Active power distribution network active and reactive power collaborative optimization method based on shared energy storage

Abstract

The invention relates to an active power distribution network active and reactive collaborative optimization method based on shared energy storage, and the method comprises the following steps: 1, building an objective function: building an active and reactive objective function in an ADN active and reactive collaborative optimization scheduling model; step 2, carrying out solution by adopting an NSGA (Non-dominated Sorting Genetic Algorithm)-II (Non-dominated Sorting Genetic Algorithm) fused with a forward-backward sweep method, namely, establishing a Multi-Objective Non-linear with Sponge-Order Conne Constraints model, namely, a multi-target non-linear programming model based on a second-order cone constraint, and carrying out solution on the multi-target non-linear programming model based on the second-order cone constraint; the method has the advantages of optimizing the network loss, improving the voltage level, improving the electric energy quality and optimizing the power consumption cost.

Description

Technical Field

[0001] The present invention belongs to the technical field of active distribution networks, and in particular relates to a method for collaboratively optimizing active and reactive power of an active distribution network based on shared energy storage. Background Art

[0002] Driven by the "dual carbon" goals, the installed capacity of distributed renewable energy (DRE) continues to rise, which has had a profound impact on the structure and operation mode of the power system. With the increase in DRE penetration, the traditional passive distribution network architecture is gradually transforming to an active distribution network (ADN) to achieve more efficient and flexible energy management and scheduling; however, in this transformation process, ADN optimization and scheduling technology faces many challenges: First, the inherent intermittent and uncertain nature of DRE has made it more difficult to balance supply and demand, increased the system's operating risks and scheduling complexity, and traditional power grid planning and scheduling methods are difficult to effectively cope with such dynamic changes; secondly, after a high proportion of DRE is connected to the grid, problems such as power flow control, voltage stability and reactive power compensation in local networks have become increasingly prominent, especially in the absence of sufficient In the case of energy storage facilities or rapid adjustment means, it is easy to cause power grid safety and stability problems; furthermore, the existing distribution network equipment and technology may not be sufficient to handle the power quality problems caused by the access of a large number of distributed power sources, such as voltage fluctuations, increased harmonics, etc.; in response to the above challenges, the traditional optimal power flow (OPF) algorithm and traditional energy storage need to be adaptively upgraded to solve how to adjust the control variables such as generator output, load demand, and energy storage device status in real time in the ADN environment, so as to achieve economic distribution of active power to reduce operating costs, and realize reactive power optimization to reduce line losses and maintain voltage levels. Therefore, it is very necessary to provide an active and reactive collaborative optimization method for active distribution networks based on shared energy storage that optimizes network losses, improves voltage levels, improves power quality, and optimizes electricity costs. Summary of the invention

[0003] The purpose of the present invention is to overcome the shortcomings of the prior art and to provide a method for collaboratively optimizing active and reactive power of an active distribution network based on shared energy storage, which optimizes network losses, improves voltage levels, improves power quality, and optimizes electricity costs.

[0004] The object of the present invention is achieved by: a method for collaborative optimization of active and reactive power of an active distribution network based on shared energy storage, the method comprising the following steps:

[0005] Step 1: Construct objective function: In the active and reactive power coordinated optimization dispatch model of ADN, construct the objective functions of active power and reactive power;

[0006] Step 2: Use NSGA-ΙΙ combined with forward and back iteration to solve the problem: establish a Multi-ObjectiveNonlinear Programming with Second-Order Cone Constraints model, that is, a multi-objective nonlinear programming model based on second-order cone constraints, and solve it.

[0007] The objective function constructed in step 1 is specifically as follows: In the active and reactive power coordinated optimization dispatch model of ADN, the active power goal is to minimize the daily dispatch cost, while the reactive power goal is to minimize the total active power loss of the branch: Where: F is the daily scheduling cost; is the amount of electricity exchanged between ADN and the main network at time t. When ADN buys electricity from the main grid, when ADN sells electricity; The time-of-use electricity purchase and sale price for the main grid, hour The electricity price for the main grid; D * are the output and cost of distributed generation at time t, respectively, where *∈(WT,PV,GT,SES); P loss is the total active power loss of the branch; r ij 、z ij , l ij They are the resistance of the ij branch, the reactance of the ij branch and the square of the current amplitude of the ij branch respectively.

[0008] The specific solution of using NSGA-II integrated with forward-backward substitution method in step 2 is as follows: the established model is Multi-Objective Nonlinear Programming with Second-Order Cone Constraints. Due to its nonlinear and convex optimization characteristics, it is difficult to obtain the optimal solution using traditional algorithms. Therefore, NSGA-II integrated with forward-backward substitution method is used for solution, wherein the NSGA-II algorithm is used to update the output of distributed energy DG, and the forward-backward substitution method is used for power flow analysis.

[0009] The NSGA-II algorithm solution process in step 2 includes the following steps:

[0010] Step 2.1: Population initialization: Initialize the number of individuals N and the number of iterations gen. Since the output of DGs will affect the power flow distribution of the entire system, its output needs to be initialized first. After the data of each DG is initialized, a population is formed: In the formula, t is the scheduling period (24 hours a day); are the output of each DGs in the initialization individual at time, the maximum output limit and the minimum output limit of each DGs at time; X 0 are the initial output matrix of each DGs and the individual output matrix respectively;

[0011] Step 2.2: Generate offspring based on the parent generation and merge the population;

[0012] Step 2.3: Forward to solve the power distribution, ignore the voltage loss, only calculate the power loss, and deduce the power at the head end of the branch and the branch network loss: Where n is the number of iterations; r ki 、z ki is the resistance and reactance of branch ki; is the power loss of branch ki; is the active power and reactive power flowing through branch ij; P i , Q i is the active load and reactive load of node i; is the square of the voltage amplitude at node i;

[0013] Step 2.4: Back-substitution to solve the voltage, extrapolate the voltage drop from the starting voltage and power to the end, and calculate the voltage of each node without calculating the power loss separately. The back-substitution calculation formula for the voltage of node i is: In the formula, I ki is the current of branch ki; is the conjugate of the complex voltage at node k; V i is the voltage of node i; j is the imaginary unit;

[0014] Step 2.5: Substitute the voltage obtained by back substitution into step 2.2 to obtain the new round of head-end power, compare it with the head-end power residual obtained in the previous round, and compare it with the set error value, and repeat steps 2.2-2.4 until it is less than the set error;

[0015] Step 2.6: Introduce the constraint into the objective function through the penalty function method, substitute the DGs output into the solution to obtain the dispatching cost, branch network loss and penalty term, divide the penalty term into two parts and add them to the two objectives to obtain the fitness of the population;

[0016] Step 2.7: Non-dominated sorting;

[0017] Step 2.8: Select individuals. Select the next generation of individuals based on non-dominated sorting and crowding distribution. The number of individuals selected should be consistent with the number of individuals in the population N. To ensure the convergence of the algorithm, solve the objective function again and compare it with the solution in step 2.6. Replace the current one with the better individual. Repeat step 2.2 until the number of iterations is reached.

[0018] The specific steps of generating offspring according to the parent generation and merging the population in step 2.2 are as follows: using crossover and mutation operations to generate offspring, but fixed crossover probability and mutation probability are not conducive to distinguishing the process of early global search and late local development. Therefore, a nonlinear dynamic probability is proposed to replace the fixed one. In the early and mid-term global search of the iteration, cr gen Larger and mr gen The algorithm dynamically explores in a larger range to find potential high-quality individuals. As the number of iterations increases, cr gen Nonlinear decreasing and mr gen The algorithm's development performance is gradually enhanced with nonlinear increment, and the algorithm searches as finely as possible around the optimal individual to balance the algorithm's global search and local development capabilities, coordinate global search and local development capabilities, and improve the algorithm's optimization ability; for each DG, crossover and mutation operations are performed separately.

[0019] The crossover operation in step 2.2 is specifically as follows: randomly generate a crossover mask CR gen , select the gene at the corresponding position from the parent generation according to the mask; the mutation operation is specifically: randomly generate a mutation mask MR gen , generate the mutation matrix A according to the mask, then multiply the mutation matrix with the upper bound of the device and add it to the offspring, and finally merge the generated offspring with the parent to form a new population:

[0020] The non-dominated sorting in step 2.7 specifically includes the following steps:

[0021] Step 2.71: Initialization: mark all solutions as the first layer;

[0022] Step 2.72: Compare solutions: For each pair of solutions, compare their performance on each objective. If solution A is not inferior to solution B on all objectives, remove B from the first layer and mark it as dominated by A. If solution A is better than solution B on at least one objective but inferior to or equal to solution B on other objectives, remove B from the first layer and mark it as partially dominated by A. If solution A is inferior to solution B on all objectives, mark A as dominated by B.

[0023] Step 2.73: Determine the Pareto front: Repeat step 2.72 until all solutions without penalty terms are classified as a layer on the Pareto front, which is the union of all non-dominated layers;

[0024] Step 2.74: Hierarchical sorting: Sort the solutions on the Pareto front in layers, with each layer containing solutions that were not dominated in the previous steps;

[0025] Step 2.75: Calculate crowding: The crowding is calculated for the solutions in each layer to provide information about the distribution of solutions in the target space.

[0026] Beneficial effects of the invention: The present invention is a method for collaborative optimization of active and reactive power of an active distribution network based on shared energy storage. In use, the present invention aims at the problem of collaborative optimization of active and reactive power of an active distribution network, establishes an active and reactive collaborative optimization model of an active distribution network based on shared energy storage, adopts a fusion-improved NSGA-II algorithm and a forward-backward substitution method to solve the problem, conducts comparative experiments with set scenarios, uses Hypervolume for analysis, and the simulation results verify the effectiveness of the proposed strategy and method; the shared energy storage and optimization algorithm of the present invention can not only optimize network losses, improve voltage levels, and improve power quality, but also further optimize electricity costs, bringing new ideas to the optimization of active distribution networks; the present invention has the advantages of optimizing network losses, improving voltage levels, improving power quality, and optimizing electricity costs. DETAILED DESCRIPTION

[0027] The present invention is further described below.

[0028] Example 1

[0029] A method for collaborative optimization of active and reactive power in an active distribution network based on shared energy storage, the method comprising the following steps:

[0030] Step 1: Construct objective function: In the active and reactive power coordinated optimization dispatch model of ADN, construct the objective functions of active power and reactive power;

[0031] Step 2: Use NSGA-ΙΙ combined with forward and back iteration to solve the problem: establish a Multi-ObjectiveNonlinear Programming with Second-Order Cone Constraints model, that is, a multi-objective nonlinear programming model based on second-order cone constraints, and solve it.

[0032] The present invention is a method for collaborative optimization of active and reactive power of an active distribution network based on shared energy storage. In use, the present invention aims at the problem of collaborative optimization of active and reactive power of an active distribution network, establishes an active and reactive collaborative optimization model of an active distribution network based on shared energy storage, adopts a fusion-improved NSGA-II algorithm and a forward-backward substitution method to solve the problem, conducts comparative experiments with set scenarios, uses Hypervolume for analysis, and the simulation results verify the effectiveness of the proposed strategy and method. The shared energy storage and optimization algorithm of the present invention can not only optimize network loss, improve voltage level, and improve power quality, but also further optimize electricity cost, bringing new ideas to the optimization of active distribution networks. The present invention has the advantages of optimizing network loss, improving voltage level, improving power quality, and optimizing electricity cost.

[0033] Example 2

[0034] A method for collaborative optimization of active and reactive power in an active distribution network based on shared energy storage, the method comprising the following steps:

[0035] Step 1: Construct objective function: In the active and reactive power coordinated optimization dispatch model of ADN, construct the objective functions of active power and reactive power;

[0036] Step 2: Use NSGA-ΙΙ combined with forward and back iteration to solve the problem: establish a Multi-ObjectiveNonlinear Programming with Second-Order Cone Constraints model, that is, a multi-objective nonlinear programming model based on second-order cone constraints, and solve it.

[0037] The objective function constructed in step 1 is specifically as follows: In the active and reactive power coordinated optimization dispatch model of ADN, the active power goal is to minimize the daily dispatch cost, while the reactive power goal is to minimize the total active power loss of the branch: Where: F is the daily scheduling cost; is the amount of electricity exchanged between ADN and the main network at time t. When ADN buys electricity from the main grid, when ADN sells electricity; The time-of-use electricity purchase and sale price for the main grid, hour The electricity price for the main grid; D * are the output and cost of distributed generation at time t, respectively, where *∈(WT,PV,GT,SES); P loss is the total active power loss of the branch; r ij 、z ij , l ij They are the resistance of the ij branch, the reactance of the ij branch and the square of the current amplitude of the ij branch respectively.

[0038] The specific solution of using NSGA-II integrated with forward-backward substitution method in step 2 is as follows: the established model is Multi-Objective Nonlinear Programming with Second-Order Cone Constraints. Due to its nonlinear and convex optimization characteristics, it is difficult to obtain the optimal solution using traditional algorithms. Therefore, NSGA-II integrated with forward-backward substitution method is used for solution, wherein the NSGA-II algorithm is used to update the output of distributed energy DG, and the forward-backward substitution method is used for power flow analysis.

[0039] The NSGA-II algorithm solution process in step 2 includes the following steps:

[0040] Step 2.1: Population initialization: Initialize the number of individuals N and the number of iterations gen. Since the output of DGs will affect the power flow distribution of the entire system, its output needs to be initialized first. After the data of each DG is initialized, a population is formed: In the formula, t is the scheduling period (24 hours a day); are the output of each DGs in the initialization individual at time, the maximum output limit and the minimum output limit of each DGs at time; X 0 are the initial output matrix of each DGs and the individual output matrix respectively;

[0041] Step 2.2: Generate offspring based on the parent generation and merge the population;

[0042] Step 2.3: Forward to solve the power distribution, ignore the voltage loss, only calculate the power loss, and deduce the power at the head end of the branch and the branch network loss: Where n is the number of iterations; r ki 、z ki is the resistance and reactance of branch ki; is the power loss of branch ki; is the active power and reactive power flowing through branch ij; P i , Q i is the active load and reactive load of node i; is the square of the voltage amplitude at node i;

[0043] Step 2.4: Back-substitution to solve the voltage, extrapolate the voltage drop from the starting voltage and power to the end, and calculate the voltage of each node without calculating the power loss separately. The back-substitution calculation formula for the voltage of node i is: In the formula, I ki is the current of branch ki; is the conjugate of the complex voltage at node k; Vi is the voltage of node i; j is the imaginary unit;

[0044] Step 2.5: Substitute the voltage obtained by back substitution into step 2.2 to obtain the new round of head-end power, compare it with the head-end power residual obtained in the previous round, and compare it with the set error value, and repeat steps 2.2-2.4 until it is less than the set error;

[0045] Step 2.6: Introduce the constraint into the objective function through the penalty function method, substitute the DGs output into the solution to obtain the dispatching cost, branch network loss and penalty term, divide the penalty term into two parts and add them to the two objectives to obtain the fitness of the population;

[0046] Step 2.7: Non-dominated sorting;

[0047] Step 2.8: Select individuals. Select the next generation of individuals based on non-dominated sorting and crowding distribution. The number of individuals selected should be consistent with the number of individuals in the population N. To ensure the convergence of the algorithm, solve the objective function again and compare it with the solution in step 2.6. Replace the current one with the better individual. Repeat step 2.2 until the number of iterations is reached.

[0048] The specific steps of generating offspring according to the parent generation and merging the population in step 2.2 are as follows: using crossover and mutation operations to generate offspring, but fixed crossover probability and mutation probability are not conducive to distinguishing the process of early global search and late local development. Therefore, a nonlinear dynamic probability is proposed to replace the fixed one. In the early and mid-term global search of the iteration, cr gen Larger and mr gen The algorithm dynamically explores in a larger range to find potential high-quality individuals. As the number of iterations increases, cr gen Nonlinear decreasing and mr gen The algorithm's development performance is gradually enhanced with nonlinear increment, and the algorithm searches as finely as possible around the optimal individual to balance the algorithm's global search and local development capabilities, coordinate global search and local development capabilities, and improve the algorithm's optimization ability; for each DG, crossover and mutation operations are performed separately.

[0049] The crossover operation in step 2.2 is specifically as follows: randomly generate a crossover mask CR gen , select the gene at the corresponding position from the parent generation according to the mask; the mutation operation is specifically: randomly generate a mutation mask MR gen , generate the mutation matrix A according to the mask, then multiply the mutation matrix with the upper bound of the device and add it to the offspring, and finally merge the generated offspring with the parent to form a new population:

[0050] The non-dominated sorting in step 2.7 specifically includes the following steps:

[0051] Step 2.71: Initialization: mark all solutions as the first layer (non-dominated layer);

[0052] Step 2.72: Compare solutions: For each pair of solutions (individuals), compare their performance on each objective. If solution A is not inferior to solution B on all objectives (i.e., A is better than or at least as good as B), then remove B from the first layer and mark it as dominated by A. If solution A is better than solution B on at least one objective but inferior to or equal to solution B on other objectives, then remove B from the first layer and mark it as partially dominated by A. If solution A is inferior to solution B on all objectives, mark A as dominated by B.

[0053] Step 2.73: Determine the Pareto front: Repeat step 2.72 until all solutions without penalty terms are classified as a layer on the Pareto front, which is the union of all non-dominated layers;

[0054] Step 2.74: Hierarchical sorting: Sort the solutions on the Pareto front in layers, with each layer containing solutions that were not dominated in the previous steps;

[0055] Step 2.75: Calculate crowding: The crowding is calculated for the solutions in each layer to provide information about the distribution of solutions in the target space.

[0056] The present invention is a method for collaborative optimization of active and reactive power of active distribution network based on shared energy storage. In use, the present invention establishes an active and reactive collaborative optimization model to realize the active and reactive collaborative optimization of ADN based on SES, wherein the active goal is to minimize the daily dispatching cost, and the reactive goal is to minimize the total active power loss of the branch. The solution is solved by a solution method integrating the improved NSGA-II algorithm and the forward-backward substitution method, and four scenarios are set for comparative analysis, and the following conclusions are drawn: compared with ES, SES can further play the role of active and reactive collaborative optimization, and provide a new idea for the optimal scheduling of ADN; the improved NSGA-II algorithm has better solution accuracy than the traditional algorithm, and can find a better Pareto frontier in multi-objective optimization; the present invention has the advantages of optimizing network loss, improving voltage level, improving power quality, and optimizing electricity cost.

Claims

1. A method for collaborative optimization of active and reactive power in active distribution networks based on shared energy storage, characterized in that: The method comprises the following steps: Step 1: Construct objective function: In the active and reactive power coordinated optimization dispatch model of ADN, construct the objective functions of active power and reactive power; Step 2: Use NSGA-ΙΙ combined with forward and back iteration to solve: Establish a Multi-Objective Nonlinear Programming with Second-Order Cone Constraints model, that is, a multi-objective nonlinear programming model based on second-order cone constraints, and solve it.

2. The method for collaborative optimization of active and reactive power in an active distribution network based on shared energy storage according to claim 1, characterized in that: The objective function constructed in step 1 is specifically as follows: In the active and reactive power coordinated optimization dispatch model of ADN, the active power goal is to minimize the daily dispatch cost, while the reactive power goal is to minimize the total active power loss of the branch: Where: F is the daily scheduling cost; is the amount of electricity exchanged between ADN and the main network at time t. When ADN buys electricity from the main grid, when ADN sells electricity; The time-of-use electricity purchase and sale price for the main grid, hour The electricity price for the main grid; D * are the output and cost of distributed generation at time t, respectively, where *∈(WT,PV,GT,SES); P loss is the total active power loss of the branch; r ij 、z ij , l ij They are the resistance of the ij branch, the reactance of the ij branch and the square of the current amplitude of the ij branch respectively.

3. The method for collaborative optimization of active and reactive power in an active distribution network based on shared energy storage according to claim 1, characterized in that: The specific solution of using NSGA-II integrated with forward-backward substitution method in step 2 is as follows: the established model is Multi-Objective Nonlinear Programming with Second-Order Cone Constraints. Due to its nonlinear and convex optimization characteristics, it is difficult to obtain the optimal solution using traditional algorithms. Therefore, NSGA-II integrated with forward-backward substitution method is used for solution, wherein the NSGA-II algorithm is used to update the output of distributed energy DG, and the forward-backward substitution method is used for power flow analysis.

4. The method for collaborative optimization of active and reactive power in an active distribution network based on shared energy storage as claimed in claim 3, characterized in that: The NSGA-II algorithm solution process in step 2 includes the following steps: Step 2.1: Population initialization: Initialize the number of individuals N and the number of iterations gen. Since the output of DGs will affect the power flow distribution of the entire system, its output needs to be initialized first. After the data of each DG is initialized, a population is formed: Where t is the scheduling period; are the output of each DGs in the initialization individual at time, the maximum output limit and the minimum output limit of each DGs at time; X 0 are the initial output matrix of each DGs and the individual output matrix respectively; Step 2.2: Generate offspring based on the parent generation and merge the population; Step 2.3: Forward to solve the power distribution, ignore the voltage loss, only calculate the power loss, and deduce the power at the head end of the branch and the branch network loss: Where n is the number of iterations; r ki 、z ki is the resistance and reactance of branch ki; is the power loss of branch ki; is the active power and reactive power flowing through branch ij; P i , Q i is the active load and reactive load of node i; is the square of the voltage amplitude at node i; Step 2.4: Back-substitution to solve the voltage, and extrapolate the voltage drop from the starting voltage and power to the end. Without calculating the power loss separately, calculate the voltage of each node. The back-substitution calculation formula for the voltage of node i is: In the formula, I ki is the current of branch ki; is the conjugate of the complex voltage at node k; V i is the voltage of node i; j is the imaginary unit; Step 2.5: Substitute the voltage obtained by back substitution into step 2.2 to obtain the new round of head-end power, compare it with the head-end power residual obtained in the previous round, and compare it with the set error value, and repeat steps 2.2-2.4 until it is less than the set error; Step 2.6: Introduce the constraint into the objective function through the penalty function method, substitute the DGs output into the solution to obtain the dispatching cost, branch network loss and penalty term, divide the penalty term into two parts and add them to the two objectives to obtain the fitness of the population; Step 2.7: Non-dominated sorting; Step 2.8: Select individuals. Select the next generation of individuals based on non-dominated sorting and crowding distribution. The number of individuals selected should be consistent with the number of individuals in the population N. To ensure the convergence of the algorithm, solve the objective function again and compare it with the solution in step 2.

6. Replace the current one with the better individual. Repeat step 2.2 until the number of iterations is reached.

5. The method for collaborative optimization of active and reactive power in active distribution network based on shared energy storage as claimed in claim 4, characterized in that: The specific steps of generating offspring according to the parent generation and merging the population in step 2.2 are as follows: using crossover and mutation operations to generate offspring, but fixed crossover probability and mutation probability are not conducive to distinguishing the process of early global search and late local development. Therefore, a nonlinear dynamic probability is proposed to replace the fixed one. In the early and mid-term global search of the iteration, cr gen Larger and mr gen The algorithm dynamically explores in a larger range to find potential high-quality individuals. As the number of iterations increases, cr gen Nonlinear decreasing and mr gen The algorithm's development performance is gradually enhanced with nonlinear increment, and the algorithm searches as finely as possible around the optimal individual to balance the algorithm's global search and local development capabilities, coordinate global search and local development capabilities, and improve the algorithm's optimization ability; for each DG, crossover and mutation operations are performed separately.

6. The method for collaborative optimization of active and reactive power in active distribution network based on shared energy storage as claimed in claim 5, characterized in that: The crossover operation in step 2.2 is specifically as follows: randomly generate a crossover mask CR gen , select the gene at the corresponding position from the parent generation according to the mask; the mutation operation is specifically: randomly generate a mutation mask MR gen , generate the mutation matrix A according to the mask, then multiply the mutation matrix with the upper bound of the device and add it to the offspring, and finally merge the generated offspring with the parent to form a new population:

7. The method for collaborative optimization of active and reactive power in an active distribution network based on shared energy storage according to claim 4, characterized in that: The non-dominated sorting in step 2.7 specifically includes the following steps: Step 2.71: Initialization: mark all solutions as the first layer; Step 2.72: Compare solutions: For each pair of solutions, compare their performance on each objective. If solution A is not inferior to solution B on all objectives, remove B from the first layer and mark it as dominated by A. If solution A is better than solution B on at least one objective but inferior to or equal to solution B on other objectives, remove B from the first layer and mark it as partially dominated by A. If solution A is inferior to solution B on all objectives, mark A as dominated by B. Step 2.73: Determine the Pareto front: Repeat step 2.72 until all solutions without penalty terms are classified as a layer on the Pareto front, which is the union of all non-dominated layers; Step 2.74: Hierarchical sorting: Sort the solutions on the Pareto front in layers, with each layer containing solutions that were not dominated in the previous steps; Step 2.75: Calculate crowding: The crowding is calculated for the solutions in each layer to provide information about the distribution of solutions in the target space.

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