A method for estimating the optimal arrangement of energy conduits within an array.

Abstract

Methods for estimating the optimal placement of energy conduits within an array and for estimating the optimal placement of energy conduits (cables) between arrays are disclosed. A nondeterministic mixed-integer nonlinear programming algorithm is used and set to generate outputs containing cable placement and associated objective function values. Multiple processors or computers run the algorithm in parallel, generating multiple outputs using the same problem definition and the same stopping criterion, but with different seed values. The outputs are analyzed to determine the best objective function value and associated energy conduit placement, taking into particular consideration the costs associated with power loss.

Description

Technical Field

[0001] The present invention relates to a method for estimating the optimal placement of energy conduits within an array, and more particularly, but not limited to, a method for determining the optimal layout of power cables between turbines in an offshore wind farm, taking into account factors such as power loss.

Background Art

[0002] In an offshore wind farm, cables are used to connect individual turbines back to an onshore location. This is typically done via one or more nodes or substations, and the cables generally connect the turbines in a chain to each other before connecting to the node (substation). The cable system that connects the turbines to the node (substation) is generally referred to as an inter-array system. Since each turbine typically generates power in megawatts, the cables that transmit this power need to be large and are costly per unit length. Further, since the turbines are typically spaced at intervals of 1 kilometer, the total length of cable used between dozens to hundreds of turbines is enormous. When one turbine is connected to another, the cable from the second turbine (closer to the substation) needs to have the ability to transmit a larger capacity than required by a single turbine in order to transmit the power from both turbines. Due to the high cost of these cables (usually exceeding 10% of the total capital investment of the wind farm), it is important that the appropriate type of cable is selected and the most efficient layout is chosen. This efficiency includes not only the total length and installation cost of the cable, but also the potential for power loss from the measured cable and the non-operational losses over the expected service life.

[0003] It is important that the mathematical model takes into account real-world phenomena and constraints as accurately as possible. If the system is inaccurately modeled, even a global optimum solution may prove insufficient or impossible to implement in the real world.

[0004] Systems used to optimize the cable layout within an array are disclosed in the following documents. CN113011090A presents a method for discovering cable connection layouts using particle swarm optimization, accompanied by the generation of Voronoi adaptive zones. A drawback of this approach is that it can be time-consuming to find the optimal cable layout, and the obtained solution may be a local maximum rather than a global maximum. CN111754035A (or WO2021253291A1) also describes a method for optimizing wind power plant layouts using a metaheuristic algorithm, focusing on the optimization of substation coordinates. This document considers nonlinear power losses, including resistive or dielectric losses, in the objective function. Further examples in the patent literature include CN114580725A, which details a genetic algorithm for a specific application of photovoltaic wiring. This algorithm is typically run for extended periods to increase the chances of finding the optimal solution.

[0005] In "A review of offshore wind farm layout optimization and electrical system design methods" by Hou et al., *Journal of Modern Power Systems and Clean Energy* (Volume 7, Issue 5, September 2019), a review of several approaches to optimizing inter-array cable layouts is presented. This review includes classical deterministic approaches that, if a solution is found, cannot be found within a reasonable timeframe, unless the problem is significantly simplified, sacrificing the accuracy of the results. Many of the methods disclosed in this review are found to be non-deterministic heuristic algorithms to avoid simplification. However, these algorithms are typically run for very long periods to obtain a solution that is as close to optimal as possible.

[0006] Cazzaro et al., “Heuristic algorithms for the Wind Farm Cable Routing problem” (https: / / www / reserachgate.net / publication / 343567214), compares several heuristic algorithms to a set of cable layout optimization problems and demonstrates the variability of results obtained through prior art probabilistic algorithms. Examples of these approaches make it difficult to know whether good results were obtained due to changes in input, changes in algorithm configuration, or simply by chance.

[0007] Another example of a system aimed at optimizing cable layout is described in Fischetto Martina et al., “Optimizing wind farm cable routing considering power losses,” European Journal of Operational Research, Volume 270, no. 3, pp. 917–930. As the title suggests, the method disclosed in this publication addresses the important consideration of power loss. [Overview of the Initiative]

[0008] Preferred embodiments of the present invention aim to overcome or mitigate the aforementioned drawbacks of the prior art.

[0009] According to an aspect of the present invention, a method for estimating the optimal arrangement of energy conduits in an array, A step of setting up a nondeterministic mixed-integer nonlinear programming algorithm in at least one computer device and optimizing the arrangement of energy conduits in an array as a nonlinear optimization problem, wherein the algorithm generates an output including at least one energy conduit arrangement and associated objective function values, The steps include: running the calculation of the algorithm multiple times in parallel on at least one of the above computer devices to generate multiple outputs using substantially the same problem definition, substantially the same stopping criteria, and different seed values; The steps include: analyzing the multiple outputs in the above-mentioned computer device or another computer device to determine substantially the best objective function value and the associated energy conduit arrangement; A method is provided that includes this.

[0010] By using a nondeterministic mixed-integer nonlinear programming algorithm and running the algorithm multiple times with substantially the same problem definition, the same stopping criteria, and different seed values, the optimized arrangement of energy conduits in an array can be estimated as a nonlinear problem. This offers the advantage that, using multiple computing devices in parallel, estimated optimizations can be generated more quickly and accurately (with more complex and precise problem definitions) than using prior art methods. Typically, when similar problems are solved using nondeterministic optimization algorithms in the prior art, a single execution of the algorithm over a long period of time is used on a single computing device. This long execution time is used to ensure a high probability of finding a near-optimal output (objective function value). However, it has been found that it is more effective to run the algorithm many times over a much shorter period of time and then determine which objective function value provided the best lifetime value for the proposed energy conduit arrangement.

[0011] Furthermore, in the present invention, computational load is used more effectively. That is, for the same amount of computational load, the reliability of returning good results is usually increased, and / or the variability of the returned results is reduced. In other words, by running more computers for less time, and especially by using algorithms with fewer compromises and simplifications, such as mixed-integer nonlinear programming algorithms, results with a higher lifetime value (closer to true optimal) can be obtained than in prior art systems.

[0012] Furthermore, by repeatedly executing the same algorithm over short periods, the algorithm can be run on multiple computing devices. This subdivides waiting time and computation time, meaning that computing power / effort is reasonably deployed to obtain good results. This also allows for reduced waiting time, which leads to increased productivity for users, enabling them to run more scenarios with different input data as needed. As a result, lead times are reduced, and the amount of engineer work required to complete the design process is decreased. This is partly due to the improved reliability of the results produced by the algorithm and the reduced time that allows more variations (e.g., substation locations) to be tried within the time constraints of the design process.

[0013] In examples of prior art, the objective function portion of the algorithm is executed many times in each iteration of the algorithm, and the time required to execute the cost model is important; as a result, simplifications are often made in the problem definition to ensure reasonable computation time. For example, more complex nonlinear problems are addressed by simplifying the algorithm to include only a linear function that serves as an estimate of the nonlinear problem. However, this carries the risk that the model will optimize far from the true optimal.

[0014] In some cases, the prior art employs parallelization in the computation of the objective function execution to solve some of these problems. However, a key difference is that in the prior art, each iteration depends on the result of the previous iteration, and parallelization occurs in the execution of the objective function for each iteration. In contrast, the present invention performs independent computations of the algorithm itself, including all iterations, in parallel, so that an iteration on one computer does not depend on the iterations of any other computer. The method of the present invention enables more sophisticated cost models, particularly those involving mixed-integer nonlinear problems. This allows for waiting for optimization to complete, resulting in optimization that is closer to a more accurate model of the real world.

[0015] In this invention, the real-world time required to complete the optimization task is significantly shorter than in the prior art, so that time-critical optimization problems (such as emergency service-related problems concerning the allocation of rescue resources in search and rescue) can typically be solved in seconds to minutes, rather than the usual hours to days. Furthermore, this high-speed optimization can also be used to perform multiple levels of optimization related to a single project. For example, in an inter-array cable system for an offshore wind farm, there are several factors related to the total capital cost. These include not only the cable arrangement but also the location of any substations within the array. Therefore, it is possible to perform multiple optimizations with respect to these different characteristics.

[0016] The method disclosed by Fischetto Martina et al. addresses the cable optimization problem using a mixed-integer linear programming algorithm. This is achieved by isolating a small portion of the network, optimizing that portion while keeping the rest constant, and then isolating a different portion and repeating the process multiple times. Although the power loss problem is nonlinear, the system by Fischetto Martina et al. addresses this problem by estimating the power loss as a linear term in the algorithm and optimizing it using mixed-integer linear programming (MILP). The compromise of estimating the power loss as a linear term to enable optimization as MILP makes it less likely that the optimized array found by the algorithm will be close to the true optimal. In other words, the linearization of power loss means that the cost model in the prior art is a low-precision model of the real-world inter-array cable system, and it is shown that the optimal in the prior art formulation is unlikely to coincide with the real-world optimal. The MILP algorithm finds a global optimal (i.e., the optimal for the algorithm), but the compromise of linearizing the nonlinear term means that it may not approach the true (real-world) optimal for the array.

[0017] It should be noted that, when attempting to approach a true real-world optimal, executing more complex (nonlinear) algorithms in a short time is counterintuitive. However, by executing complex algorithms many times in short execution times, outputs closer to the true optimal than those seen in prior art can be produced. This is partly because the optimal found by each calculation depends on the quality of the starting seed value and the ease of the path to the calculated optimal. By performing a statistically significant number of calculations from different seed values, resulting in different paths through the computational space, the system of the present invention is ultimately likely to produce results closer to the real-world optimal, particularly because it can use more complex (closer to real-world) algorithms. The statistically significant minimum sample size depends heavily on the formulation of the problem, the desired outcome, and the predictions to be made. Therefore, the statistically significant minimum sample size should be determined by a qualified statistician using methods such as goodness-of-fit tests (Kolmogorov-Smirnov, Anderson-Darling), one-sample and two-sample t-tests, statistical power analysis, and statistical significance analysis. The above examples are a non-exhaustive list of appropriate technologies, and other technologies may be available that can be selected at the discretion of a person skilled in the art.

[0018] In a preferred embodiment, at least one mixed-integer nonlinear programming algorithm comprises optimizing the energy loss in the energy conduit.

[0019] Energy loss is an example of a nonlinear problem related to cable array optimization. Therefore, as mentioned above, using a mixed-integer nonlinear programming algorithm allows for more accurate optimization, i.e., optimization closer to the true optimal than the estimates used in prior art. In particular, creating a linear term to model energy loss, which is actually nonlinear, in a mixed-integer linear programming algorithm is an inaccurate optimization method.

[0020] The method may further include analyzing the above-mentioned multiple objective function values ​​to obtain at least one of the following: the probability distribution function of the objective function values, the probability distribution function of the computation time of the selected algorithm, summary statistics of the objective function values, and summary statistics of the computation time.

[0021] The method may also be further equipped with the ability to estimate the substantially best probabilistic boundary for the objective function value using the above-mentioned probability distribution function or summary statistics for the objective function value.

[0022] Using probability distribution functions or summary statistics such as mean, percentile, median, minimum, maximum, standard deviation, and variance indices provides additional advantages over prior art. Because each calculation is independent, statistical distributions can be generated that allow us to estimate "how lucky" we were compared to what might happen if the algorithm were run again. As a result, we can determine whether the objective function value returned at the end of the study was sufficiently lucky, or whether it is justifiable to rerun the study in the expectation of better results.

[0023] Furthermore, the constructed probability distribution can be used to precisely indicate how many additional calculations are needed to achieve a desired probability of exceeding a specific percentile in the distribution of the objective function value. Thus, the exact amount of computational resources required can be used to obtain the desired result, avoiding wasted time and cost. Quantifying the improvement probability reduces the inherent uncertainty in the optimality of the heuristic algorithm's results. Therefore, this can be incorporated into the algorithm itself as a stopping criterion, automatically halting a large number of calculations once the desired percentile, or "luck level," is reached.

[0024] A derivative advantage of the above is that the resulting probability distribution is far more robust to random chance than simply using the best results found during the study. This means that changes to the model inputs can be linked more confidently to subsequent changes in the best possible solution achievable with those inputs. This is further illustrated below with reference to Figure 4 in the diagram.

[0025] The method performs, before performing the method for estimating the optimal arrangement of the energy conduits in the array described above, in a plurality of studies each using a different stopping criterion, executing the analysis step of the method described above, and using at least one of the above distribution functions to predict the probabilistic bounds at different numbers of calculations and estimating the best balance between the number of calculations and the stopping criterion for efficiently estimating the optimal arrangement of the energy conduits in the array, and performing further studies according to any of the preceding claims and estimating the optimal arrangement of the energy conduits in the array with the best balance between the number of calculations and the stopping criterion may further be provided.

[0026] In a preferred embodiment, at least one empirical relationship between the stopping criterion and the distribution function and / or the above summary statistics is determined.

[0027] By performing a series of pre-study calculations, the results of these calculations provide the advantage of enabling the estimation of the probabilistic bounds of the objective function values for any stopping criterion and any number of calculations. Thereby, it is possible to determine the most appropriate stopping criterion and number of calculations before performing the study for estimating the optimal arrangement of the energy conduits in the array.

[0028] In a preferred embodiment, the cost of the energy conduits, the energy loss in the energy conduits, the cost of maintaining the energy conduits, the cost of repairing the energy conduits, the energy loss due to power loss in the energy conduits, the energy loss due to power outages, the amount of energy reaching a given point in the array or in the energy conduits, the energy value passing through the energy conduits, and the cost of installing the energy conduits An objective function value is determined that includes at least one of the following.

[0029] In a further preferred embodiment, multiple outputs are used to generate a heatmap of the arrangement to provide the estimated optimal arrangement.

[0030] Having a heatmap output makes it possible to provide reasonable estimates of likely locations for components (e.g., cables or substations), allowing for initial investigations to determine the suitability of proposed locations before further optimization work is carried out. In prior art, attempts to conduct similar research result in unfeasible time spent performing the calculations required to generate the heatmap. In the case of substations, the heatmap is used to create recommended substation locations that are most likely to reduce the overall cost of inter-array cables.

[0031] In a preferred embodiment, the energy conduit includes a cable.

[0032] In another preferred embodiment, the cable connects multiple energy generating devices to at least one node.

[0033] In a further preferred embodiment, the energy generation device comprises a wind turbine.

[0034] According to another aspect of the present invention, a method for installing an array of interconnected energy-related devices, The steps include determining the location of an array of energy-related devices, The steps include: estimating the optimal arrangement of energy conduits interconnecting the above arrays using the method described above; The steps include installing the energy-related equipment and the energy conduits arranged as described above. A method is provided that includes this.

[0035] In a preferred embodiment, the energy conduit includes a cable.

[0036] In another preferred embodiment, the energy-related device comprises an energy-generating device.

[0037] In a further preferred embodiment, the energy generation device comprises a wind turbine.

[0038] In another preferred embodiment, the termination criteria result in an average execution time of less than 60 minutes, less than 30 minutes, less than 15 minutes, or less than 10 minutes for a single run of the algorithm.

[0039] In a further preferred embodiment, the plurality of computers comprises at least 10 computers, or at least 25 computers, or at least 32 computers, or at least 50 computers, or at least 100 computers, or at least 1000 computers.

[0040] According to a further aspect of the present invention, a method for estimating a solution to a nondeterministic problem, A step of setting up a nondeterministic mixed-integer nonlinear programming algorithm on at least one computer device and optimizing a nonlinear optimization problem, wherein the algorithm generates an output including a solution to the problem and associated objective function values, The steps include: running the calculation of the algorithm multiple times in parallel on at least one of the above computer devices to generate multiple outputs using substantially the same problem definition, substantially the same stopping criteria, and different seed values; The steps include: analyzing the multiple outputs in the above-mentioned computer device or another computer device to determine the substantially best objective function value and the associated solution; A method is provided that includes this.

[0041] The present invention provides the advantages described above. In addition, it is also suitable for providing estimated solutions to nonlinear optimization problems other than the problem of cable positioning between arrays, which provide some or all of the advantages described above. For example, the present invention provides sufficient reliability in optimizations involving nonlinear functions, enabling rapid decisions in situations where optimization was previously not used due to the long time required to complete the optimization. As a result, such optimizations involving nonlinear functions can be used in emergencies, such as the allocation of rescue resources in search and rescue operations. Furthermore, complex problems that are short in duration and regularly scheduled, such as the selection of delivery routes, can be rapidly optimized with high confidence that the proposed route is close to the true optimal.

[0042] The method may further include analyzing the above-mentioned multiple objective function values ​​to obtain at least one of the following: the probability distribution function of the objective function values, the probability distribution function of the computation time of the selected algorithm, summary statistics of the objective function values, and summary statistics of the computation time.

[0043] The method may further provide for estimating the substantially best probabilistic boundary for the objective function value using the above probability distribution function or summary statistics for the objective function value.

[0044] The method involves, before performing the method for estimating the solution to the non-deterministic problem described above, In multiple studies, each using different stopping criteria, the analysis steps described above are performed, Using at least one of the above distribution functions, we predict the stochastic boundary at different computation counts and estimate the best balance between computation count and stopping criteria for efficiently estimating the optimal solution to the problem. To perform further research as described in any of the above claims and to estimate the optimal solution to the problem with the best balance between the number of calculations and the stopping criteria, You may want to prepare even more.

[0045] In a preferred embodiment, at least one empirical relationship is determined between the stopping criterion and the distribution function and / or the summary statistics.

[0046] In another preferred embodiment, the termination criteria result in an average execution time of less than 60 minutes, less than 30 minutes, less than 15 minutes, or less than 10 minutes for a single run of the algorithm.

[0047] In a further preferred embodiment, the plurality of computers comprises at least 10 computers, or at least 25 computers, or at least 32 computers, or at least 50 computers, or at least 100 computers, or at least 1000 computers.

[0048] Preferred embodiments of the present invention will be described below with reference to the accompanying drawings, not in any way limited to their scope. [Brief explanation of the drawing]

[0049] [Figure 1a] This is a schematic diagram of the offshore wind farm layout. [Figure 1b] This is a pair of enlarged views of the layout in Figure 1a. [Figure 1c] This is a pair of enlarged views of the layout in Figure 1a. [Figure 2a] This is an example of a heatmap showing the output used in the present invention. [Figure 2b] This is an example of a heatmap showing the output used in the present invention. [Figure 2c] This is an example of a heatmap showing the output used in the present invention. [Figure 2d] This is an example of a heatmap showing the output used in the present invention. [Figure 3] This is a flowchart of the process of the present invention. [Figure 4] This graph shows the main advantages over prior art. [Figure 5]This graph shows the main advantages over prior art. [Figure 6] This graph shows the main advantages over prior art. [Modes for carrying out the invention]

[0050] The method of the present invention is used to estimate the optimal arrangement of energy conduits within an array. An example in which the present invention is used is determining the optimal arrangement of inter-array cables in an offshore wind farm consisting of an array of wind turbines for power generation. Referring to Figure 1a, the schematic diagram of the illustrated wind farm 10 has multiple energy-related devices in the form of turbines represented by point 12, which are connected to adjacent turbines by energy conduits in the form of cables represented by line 14. Figures 1b and 1c show two of several possible arrangements of cables for connecting the turbines. Multiple turbines are connected to localized substations or nodes 16 that store the generated power before it is transmitted to land. Nodes 16 (substations) are generally located offshore, but are not limited to, and export cables connect offshore nodes(s) to onshore substations for broad transmission to where the energy is used. Theoretically, cables could connect each turbine directly to a node. However, this is not done in practice, and connections are made from one turbine to an adjacent turbine and finally to a node.

[0051] The cables between turbines must be capable of transmitting the power generated by each turbine. Considering the high wattage that can be generated by the turbines, the cost per unit length of these connecting cables is significant. When one turbine is connected to another, the cable from the second turbine (closer to the substation) carries power from both turbines, thus requiring a larger and potentially more expensive cable. Since power loss in the cables is inevitable, determining the optimal cable placement and ensuring that the chosen placement balances capital, installation, and operating costs in minimizing these losses is a crucial factor. Because these power losses are an example of nonlinear factors, optimizing the cable array becomes a nonlinear problem, best addressed using mixed-integer nonlinear programming algorithms. As a result, it becomes clear that there are potentially competing factors in determining the optimal cable placement. Capital and installation costs can be saved by resulting in an optimized cable placement that minimizes total cable costs. Furthermore, the efficiency of power transmission from turbines to nodes before transmission to land can be improved. These factors, along with others, provide a so-called lifetime value for a particular cable placement between turbines and nodes. The objective of this invention is to determine the best estimate for the optimal cable placement in order to provide the maximum lifetime value or a similar lifecycle cost metric that yields the maximum return on investment in a wind power plant. Therefore, this lifetime value is the objective function value of the algorithm used to estimate the optimal cable placement within the array.

[0052] The problem to be solved in estimating the optimal cable arrangement in an array of wind turbines for power generation at an offshore wind farm is a nonlinear mixed-integer problem, and the solution can be obtained using a nondeterministic optimization algorithm, specifically the mixed-integer nonlinear programming (MINLP) algorithm. The output of the algorithm includes the arrangement of energy conduits in the array and the associated objective function values ​​related to that arrangement.

[0053] First, the problem must be defined. Problem definition involves defining the objective function, identifying the decision variables to be optimized, identifying any fixed input variables, and defining any constraints that must be adhered to in any acceptable solution. This objective function value (which may also be called a fitness value, cost function value, or lifetime value, depending on the problem being solved) is a numerical value related to an output array that represents the benefits obtained from the arrangement. Next, the problem definition is determined, which includes defining the problem itself. Examples of factors used in defining the problem include, but are not limited to, the proposed turbine locations, the maximum turbine output, the node / substation locations, the type and capacity of the cables used to connect the turbines, the seabed topography and features through which the cable routes pass, and the location of substations to onshore cables. It is also necessary to define constraints, which include, but are not limited to, the maximum number of turbines that can be connected to strings from a current collection node, the maximum number of strings (cables) that can be connected to a current collection node (turbine or substation), the cables required to avoid a particular area, and whether or not cable crossings are permitted.

[0054] Termination criteria for algorithm execution are also defined. Details on how to select termination criteria will be described later.

[0055] Finally, once all input variables and other variables are determined, a suitable nondeterministic optimization algorithm can be defined or selected to address the optimization problem. Examples of suitable nondeterministic optimization algorithms include, but are not limited to, tabu search algorithms, simulated annealing algorithms, genetic algorithms, ant colony optimization algorithms, particle swarm optimization algorithms, artificial fish swarm algorithms, artificial bee swarm algorithms, greedy random adaptive search algorithms, wide nearest neighbor search algorithms, and artificial neural network algorithms. If factors that introduce a nonlinear problem are defined, such as including power loss in a cable, a nondeterministic mixed-integer nonlinear programming algorithm may be used to perform the optimization.

[0056] Once the algorithm is set, a study is conducted, the output of which is, for example, the optimal cable placement and / or the associated objective function value. This study consists of numerous calculations, each of which is the execution of a selected nondeterministic optimization algorithm through numerous iterations until a stopping criterion is reached. When the study is conducted using a nondeterministic optimization algorithm, the output of each calculation is the objective function value. This objective function value is an evaluation of the cable placement, taking into account a combination of factors. These factors include, but are not limited to, the cable cost per unit length of different sized cables, the laying cost of different cables per unit length, the assumed power loss over the cable life with associated costs, and the cable degradation rate. Other factors include, but are not limited to, the cost of connecting between turbines, the laying cost per unit length of cables, the maintenance cost of energy conduits, the repair cost of energy conduits, energy loss due to power loss within energy conduits, energy loss due to power outages, the amount of energy reaching a given point in the array or energy conduit, the value of energy passing through the energy conduit, and the installation cost of the energy conduits.

[0057] Compared to prior art systems that use a nondeterministic optimization algorithm to solve a nonlinear mixed-integer problem of estimating the optimal cable arrangement in a wind turbine array, the algorithm of the present invention is ideally executed and re-executed multiple times using parallel processing on multiple computer devices, but is stopped after a short period, whereas the prior art system executes the algorithm once over a long period. Since substantially the same initial variables are used in each calculation, the same problem is defined in both cases. The algorithm is executed through calculations using the same stopping criterion, but different seed values ​​are used in each case. By using different seed values ​​for each calculation, the nondeterministic optimization algorithm may follow different paths through the solution space and arrive at solutions that are potentially different in reaching the stopping criterion but are equally valid.

[0058] As mentioned above, when trying to approach the true real-world optimality, it is counterintuitive to execute more complex (nonlinear) algorithms in a short amount of time. Examples of appropriate computation times include, but are not limited to, less than 60 minutes. Preferably less than 30 minutes. More preferably less than 15 minutes. Even more preferably less than 10 minutes.

[0059] Similarly, the number of computers running the algorithm should exceed 10. Preferably, 25 to 32 computers should be used to ensure statistically significant results. More preferably, 50 computers should be used. Even more preferably, 100 computers should be used. Ideally, more than 1000 computers may be used. While it is important, it is not essential that the number of computers used provides statistically significant results.

[0060] More specifically, this means that each computation begins with a different initial arrangement of cables in the array. The seed value is a randomly generated number, or a number in a known sequence of random numbers, which prompts or initializes the random propagation of cables in the array's starting arrangement at the beginning of each computation. This cable arrangement is then randomly modified by changing the positions of one or more cables, also controlled by the seed value. Typically, when cable positions are moved, other cables must also be moved to maintain an array that ensures a feasible solution that connects all turbines and does not violate any of the constraints identified in the problem definition. This new arrangement is the result of a single iteration of a chosen non-deterministic optimization algorithm and is tested to determine the objective function value. As an example of a very simple version, the objective function value may be the total cost of cables used, assuming that a single type of cable connects all turbines. Total length can be translated into cost. If the total cost of cables required after iterations of the algorithm is lower than that of the previous arrangement, this arrangement is superior (cheaper) and yields a better objective function value.

[0061] However, because the array connects turbines in series, it is advantageous to use low-capacity cables between turbines at the ends of the string, while if more turbines are connected, the cable capacity needs to be larger. This is known as tapering, as the cable becomes thinner towards the ends of the string. Using the method of the present invention, it is possible to determine how much of each different capacity cable is needed and the total cost to be calculated. This alternative objective function value provides more useful information than the simplest version described above. Further factors are added during the definition of the problem to generate an objective function value that reflects the lifetime cost of the proposed array, i.e., the total cost over the life of the wind farm.

[0062] The objective function value of a new cable arrangement in the array is compared to the objective function value of the previous arrangement. If the new arrangement does not provide a better objective function value, it is discarded. However, if the objective function value of the new arrangement is better, it is retained and used as the starting point for the next iteration. This process is repeated through iterations until a stopping criterion is reached. When the stopping criterion (end of calculation) is reached, the objective function value of that calculation is stored along with the final (best-improved) arrangement of cables in the array. In the example above, the optimization algorithm is searching among arrays that provide the best objective function value. The best objective function may be the minimum output, such as cable cost, installation cost, or energy loss. Similarly, the best objective function value may be the maximum output, such as the energy received on land after accounting for losses and capital expenditures, revenue or energy received at a substation, or power availability on land or at a substation.

[0063] Typically, a second calculation is performed in parallel with the first calculation, using the process described above and a new seed value (initial arrangement of cables in the array). When the stopping criterion for this second calculation is reached (almost simultaneously with the first calculation), the final objective function value of the new calculation is compared with the final objective function value of the first calculation, and the better result (better objective function value) is retained. Because multiple calculations (not just two) are performed in parallel, comparisons between all objective function values ​​are performed at once.

[0064] Instead of the seed value determining a new initial cable placement in each calculation, the same initial cable placement can be used each time, but the difference (seed value) is that which cable is initially moved to the alternative position is changed. This method yields similar results to the one described above, but this version has the drawback that the results may not be good, or it may take a long time to obtain satisfactory results (increased computer time due to the large number of calculations). If the selected first cable placement is far from the optimal solution, the results of each calculation may not be as good as when a new placement is used each time. There is also the problem that all calculations may fall to the same local minimum in the graph of the improved objective function value over the iterations, and the stopping criterion may be triggered before true optimization is reached.

[0065] The stopping criterion determines the duration for which the algorithm runs, and a stopping criterion is selected that significantly reduces the time required for each run compared to a prior art single-run system. An example of a suitable stopping criterion is selecting the number of iterations the algorithm will perform before the computation is stopped. Since the same stopping criterion is used for each run of the algorithm, if the number of iterations is the stopping criterion, the algorithm will run for substantially the same number of iterations each time. Alternative stopping criteria include, but are not limited to, a set number of iterations since the last improvement in the objective function value, reaching an allocated budget (computation time / cost), no improvement in the entire population (or the top N individuals within the population) or no improvement exceeding a certain tolerance over a specific number of iterations, no significant change in the surviving individual's determinant variable (i.e., a change that may be below the achievable real-world resolution) over a specific number of iterations, or a combination of these.

[0066] This research is most efficiently conducted by performing calculations on multiple computer devices running in parallel. Thus, each of the different seed values, which are the initial starting point configurations of the cables in the array, is programmed onto a different machine, and these computers, by performing their respective calculations substantially simultaneously, also produce their respective outputs roughly simultaneously. More specifically, the seed values ​​are typically randomly generated numbers, or numbers in a known sequence of random numbers, that facilitate the random propagation of the cables throughout each calculation.

[0067] Once multiple runs are complete, the outputs are collectively adjusted and analyzed to determine the optimal objective function value. This best-performing cable configuration is then used to provide the best return on investment in cable installation. In its simplest form, this involves using the cable configuration that yields the highest objective function value.

[0068] Using the determined (or estimated) optimal cable layout, the process of constructing a wind farm involves installing turbines in designated locations, connecting them to each other with specified cable types, and connecting them to substations that are connected to onshore transmission cables.

[0069] In many situations, it is advantageous to perform some preliminary calculations before running a study to help determine the number of calculations and stopping criteria that strike the best balance between the computer time used to run the study and the quality of the results obtained. In particular, multiple studies are performed and analyzed using the objective function value to calculate at least one of the following: the probability distribution function of the objective function value, the probability distribution function of the computation time of the selected algorithm, the summary statistics of the objective function value, and the summary statistics of the computation time. These probability distribution functions and / or summary statistics are then used to estimate the probabilistic boundary of the objective function value that is substantially best. The probabilistic boundary is then used with different numbers of calculations to estimate the best balance between the number of calculations and the stopping criteria for effectively estimating the optimal arrangement of energy conduits in the array. That is, the probabilistic boundary is used to determine the empirical relationship between the stopping criteria and the distribution function or summary statistics, allowing a decision to be made (by an operator or via automated logic) regarding the best balance between the number of calculations (cumulatively equal to the total computer time) and the selected stopping criteria, which is also a factor in the time each computer takes to complete the calculation.

[0070] Figures 2a, 2b, 2c, and 2d are examples of heatmap outputs from the present invention's method, which can be used as part of the process of planning cable arrays in an offshore wind farm. In Figure 2a, studies to determine the optimal placement of cable arrays were performed through multiple calculations, and the placement obtained at the end of each study was used to create the heatmap. Each study may or may not have different input data for uncertain values, such as substation locations. In the heatmap, lighter colors indicate the locations where connections between turbines are most likely to be present at the end of the calculation, and darker colors indicate the locations where they are least likely to be present. This data can be interpreted as indicating that the locations where lighter colors are present are most likely to require connections, and therefore it is worth considering conducting an early-stage seabed survey in this area as part of a data collection process that will later be used in the final problem definition. Furthermore, the map also shows examples of locations where cables between turbines are least likely to be present, so it can be used to identify likely routes for cables connecting substations to the shore. Generally, it is undesirable for cables to cross each other, and the heatmap in Figure 2a can be used to determine routes for land cables where inter-turbine cables are less likely to cross, i.e., to reduce the constraints of the power transmission system that may arise when optimizing the inter-array cable layout considering the power transmission route.

[0071] Figures 2b and 2c show cable arrays at different locations. This example involves more calculations in the study, but uses a different scale, with each pixel representing a wider seafloor area. In Figure 2c, the turbine locations are indicated by circles. Similar to Figure 2a, brighter colors indicate a higher probability of cable presence in the seafloor represented by these pixels. The scale (Figure 2b) shows this probability as a probability.

[0072] In Figure 2d, a suitable location for substation 16 is determined. In this example, a study is performed for a given substation location to obtain a cable layout and its objective function value. Subsequently, further studies are performed, but with the substation location moved, and a different cable layout and its lifetime value are obtained. This process is repeated for numerous studies, each with a different substation location, and each study yields a cable layout and its lifetime value. Part of Figure 2d labeled 20 shows the location of each substation, represented by circles. The color of the circles (or the shadow in the black and white figure) represents the (relative) lifetime value of the optimal cable layout associated with that substation location. For example, a dark dot near the center of part 20 of the image indicates that by placing the substation at that coordinate and performing a study, a cable layout with lower cost or better lifetime value was obtained compared to other locations under consideration. Thus, this darkest shaded location at the center of the image represents the most promising location for the substation. This may be fixed, and then a study is performed to determine the optimal cable layout for this optimal substation location.

[0073] The method of the present invention will now be explained with further reference to Figure 3. The first step of the method of the present invention is to define the problem to be solved (step S1). Defining the problem includes defining the objective function, decision variables, and any constraints. This includes defining fixed parameters, such as turbine location, substation location, and onshore cable route, if these are not determined and are not part of the study being conducted. In step S2, a non-deterministic optimization algorithm is selected or defined and set. Next, in step S3, the stopping criteria for each calculation are set. The total number of calculations to be performed in the study is also set (step S4), and the calculation count variable is set to zero (step S5).

[0074] Next, in step S6, a seed value is set for the first calculation. The present invention is most efficiently executed using one or more servers, each having multiple or many processors, each performing the calculation. Ideally, this group of computers should contain a statistically significant number of processors that execute the algorithm. Statistical significance usually starts with 25 to 32 computers, but more (50, 100, 1000 or more) are preferable. Also, even a small number, such as 10, can produce useful results. In step S7, the computers (processors in the server) begin executing iterations of the calculation. The calculation count variable is incremented by 1 (step S8), and verification of whether sufficient calculations have been performed begins by checking whether the calculation count variable has reached the required total number of calculations (step S9). If more calculations are needed, the process loops through steps S6 to S9 again, with each more processor starting to perform more calculations with a new seed value until the required total number of calculations has been reached.

[0075] Once all computers have executed the algorithm multiple times (step S10), it is necessary to wait for all computers to reach the stopping criterion (step S11) and store the objective function value obtained from each calculation along with the associated cable layout (step S12). The best objective function value obtained from all calculations is selected and output along with the associated decision variables, and this is the best result recommended as the best estimate of the optimal cable layout between arrays.

[0076] Optionally, in step S14, an analysis of the objective function values ​​and the stopping criteria and number of calculations used to derive them may be performed to obtain additional information regarding the quality of the study and the results obtained. For example, a probability distribution function and / or summary statistics for the objective function values ​​may be calculated. Similarly, a probability distribution function and / or summary statistics for the calculation time may be determined. Examples of summary statistics include, but are not limited to, the mean, arbitrary percentile values, median, minimum and / or maximum values, standard deviation, and variance indicators. Since each calculation in the study is independent, the statistical distribution thus generated makes it possible to estimate "how lucky" one is compared to what might happen if the algorithm were run again. As a result, it may be determined whether the objective function values ​​returned at the end of the study were sufficiently lucky, or whether the study should be extended by performing further calculations. This also means that if the impact of changing the input data on the optimization results is examined, a stochastic boundary may be used instead of an optimal objective function value found in isolation, and it may be possible to determine the relationship without affecting the stochastic nature of the algorithm that introduces uncertainty into the results.

[0077] The resulting advantages are shown in Figure 4 by comparing two example results obtained using a prior art system (e.g., Fischetto Martina et al.), shown using circles, with two result sets of the present invention, shown using bell curve lines. The selection of input parameters has a significant impact on the achievable optimal solution. Comparing the two input sets, A (dashed line) and B (solid line), the prior art randomly samples individual optimized objective function values ​​from the solution space, as shown by circles in Figure 4. However, as described above, the present invention obtains a probabilistic boundary to the best reasonably achievable objective function value through a large number of calculations. In the scenario shown in Figure 4, the prior art is unknowingly unlucky when using input set B, and very lucky with a single calculation when using input set A. That is, the objective function value of the prior art B sample (solid circle) is coincidentally higher than the objective function value of the prior art A sample (dashed circle). Here, using the prior art method, the user concludes that input set A is optimal because it yields the highest objective function value after iterative calculations. However, the present invention precisely concludes that input set B is optimal because it is more likely to reach a more optimal objective function value. This is true even if the best individual calculation using input set A is better than all calculations using input set B, thus improving upon the method of simply iterating through the prior art to obtain the optimal solution.

[0078] Figure 5 shows the results of a typical study of the present invention compared to a prior art study. The prior art employs long computation times as the primary method for achieving higher lifetime value. Some embodiments of the prior art recognize the influence of the initial starting point of the optimization. When recognized, the prior art typically aims to avoid the influence of particularly unlucky seed values ​​producing abnormally poor solutions by performing several restarts. The present invention, on the other hand, aims to actively utilize this phenomenon by performing many more restarts to obtain particularly lucky seed values ​​that achieve very high objective function values ​​in a very short time. For example, where the prior art performs a single one-hour calculation, the present invention may prefer to perform 60 calculations of one minute each. In this example, both methods use the same computational resources, but the restarts in the present invention are performed in parallel, allowing the objective function value to be decomposed from the computation time. This enables the use of more complex objective functions that can better represent real-world problems. As a result, the reduction in computation time yields the counterintuitive advantage of being able to solve more complex problem formulations.

[0079] In addition to enabling the use of more complex objective function values, iterative studies using the present invention result in a narrower range of objective function values ​​centered on higher values, as shown in Figure 6. The solid line in Figure 6 (prior art studies) is the probability distribution representing the expected results obtained when iterative studies are performed using the prior art. Execution over long optimization times means the distribution is skewed to the right because the probability of very low objective function values ​​decreases. The dashed line represents a single calculation in a study of the present invention (one calculation of the present invention). As mentioned above, individual calculations are performed in very short times, meaning the range of objective function values ​​they are expected to return individually is skewed towards lower values. However, studies in the present invention consist of numerous short studies. As mentioned above with reference to Figure 5, by dividing computational resources into numerous restarts rather than a few long optimizations, time is saved by enabling parallelization, and there is a higher probability of reaching a better solution with the same computational resources. Furthermore, due to the law of large numbers, each study using the present invention more consistently achieves higher percentiles in the distribution of objective function values. As a result of all these factors, the predicted objective function value becomes even more skewed to the right, as shown by the dashed and dotted lines in Figure 6 (research of the present invention).

[0080] These analytical steps can also be used in pre-study research to help determine the appropriate balance between the number of calculations to be performed and the stopping criteria to be selected. Multiple studies are performed, each involving a large number of calculations but usually fewer than the overall study, and each study uses a different stopping criterion. Next, a distribution function for the stopping criteria is determined. These are then used to estimate the best balance between the number of calculations and the stopping criteria by determining the empirical relationship between the stopping criteria and the number of calculations, in order to efficiently estimate the optimal arrangement of energy conduits in the array. Once an effective balance between the number of calculations and the stopping criteria is determined, a study is performed using this number of calculations and stopping criteria to determine the optimized array.

[0081] Those skilled in the art will understand that the embodiments described above are merely illustrative and not restrictive, and that various changes and modifications are possible without departing from the scope of protection defined by the appended claims. For example, the embodiments described above should not be read as separate disclosures of independently functioning inventions. In particular, where various embodiments and aspects of the present invention are described above, the features and steps of the apparatus and methods are interchangeable between embodiments and aspects of the present invention. For example, where dimensions and numbers are given in the embodiments described above, these are examples and should not be taken as indicating that they are essential to the performance of the present invention.

[0082] Furthermore, the method of the present invention can be used to address other nonlinear problems. Such problems include, but are not limited to, route planning for delivery and / or collection, programming the movement of robotic arms during manufacturing processes, planning the layout of factory floors, modeling global temperature changes in climatology, and portfolio optimization in financial mathematics. This process is similar to that described above in that a nondeterministic optimization algorithm is set up and executed multiple times using substantially the same problem definition, substantially the same stopping criteria, and different seed values. The output of each computation of the algorithm in the study is the result of the determined decision variables and associated objective function values. The output with substantially the best objective function value is selected as the recommended solution to the problem, and the associated variables are provided.

[0083] The process described above is ideally carried out using numerous computers that perform these calculations in parallel. However, because the execution time is significantly shorter than that of prior art systems (usually measured in seconds or minutes rather than hours or days), it is possible to reduce the number of computers. For example, if half the number of computers are used, the total time required is still significantly shorter than that of a similar prior art system, even if the first half of the research is performed in the first computation run followed by the second half.

[0084] In the main embodiments of the present invention disclosed above, the example described is an offshore wind power plant. However, the present invention can be equally applied to other power generation systems, including but not limited to onshore wind power plants, wave and tidal power generators, and solar panel arrays. Also, in the above example, a wind turbine is given as an example of an energy-related device, and cables are given as an example of energy conduits. However, the present invention is equally applicable to other forms of energy-related devices. For example, the cable arrangement between numerous vehicle charging terminals in a large parking lot benefits from the optimization of connecting cables. In this example, the energy-related device is a vehicle charging point that consumes rather than generates electricity, and the cables are significantly shorter and have a lower transmission wattage than the cables for the wind turbines for power generation described above. As an alternative to cables being energy conduits, it has been proposed to transmit other forms of energy that can be used as energy sources, such as compressed air, hydrogen, or other fluid chemicals (e.g., ammonia). As an example, processes such as hydrolysis, which uses electricity generated by a turbine to decompose water into hydrogen and oxygen, are used. The hydrogen is then piped onshore in a series of pipes arranged similarly to the cables described above. It should be noted that the process of the present invention works whether the calculations in the study are performed sequentially or in parallel. However, parallel execution yields optimization results in a shorter timeframe.

Claims

1. A method for estimating the optimal arrangement of energy conduits within an array, A step of setting up a nondeterministic mixed-integer nonlinear programming algorithm in at least one computer device and optimizing the arrangement of energy conduits in an array as a nonlinear optimization problem, wherein the algorithm generates an output including at least one energy conduit arrangement and associated objective function values, The steps include: executing the calculation of the algorithm multiple times in parallel on at least one computer device to generate multiple outputs using substantially the same problem definition, substantially the same stopping criteria, and different seed values; A step of analyzing the plurality of outputs in the aforementioned computer device or another computer device and determining substantially the best objective function value and the associated energy conduit arrangement. A method for providing this.

2. The method according to claim 1, wherein the at least one mixed-integer nonlinear programming algorithm comprises optimizing the energy loss in the energy conduit.

3. The method according to claim 1 or 2, further comprising analyzing the plurality of objective function values ​​to obtain at least one of the probability distribution function of the objective function values, the probability distribution function of the computation time of the selected algorithm, summary statistics of the objective function values, and summary statistics of the computation time.

4. The method according to claim 3, further comprising estimating a probabilistic boundary for substantially the best objective function value using the probability distribution function or summary statistics of the objective function value.

5. Before performing the method according to claim 1, In multiple studies, each using different stopping criteria, the steps of the method according to claim 1 and claim 3 are performed, Using at least one of the aforementioned distribution functions, predict the stochastic boundary at different computation counts, and estimate the best balance between the number of computations and the stopping criterion for efficiently estimating the optimal arrangement of energy conduits in the array. To perform further research as described in any of claims 1 to 4, and to estimate the optimal arrangement of energy conduits in the array with the best balance between the number of calculations and the stopping criteria, The method according to any one of claims 1 to 4, further comprising:

6. The method according to claim 5, wherein at least one empirical relationship between the stopping criterion and the distribution function and / or the summary statistics is determined.

7. Cost of energy conduits, Energy loss in the energy conduit, Energy loss due to power loss in the energy conduit, Energy loss due to power outages, The cost of maintaining the aforementioned energy conduit, The cost of repairing the aforementioned energy conduit, The amount of energy reaching a given point in the array or energy conduit, The energy value passing through the energy conduit, and Cost of installing the aforementioned energy conduit The method according to any one of claims 1 to 6, wherein the objective function value is determined to include at least one of the following.

8. a) The plurality of outputs are used to generate a heatmap of the arrangement to provide the estimated optimal arrangement, b) The energy conduit includes a cable, c) The cable connects multiple energy generating devices to at least one node, d) The energy generation device comprises a wind turbine. The method according to any one of claims 1 to 7, further comprising one or more of the following features.

9. The method according to any one of claims 1 to 8, wherein, according to the termination criteria, the average execution time for a single execution of the algorithm is less than 60 minutes, less than 30 minutes, less than 15 minutes, or less than 10 minutes.

10. The method according to any one of claims 1 to 9, wherein the plurality of computers comprises at least 10 computers, or at least 25 computers, or at least 32 computers, or at least 50 computers, or at least 100 computers, or at least 1000 computers.

11. A method for installing an array of interconnected energy-related devices, The steps include determining the location of an array of energy-related devices, A step of using at least one computing device to operate the method according to any one of claims 1 to 10 to estimate the optimal arrangement of energy conduits interconnecting the array, The steps include installing the energy-related device and the energy conduit arranged as described above. A method for providing this.

12. The method according to claim 11, wherein the energy conduit comprises a cable.

13. The method according to claim 11 or 12, wherein the energy-related device comprises an energy generating device.

14. A method for estimating solutions to non-deterministic problems, A step of setting up a nondeterministic mixed-integer nonlinear programming algorithm on at least one computer device and optimizing a nonlinear optimization problem, wherein the algorithm generates an output including a solution to the problem and associated objective function values, The steps include: executing the calculation of the algorithm multiple times in parallel on at least one computer device to generate multiple outputs using substantially the same problem definition, substantially the same stopping criteria, and different seed values; A step of analyzing the multiple outputs in the aforementioned computer device or another computer device and determining substantially the best objective function value and the associated solution. A method for providing this.

15. The method according to claim 14, further comprising analyzing the plurality of objective function values ​​to obtain at least one of the probability distribution function of the objective function values, the probability distribution function of the computation time of the selected algorithm, summary statistics of the objective function values, and summary statistics of the computation time.

16. The method according to claim 15, further comprising estimating the substantially best probabilistic boundary of the objective function value using the probability distribution function or summary statistics of the objective function value.

17. Before performing the method according to claim 12, In multiple studies, each using different stopping criteria, the steps of the method according to claim 15 and claim 16 are performed, Using at least one of the distribution functions, predict the stochastic boundary at different computation counts, and estimate the best balance between the number of computations and the stopping criterion for efficiently estimating the optimal solution to the problem. To perform further research as described in any of claims 14 to 16, and to estimate the optimal solution to the problem with the best balance between the number of calculations and the stopping criteria. The method according to any one of claims 14 to 16, further comprising:

18. The method according to claim 17, wherein at least one empirical relationship between the stopping criterion and the distribution function and / or the summary statistics is determined.

19. The method according to any one of claims 14 to 18, wherein, according to the termination criteria, the average execution time for a single execution of the algorithm is less than 60 minutes, less than 30 minutes, less than 15 minutes, or less than 10 minutes.

20. The method according to any one of claims 14 to 18, wherein the plurality of computers comprises at least 10 computers, or at least 25 computers, or at least 32 computers, or at least 50 computers, or at least 100 computers, or at least 1000 computers.

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