention;

[0045] Figure 2 is the technical roadmap of a non - convex optimization scheduling method for water supply pumping stations based on the combined homotopy interior - point method (CHIPM) of the present invention;

[0046] Figure 3 is the technical roadmap of the Yangtze River water diversion model;

[0047] Figure 4 is the characteristic curve graph of the pump in the station;

[0048] Figure 5 is the optimized scheduling operation result of the Yangtze River water diversion mode;

[0049] Figure 6 is the change graph of the water conveyance flow of the pump in the Yangtze River water diversion mode. DETAILED DESCRIPTION OF THE INVENTION

[0050] The present invention will be further described below in conjunction with the drawings and specific embodiments, but it shall not be used as a basis for limiting the present invention.

[0051] As Figure 1 shown, it is the overall process of the convexification solution method for the daily optimization scheduling of urban water supply pumping stations based on the gradient (CHIPM) of the present invention, which specifically includes the following steps:

[0052] Step 1: Problem Transformation; Using the homotopy method, transform the original non-convex optimization problem into an equivalent convex optimization problem. The key step is to define the homotopy equation to ensure the convexity of the objective function and constraints during the homotopy parameterization process, and then solve it using convex optimization algorithms; A numerical technique for solving complex mathematical problems by constructing a continuous deformation path, mainly applied in fields such as optimization and nonlinear equation solving. Its core idea is to connect the original difficult problem with a simple problem through a parameterized path and gradually approach the solution;

[0053] Step 2: Construct the Homotopy Path; The homotopy path is the path connecting the original problem and the simplified problem. Use a parameterized function to represent the homotopy path. The homotopy parameter θ varies from 0 to 1. By gradually changing the value of the homotopy parameter θ, starting from the simplified problem, gradually transition from the simplified problem to the original non-linear equation problem;

[0054] Step 3: Set the Initial Point. By solving the homotopy equation, set θ = 0, and the homotopy equation HM(x, 0) = G(x) = 0 to obtain the simplified problem of the original problem and determine the initial point on the homotopy path, which corresponds to the solution of the simplified problem;

[0055] Step 4: Iterative Solution Using the Primal-Dual Interior Point Method; Starting from the initial point, use the primal-dual interior point method to iteratively solve the optimization problem on the homotopy path. The primal-dual interior point method uses the interior point method to search for solutions within the feasible region and gradually approaches the optimal solution by continuously adjusting the relaxation factor;

[0056] Step 5: Update the Homotopy Parameter; Gradually increase the value of the homotopy parameter θ, gradually transition from the simplified problem to the complex problem, and gradually approach the real situation of the complex problem;

[0057] Step 6: Termination Condition; If the homotopy parameter θ < 1, return to Step 2 and continue the iteration; If the homotopy parameter θ = 1, the corresponding homotopy problem will be exactly the same as the original problem, then the algorithm stops and outputs the calculation result.

[0058] Step 7: Result Analysis; Analyze the final calculation result, check the feasibility of the solution, evaluate the quality of the objective function value, and verify whether the constraint conditions are satisfied, etc.

[0059] Specifically, the homotopy method described in Steps 1 and 3 introduces a continuously varying parameter between the complex problem and the simplified problem, establishes the connection between the complex problem and the simplified problem, and gradually approaches the real solution of the complex problem by gradually updating the relevant homotopy parameters. The basic idea of the homotopy method is as follows: By introducing the affine parameter θ, construct the homotopy mapping HM(x, θ): Ω × [0, 1] → R n , and the homotopy equation is as shown in the formula:

[0060] HM(x, θ) = (1 - θ)G(x) + θF(x), where θ ∈ [0, 1]

[0061] HM(x, 0) = G(x)

[0062] HM(x, 1) = F(x)

[0063] where: the solution x of G(x) = 0 0 is a known solution. Under certain conditions, the homotopy method can be regarded as a path-tracking problem. Starting from the simplified problem HM(x, 0) = 0 and ending with the complex problem HM(x, 1) = 0, a smooth homotopy path curve is determined. By gradually increasing the homotopy parameter, the solution of the complex problem F(x) is sought.

[0064] The primal-dual interior point method in Step 4 is a numerical optimization algorithm for solving convex optimization problems. Its basic idea is to introduce a barrier function, construct penalty terms for the objective function and constraint functions, and iteratively approximate the optimal solution from the interior points of the feasible region by adjusting the parameters of the barrier function.

[0065] The barrier function usually adopts a logarithmic barrier function, expresses the constraint conditions as penalty terms, and introduces dual variables to implement the penalty of the constraints. After introducing the logarithmic barrier function, the convex optimization problem can be transformed into the following form:

[0066]

[0067] s.t. c(x) = 0

[0068] x ≥ 0;

[0069] where: μ is the parameter of the barrier function, and x i are the components of x.

[0070] By introducing dual variables and constructing penalty terms for the constraint conditions, the Lagrangian function is as shown in the equation:

[0071]

[0072] where: μ is the Lagrange multiplier vector.

[0073] The primal-dual interior point method updates the decision variable values and dual variable values by solving a series of linear equations; the iterative steps include operations such as calculating the gradient of the objective function, calculating the Jacobian matrix of the constraint conditions, and solving linear equations. During the iteration process, the parameter μ plays a role in balancing the objective function and the constraint conditions. As the iteration progresses, the value of the parameter μ gradually decreases, accelerating the approximation to the optimal solution. The decrease of the parameter μ also requires the iteration process to be closer to the boundary of the feasible region. The primal-dual interior point method usually starts the iteration from the interior points within the feasible region and gradually approaches the boundary of the feasible region, which helps to find the optimal solution faster.

[0074] Taking a certain pumping station as an example, considering constraints such as the operating efficiency of pumps, the water purification capacity of water plants, and the water conveyance capacity of pipelines, an optimal scheduling model for urban water supply pumping stations is established. The CHIPM non-convex optimization algorithm is used to solve the model to determine the opening and closing conditions, speed ratios, and water conveyance flows of pumps during the 24-hour period within the scheduling cycle. The urban water supply is from the Yangtze River diversion. A daily optimal scheduling model is established to carry out research on the optimal scheduling of urban water supply pumping stations. The specific steps are as follows:

[0075] In Step 1 mentioned above, the establishment of the optimal scheduling model for urban water supply pumping stations is specifically described as follows:

[0076] a) Design the objective function: Taking the energy consumption cost F of the water supply pumping station price as the objective function, whose input is, on the premise of ensuring the annual water volume distribution pattern and water demand, a vector composed of decision variables, such as decision-making methods including peak-shaving water supply, optimizing the matching of pump units in the pumping station, and reasonably regulating the storage capacity of the regulating pond in the pumping station, to seek an energy-saving combined operation plan for the pumping stations in the water supply system, and its output is an evaluation index in scalar form. The evaluation index is the minimization of the objective function to achieve the reduction of operating costs. The objective function is shown as follows:

[0077]

[0078] Furthermore, considering the pipeline head loss for the objective function, a new objective function is obtained. The Hazen-Williams formula is an empirical formula established based on a large number of test data of industrial pipelines with a diameter ≤ 3.66 m and is applicable to water conveyance pipelines at normal temperature. In this step, the Hazen-Williams formula is used to establish the pipeline head loss equation, as shown below:

[0079]

[0080] In the formula, h f，hw represents the pipeline head loss, C pipe represents the Hazen-Williams coefficient, that is, the pipeline roughness coefficient, which is related to the roughness of the pipe wall surface of different pipe materials, D pipe represents the pipeline diameter, L pipe represents the pipeline length, F price represents the operating cost of the pumping stations in the water supply system; T represents the total number of time periods within the scheduling cycle; M represents the total number of pumps in the water supply system, PR t represents the power supply price in the t-th time period; CV t represents the conversion coefficient between energy consumption and electricity price in the t-th time period, Q m,t , H m,t respectively represent the flow rate and head of the m-th pump in the t-th time period, and η m,t represents the operating efficiency of the m-th pump in the t-th time period.

[0081] b) Conduct constraint design: Considering factors such as the water conveyance capacity of the project and the operation of pumps, the optimized model constraints include water balance constraints, project scale constraints, pump operation constraints, variable non-negativity constraints, etc., to ensure the safe operation of the water supply system. Each constraint is shown as follows:

[0082] i. Water balance constraint of the regulating pond or reservoir, shown as follows:

[0083] VR n,t = VR n,t-1 + QI n,t - QO n,t (3)

[0084] VR n = V relation,n (HR n ) (4)

[0085]

[0086] In the formula, VR n,t is the water storage volume of the nth regulating pond (reservoir) at the tth time, VR n,t-1 is the water storage volume of the nth regulating pond (reservoir) at the (t - 1)th time, QI n,t is the inflow water volume of the nth regulating pond (reservoir) at the tth time, QO n,t is the outflow water volume in the t time period, VR n is the water storage volume of the nth regulating pond (reservoir), V relation,n () is the water level - storage volume relationship function of the nth regulating pond (reservoir), HR n is the water level of the nth regulating pond (reservoir), is the total amount of the outflow water volume during the entire T time period;

[0087] ii. Project scale constraints of the reservoir, regulating pond and water pipeline, shown as follows:

[0088] Q pump,t,k ≤ GC k (6)

[0089] VR min,n,t ≤ VR n,t ≤ VR max,n,t (7)

[0090] CL t,l ≤ CL max,n,t (8)

[0091] In the formula, Q pump,t,k is the water conveyance volume of the kth water pipeline at the tth time, GC k is the maximum water conveyance capacity of the kth water pipeline, VRmin,n,t , VR max,n,t are the minimum water storage and maximum water storage of the nth regulating pond (reservoir) at the tth time, respectively. VR n,t is the water storage of the nth regulating pond (reservoir) at the tth time, CL max,n,t is the maximum water purification capacity of the lth water plant at the tth time, CL t,l is the water purification volume of the lth water plant at the tth time, η m,k represents the operating efficiency of the kth pump, η minm,k represents the minimum operating efficiency of the kth pump, n minm,k represents the maximum operating speed ratio of the kth pump, η minm,k represents the minimum operating speed ratio of the kth pump, n m,k represents the operating speed ratio of the kth pump. m represents the water pump, η m,k represents the operating efficiency of the kth pump, η m,t represents the efficiency of the mth water pump in the tth time period.

[0092] iii. Water pump operation constraints are shown as follows:

[0093] η minm,k < η m,k (9)

[0094] n minm,k < n m,k < n maxm,k (10)

[0095] In the formula, η m,k is the operating efficiency of the kth pump, η minm,k is the minimum operating efficiency of the kth pump, n maxm,k is the maximum operating speed ratio of the kth pump, n minm,k is the minimum operating speed ratio of the kth pump, n m,k is the operating speed ratio of the kth pump;

[0096] The constraint conditions of i to iii are linear approximation terms and do not require convex optimization processing. And non-linear terms,

[0097] Step 2: Conduct convex optimization processing on the convex optimization model of the urban water supply pumping station to obtain the homotopy equation of the objective function:

[0098] a) Conduct convex optimization processing on the new objective function of the urban water supply pumping station optimization model only for non-linear terms:

[0099] i. The convex optimization processing result of the pipeline head loss equation is shown as follows:

[0100] h f,hw (Q, θ) = (1 - θ)h f,hw,0 + θh f,hw,t(11)

[0101] The homotopy equation of the pipeline head loss equation is shown as follows:

[0102]

[0103] In the formula, h f,hw,0 is the linear approximation term of the homotopy equation of the Hazen-Williams pipeline head loss equation, h f,hw,t represents the non-linear term of the homotopy equation of the Hazen-Williams pipeline head loss, θ is the homotopy parameter, C pipe represents the Hazen-Williams coefficient, that is, the pipeline roughness coefficient, which is related to the roughness of the pipe wall surface of different pipe materials, D pipe represents the pipeline diameter, L pipe represents the pipeline length;

[0104] ii. Establish a convex optimization model for the energy-saving objective function of the new water supply system pump station to obtain the homotopy equation of the energy-saving objective function of the new water supply system pump station;

[0105] That is, considering the head difference between upstream and downstream and the pipeline head loss, the convex optimization model of the energy-saving objective function of the new water supply system pump station is shown as follows:

[0106]

[0107] In the formula, H d，m，t and H u，m，t respectively represent the downstream water level and the upstream water level corresponding to the mth pump in the tth period, corresponding to the relevant regulating pool or reservoir respectively;

[0108] Perform homotopy processing on the non-linear term, and the homotopy equation of the new objective function is shown as follows:

[0109]

[0110] In the formula, θ is the homotopy parameter, H m,0 represents the linear approximation term of the head difference between the upstream and downstream of the mth pump, H m,t represents the head difference between the upstream and downstream in the tth period, η m,0 represents the linear approximation term of the pump operation efficiency, η m,t represents the pump operation efficiency in the tth period. Within the normal operation range, the pump operation efficiency is a function of the water supply flow and the head, h f,hw,m,0 represents the linear approximation term of the pipeline head loss homotopy equation corresponding to the mth pump, h f,hw,m,t represents the non-linear term of the pipeline head loss homotopy equation corresponding to the mth pump. According to formulas (9), (10), and (11), h in formula (12) is obtained f,hw,m,t ;

[0111] a) Perform convex optimization with three constraint parameters, including the water level - storage capacity relationship constraint equation, the pump operation power equation, and the flow equation; specifically, the three parameters of water level - storage capacity relationship, pump operation power, and flow are the main objects of the constraint conditions in Step 1. By adjusting the homotopy parameter θ, the solution of the nonlinear equation is obtained to achieve convex optimization. The convex optimization of the constraint conditions in Step 2 is for the equations, that is, the variable equations involved in Step 1. The problem of the nonlinearity of the constraint conditions in Step 1 can be solved through the homotopy method in Step 2;

[0112] i. The convex optimization result of the water level - storage capacity relationship constraint equation is shown as follows:

[0113] HR n,t (t,θ)=(1 - θ)(α Γ VR n,t +β Γ )+θVR n,t (16)

[0114] In the formula, α Γ and β Γ represent the approximate parameters of the water level - storage capacity relationship. The water level of the regulating pond or reservoir strictly increases with the increase of the storage capacity, and α Γ is a positive number;

[0115] ii. The convex optimization result of the pump operation power equation is shown as follows:

[0116]

[0117] In the formula, P pump (t - θ) represents the homotopy equation expression of the pump operation efficiency, ρ represents the density of water; g represents the acceleration due to gravity, Q m,t (H m,0 +h f,hw,m,0 ) represents the output power of the pump, and η m,t represents the input efficiency of the m - th pump in the t - th period;

[0118] iii. The convex optimization result of the flow equation is shown as follows:

[0119]

[0120] In the formula, Q n,i (t) represents the inflow of the n - th reservoir (regulating pond) in the t - th period, and Q n,o (t) represents the outflow of the n - th reservoir (regulating pond) in the t - th period;

[0121] Step 2 transforms the original non-convex optimization problem into an equivalent convex optimization problem using the homotopy method, ensuring the convexity of the objective function and the constraints during the homotopy parameterization process.

[0122] Step 3: Use the primal-dual interior point method to iteratively solve the optimization problem on the homotopy path of the new objective function homotopy equation. The specific process is as follows:

[0123] Construct the homotopy path: The homotopy path is the path connecting the original problem and the simplified problem. The homotopy path is represented by a parameterization function. The homotopy parameter θ varies from 0 to 1. By gradually changing the value of the homotopy parameter θ, starting from the simplified problem, it gradually transitions from the simplified problem to the original non-linear equation problem;

[0124] Set the initial point: Solve the homotopy equation. Let θ = 0, and the homotopy equation HM(x,0) = G(x) = 0 to obtain the simplified problem of the original problem, and determine the initial point on the homotopy path. This point corresponds to the solution of the simplified problem;

[0125] Primal-dual interior point method iterative solution: Starting from the initial point, use the primal-dual interior point method to iteratively solve the optimization problem on the homotopy path. The primal-dual interior point method uses the interior point method to search for solutions within the feasible region and approaches the optimal solution by continuously adjusting the relaxation factor;

[0126] Homotopy parameter update: Gradually increase the value of the homotopy parameter θ, gradually transitioning from the simplified problem to the complex problem, and gradually approaching the real situation of the complex problem. Termination condition: If the homotopy parameter θ < 1, then return and continue the iteration; if the homotopy parameter θ = 1, the corresponding homotopy problem will be exactly the same as the original problem, then the algorithm stops and outputs the calculation result.

[0127] Obtain the optimal solution; The obtained optimal solution is when the homotopy parameter θ = 1 and the barrier parameter μ approaches 0; The introduced barrier function is a logarithmic function (the theoretical explanation has been given in the previous part). Consider the general convex optimization problem:

[0128]

[0129] s.t.c(x) = 0

[0130] x≥0;

[0131] where f(x) is a convex function, c(x) = 0 is a linear equality constraint, and x i is an inequality constraint; Transform x i (i = 1,……m) into a logarithmic barrier term: The objective function becomes:

[0132]

[0133] s.t. c(x) = 0

[0134] x ≥ 0;

[0135] where μ > 0 represents the barrier parameter, controlling the weight of the barrier term, and x i represents the inequality constraint component, such as in equations (6), (7), (8), (9), (10);

[0136] For the primal - dual interior - point method, let the homotopy parameter θ take values in the range [0, 1], the step size be set to 0.05, the maximum number of iterations be set to 200 times, and the tolerance be 1e - 6. Starting from the initial point, the primal - dual interior - point method searches for the solution within the feasible region on the homotopy path of the homotopy equation of the objective function in an interior - point manner and approaches the optimal solution by continuously adjusting the relaxation factor;

[0137] The primal - dual interior - point method updates the decision variable values and dual variable values by solving a series of linear equations; the iterative steps include operations such as calculating the gradient of the objective function, calculating the Jacobian matrix of the constraint conditions, and solving the linear equations. During the iteration process, the parameter μ plays a role in balancing the objective function and the constraint conditions. As the iteration progresses, the value of the parameter μ gradually decreases, accelerating the approach to the optimal solution. The decrease of the parameter μ also requires the iteration process to be closer to the boundary of the feasible region. The primal - dual interior - point method usually starts the iteration from an interior point within the feasible region and gradually approaches the boundary of the feasible region, which helps to find the optimal solution faster;

[0138] Step 4: Build the following applicable - condition models and set the parameters in each model:

[0139] a) Water diversion and water - use plan model, including the water diversion and water - use plan, which includes the external water - source and water - storage reservoir water - diversion plan, and the water - use plan of the water purification plant. In this example, the Yangtze River water - diversion supply mode is considered, as Figure 3 shown in the technical route map of the Yangtze River water - diversion supply model.

[0140] b) Time - of - use electricity price policy model, including that the flat - rate electricity price of the user's peak - valley - flat electricity price policy is based on the local current electricity price level, and the electricity prices in the peak, flat, and valley periods are 0.960, 0.668, and 0.353 yuan / kWh respectively; the peak - valley - flat time periods are peak hours (9:00 to 12:00, 16:00 to 21:00), flat hours (7:00 to 9:00, 12:00 to 16:00, 21:00 to 23:00), and valley hours (23:00 to 7:00).

[0141] c) Pump operation parameter and characteristic curve model, including that the main function of a certain pumping station is to supply water to a certain water purification plant and a certain reservoir. The details of the pump include the pump operation parameters such as rated power, rated head, rated flow, and rated speed, as shown in Table 1, which is the list of pump operation parameters. AsFigure 4 As shown, it is the characteristic curve of the water pump.

[0142] Table 1

[0143]

[0144] d) Engineering water conveyance capacity model, including pipeline design for engineering water conveyance capacity. As shown in Table 2, they are the pipeline design parameters. Combining the water passing data and pipeline design materials, the Hazen-Williams coefficient of the Haicheng-Williams pipeline of the South Main Line is taken as 83.

[0145] Table 2

[0146]

[0147] e) Regulation tank and water purification capacity model of the water plant, including that after the upstream incoming water enters the water purification plant, it usually passes through a pressure stabilizing well or a water distribution well, a reaction tank, a sedimentation tank, a filter tank, enters the clear water tank, and then enters the water supply pumping station. The raw water purification process has little impact on the optimal dispatching research. Ignoring the unpressurized flow part after the raw water enters the water purification plant, the pre-sedimentation tank and the clear water tank of the water purification plant are unified and combined as a regulation tank for treatment. As shown in Table 3, it is the statistical table of the parameters of the raw water plant and the regulation tank of the water purification plant.

[0148] Table 3

[0149]

[0150] Step Five: According to the specific situation, combine one or more applicable condition models in Step Four to conduct the result analysis of the optimal solution of the objective function.

[0151] a) Analysis of the optimal dispatching operation results: For the Yangtze River water diversion water supply mode, based on the annual urban water supply dispatching plan and the short-term water volume prediction results, and based on the peak-valley flat time-of-use electricity price rule, with the lowest energy consumption cost of the water supply pumping station as the optimization goal, solve the water delivery flow, pump opening and closing, and frequency modulation status of the raw water plant in each period. The optimal dispatching operation results of the urban water supply pumping station are as Figure 4 shown. It can be seen from the dispatching calculation results of the Yangtze River water diversion water supply mode on May 21, 2022 that the water delivery flow of a certain village pumping station is 6.9 - 9.3 m3 / s, the water supply flow of a certain village pumping station in the direction of Jinbin Water Plant is 3.8 - 5.4 m3 / s, and the water supply flow of a certain village pumping station in the direction of Beitang Reservoir is 3.1 - 4.1 m 3 / s. The dispatching plan can meet the water use demand of Jinbin Water Plant and the operation demand of Beitang Reservoir. As Figure 5As shown in the figure, it is the optimized scheduling operation result of the Yangtze River water diversion supply mode. By comparing and analyzing the pump speed ratio and the water supply flow process of the actual scheduling and the optimized scheduling, and the comparison between the optimized water supply flow of the pumping station and the actual water supply flow, the optimized scheduling operation result of the Yangtze River water diversion supply mode is shown in (5A) and (5C), and the comparison between the optimized speed ratio and the actual speed ratio of the pumping station is shown in (5B) and (5D). The relationship between the change of the water conveyance volume of the pumping station and the time-of-use electricity price is relatively close. The optimized water conveyance volume of the pumping station fluctuates significantly with the change of the time-of-use electricity price: the water conveyance volume of the pumping station is larger in the valley electricity price stage; the water conveyance volume of the pumping station is smaller in the peak electricity price stage. Considering the influence of the pump operation efficiency limit and the change of the water demand of the waterworks, each regulating pool does not completely store water in the valley electricity price stage and release water in the peak electricity price stage.

[0152] b) Comparative analysis of water supply energy consumption cost.

[0153] Compare and analyze the energy consumption costs of the actual scheduling and the optimized scheduling to verify the superiority of the daily optimized scheduling model of the water supply pumping station. The comparison results are shown in Table 4.

[0154] As shown in Table 4, it is the comparative result of the energy consumption analysis of the optimized scheduling scheme. According to the data in Table 4, the electricity cost consumed by the optimized scheduling of the Yangtze River water diversion supply mode for 24 hours is 30,778.9 yuan, saving 1,088.3 yuan compared with that before optimization, and the energy consumption cost decreases by 3.41%; the energy consumption cost of the optimized scheme decreases by about 12.73% in the time period of 7-12h, and the energy-saving effect is the most significant in the time period of 7-12h. Through the energy consumption analysis and comparison, the daily optimized scheduling scheme proposed in this study realizes the energy-saving optimized operation of the pumping station on the premise of ensuring the water demand and safe operation, and effectively reduces the power consumption cost of the original water pumping station. When adopting the time-of-use electricity price policy, the optimized operation scheme makes full use of the storage capacity of the clear water tank and the electricity price difference, stores as much water as possible in the low valley electricity price period, reduces the operation of the primary raw water pumping station as much as possible in the peak electricity price period, and uses part of the stored water in the clear water tank to meet the water supply requirements of the secondary water purification plant pumping station, realizing the peak-shifting operation of the primary raw water pumping station and effectively reducing the operation electricity cost of the original water pumping station.

[0155] Table 4

[0156]

[0157] C) Calculation stability and speed analysis.

[0158] When solving the optimization problem according to the homotopy path, the water conveyance flow of the pump becomes a function of the homotopy parameter θ and the time period t, and the change of the water conveyance flow of the pump is as Figure 6As shown, (6A) are pumps 1# and 2# of a certain village pumping station, and (6B) are pumps 3# and 4# of the same pumping station. By analyzing the solutions corresponding to the changes in homotopy parameters, the stability of the algorithm during the solution process is evaluated. When the homotopy parameter θ is greater than 0.5, as the homotopy parameter gradually increases, the changes in the corresponding solutions are relatively small, and the CHIPM algorithm demonstrates good stability. Regarding the analysis of the computational speed for the water supply optimization scheduling problem, with a scheduling duration of 24 hours, the solution time corresponding to the Yangtze River water diversion supply mode is 26.6 seconds. The optimization algorithm has a relatively fast computational speed and can quickly converge to the optimal solution. The water supply optimization algorithm proposed in this paper has a fast computational speed and good convergence performance. It improves the deficiencies of the meta-heuristic method, such as being sensitive to the initial value and difficult to handle ill-conditioned conditions, maintains good computational accuracy and efficiency, and has the potential for online application.

[0159] In summary, the homotopy equation interior point method provides a solution framework that balances efficiency and globality for non-convex optimization problems by combining the advantages of path tracking and the interior point method. Its core lies in decomposing complex problems into manageable sub-problems through continuous deformation, efficiently solving each sub-problem using the interior point method, and finally approaching the solution of the original problem along the path. Despite the challenges of computational complexity and path design, it shows significant potential in specific non-convex problems, especially in scenarios where the homotopy mapping can be reasonably constructed.

[0160] The above is a detailed introduction to a non-convex optimization scheduling method for water supply pumping stations based on CHIPM provided by this application. It should be noted that the present invention is not limited to the above method steps and calculation processes. The above specific implementation manners are merely illustrative and not restrictive. Researchers in the field can make formal changes and modifications to the present invention under the inspiration of the present invention without departing from the purpose of the present invention and the scope of protection of the claims, and these all fall within the protection scope of the present invention.

Claims

1. A non-convex optimal scheduling method for a water supply pumping station based on the combined homotopy interior point method, characterized in that, Including: including the following steps: Establish an optimal scheduling model for urban water supply pumping stations. The inputs of the model are decision variables for ensuring the annual water volume allocation pattern and water demand, and the optimal scheduling constraints of the water supply pumping stations. The output of the model is the energy-saving objective function of the water supply system pumping stations. According to the optimal scheduling model for urban water supply pumping stations, establish a convex optimization model and a homotopy equation for the energy-saving objective function of the new water supply system pumping stations considering pipeline head loss. According to the homotopy equation of the energy-saving objective function of the new water supply system pumping stations and the optimal scheduling constraints of the water supply pumping stations, obtain the optimal solution of the optimal scheduling model for urban water supply pumping stations on the homotopy path.

2. The non-convex optimal scheduling method for a water supply pumping station based on the combined homotopy interior point method according to claim 1, wherein This method further includes the following steps: Conduct modeling of the applicable condition model, where the applicable condition model includes a water transfer and water use plan model, a time-of-use electricity price policy model, a pump operation parameter and characteristic curve model, an engineering water conveyance capacity model, and a regulating pond and water treatment plant purification capacity model. Combine the applicable condition model to analyze the optimal solution of the optimal scheduling model for urban water supply pumping stations.

3. A non-convex optimal scheduling method for a water supply pumping station based on a combined homotopy interior point method according to claim 1, characterized in that, The energy-saving objective function of the water supply system pumping stations is shown as follows: The pipeline head loss equation is shown as follows: where h f，hw represents the head loss of the pipeline, C pipe represents the Hazen-Williams coefficient, D pipe represents the pipeline diameter, L pipe represents the pipeline length, F price represents the operating cost of the pump station in the water supply system, T represents the total number of time periods within the scheduling cycle; M represents the total number of pumps in the water supply system, PR t represents the power supply price in the t-th time period, CV t represents the conversion coefficient between energy consumption and electricity price in the t-th time period, Q m,t , H m,t respectively represent the flow rate and head of the m-th pump in the t-th time period, η m,t represents the operating efficiency of the m-th pump in the t-th time period.

4. A non-convex optimal scheduling method for a water supply pumping station based on a combined homotopy interior point method according to claim 1, characterized in that The convex optimization model of the energy-saving objective function of the new water supply system pumping stations is shown as follows: Where, H d，m，t and H u，m，t represent the downstream water level and the upstream water level corresponding to the m-th water pump in the t-th time period, respectively corresponding to the relevant regulating pond or reservoir, and h f，hw represents the pipeline head loss; The homotopy equation of the energy-saving objective function of the new water supply system pumping stations is shown as follows: where θ is the homotopy parameter, H m,0 represents the linear approximation term of the head difference between the upstream and downstream of the m-th pump, H m,t represents the head difference between the upstream and downstream at the t-th time period, η m,0 represents the linear approximation term of the pump operation efficiency, η m,t represents the pump operation efficiency at the t-th time period. Within the normal operation range, the pump operation efficiency is a function of the water delivery flow rate and the head, h f,hw,m,0 represents the linear approximation term of the homotopy equation of the pipeline head loss corresponding to the m-th pump, h f,hw,m,t represents the non-linear term of the homotopy equation of the pipeline head loss corresponding to the m-th pump. According to Eqs. (9), (10), and (11), h in Eq. (12) is obtained f,hw,m,t .

5. A non-convex optimal scheduling method for a water supply pumping station based on the combined homotopy interior point method, characterized in that, The optimal scheduling constraints of the water supply pumping stations include linear approximate constraint terms and non-linear term constraints.

6. A non-convex optimal scheduling method for a water supply pumping station based on a combined homotopy interior point method, characterized in that, The linear approximate constraint terms further include the water volume balance constraint of the regulating pond or reservoir, the engineering scale constraints of the reservoir, regulating pond, and water pipeline, and the pump operation constraints.

7. A non-convex optimal scheduling method for a water supply pumping station based on a combined homotopy interior point method, characterized in that, Establishing the non-linear terms further includes the water level-reservoir capacity relationship constraint, the pump operation power constraint, and the flow rate constraint. Conduct convex optimization processing on the non-linear terms and establish a homotopy equation for the non-linear terms, including: i. The convex optimization processing result of the water level-reservoir capacity relationship constraint is shown as follows: HR n,t (t,θ) = (1 - θ)(α Γ VR n,t +β Γ ) + θVR n,t where α Γ and β Γ represent approximate parameters of the water level - storage capacity relationship; ii. The convex optimization processing result of the pump operation power constraint is shown as follows: P pump (t - θ) = (1 - θ)ρgQ m,t (H m,0 + h f,hw,m,0 ) / η m,0 + θρgQ m,t (H m,t + h f,hw,m,t ) / η m,t Where, P pump (t - θ) represents the homotopy equation expression of the pump operation efficiency, ρ represents the density of water; g represents the acceleration due to gravity, Q m,t (H m,0 + h f,hw,m,0 ) represents the output power of the pump, η m,t represents the input efficiency of the m-th pump at the t-th time period; iii. The convex optimization processing result of the flow rate constraint is shown as follows: where Q n,i (t) represents the inflow of the nth reservoir at the tth time period, and Q n,o (t) represents the outflow of the nth reservoir at the tth time period.

8. A non-convex optimal scheduling method for a water supply pumping station based on the combined homotopy interior point method, characterized in that The optimal scheduling constraints of the water supply pumping stations for obtaining the optimal solution of the optimal scheduling model for urban water supply pumping stations on the homotopy path according to the homotopy equation of the energy-saving objective function of the new water supply system pumping stations and the optimal scheduling constraints of the water supply pumping stations further include: starting from the initial point, using the interior point method in the interior point manner to search for the corresponding solution within the feasible region formed by the homotopy equation of the energy-saving objective function of the new water supply system pumping stations and the homotopy equation of the linear terms, and approaching the optimal solution by adjusting the relaxation factor, and iteratively completing the solution of the optimization problem on the homotopy path.

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