Optimization Method for Economic Load Dispatch of Power System Based on Improved Golden Eagle Optimization Algorithm

Abstract

The present invention discloses an optimization method for economic load dispatch of power systems based on an improved Golden Eagle Optimization (GEO) algorithm. By utilizing the characteristics of Tent chaotic mapping, which has uniformity and orderliness, the diversity of the initial population is enhanced. By using the elite reverse group strategy, which can make full use of excellent individuals, the search space of the population is expanded. The linear attack and cruise weight of the original GEO algorithm are nonlinearly improved to increase the convergence speed of the algorithm. Finally, the improved GEO algorithm is applied to solve the economic load dispatch optimization model. The simulation results of test functions and IEEE 15-machine and IEEE 40-machine test systems show that the proposed scheme is reasonable and effective. Compared with the multi-machine power systems under existing swarm intelligence optimization algorithms, the multi-machine and multi-node power systems of economic load dispatch optimized by the improved Golden Eagle Optimization (IGEO) algorithm have more excellent system dynamic performance.

Description

Technical Field

[0001] The present invention relates to the technical field of power systems, and in particular to an economic load dispatch optimization method for power systems based on an improved golden eagle optimization algorithm. Background Technique

[0002] With the increasing complexity of the economic load dispatch (ELD) problem in power systems and the improvement of the requirements for power generation cost - effectiveness, complex computational time problems are faced when solving the economic load dispatch (ELD) problem of the economic operation of power system loads. Using additional methods such as gradient - based quadratic programming to handle the ELD problem has defects and deficiencies such as high computational time complexity, insufficient accuracy, slow convergence speed, and being prone to falling into local optima.

[0003] The load dispatch of a power system determines the output of each generator set within a given time period. By providing the load demand, each branch can share the load demand while meeting the actual limits, reducing the total operating cost of the entire system. Profitable load dispatch is a key issue in modern power systems and plays a crucial role in improving the economy of power systems. An economic load dispatch model for power systems considering upper and lower limits and power balance requirements. In practical applications, the economic load dispatch (ELD) problem also considers many actual non - linear constraints in the operation of power systems, such as generator forbidden zone constraints, valve - point effects, and active power output constraints, etc. Therefore, the economic load dispatch problem of power systems is a non - linear, highly non - linear multi - constraint optimization problem. The valve - point effect makes the solution of the ELD problem complex by introducing a large number of local optimizations in the process.

[0004] Swarm intelligence optimization algorithms are one of the most effective methods for solving complex non - linear problems. Swarm intelligence optimization algorithms are widely used in various fields such as engineering technology, medicine, society, and finance. In recent years, in the field of electrical engineering, such as in the economic load dispatch model of power systems, intelligent swarm optimization methods have become one of the most promising optimization techniques for solving severely constrained non - linear and non - convex optimization problems. In modern power system research, a popular field is to apply various algorithms to solve the profitable dispatch challenges in power systems while improving accuracy and practicality.

[0005] Typical swarm intelligence algorithms include Particle Swarm Optimization (PSO), Differential Evolution (DE), Artificial Immune Algorithm (AIA), and Moth-Flame Optimization (MFO), which have been widely applied in computational optimization. The Golden Eagle Optimization (GEO) algorithm is a newly proposed swarm intelligence optimization algorithm. The core inspiration of GEO is the strategy of golden eagles adjusting their speeds at different stages of spiral trajectories for hunting. The GEO algorithm has advantages such as few parameters and easy implementation, and can well solve non-linear equilibrium problems such as economic load dispatch optimization of large power systems. However, like other intelligent algorithms, as the number of iterations of the GEO algorithm increases, the population diversity of golden eagles decreases, the algorithm is prone to falling into local optima, the global search ability decreases, and ultimately the algorithm has a slow convergence speed and low convergence accuracy. Summary of the Invention

[0006] The purpose of the present invention is to provide an economic load dispatch optimization method for power systems based on an improved Golden Eagle Optimization algorithm.

[0007] The technical solution adopted by the present invention is as follows:

[0008] An economic load dispatch optimization method for power systems based on an improved Golden Eagle Optimization algorithm, which includes the following steps:

[0009] Step S1, taking the minimum total generation cost of the generator set as the objective function, establish a mathematical model for economic load dispatch of the power system; the mathematical model for economic load dispatch of the power system includes generation cost and constraint conditions;

[0010] Step S2, introduce the Tent chaotic mapping strategy into the Golden Eagle Optimization algorithm, and initialize the positions of golden eagles and relevant initial data;

[0011] The iterative formula of the Tent mapping is:

[0012]

[0013] In the formula: in the present invention, β takes 0.4, and after iteration, a chaotic sequence [z1, z2,..., z p is generated, z t is the t-th value of the chaotic sequence generated after iteration, and finally a uniformly distributed initial population is generated, and the expression is:

[0014] X i =l s +z i (u s -l s )

[0015] In the formula: l s and u s respectively represent the lower and upper limits of the search space; z i is the i-th value in the Tent chaotic sequence; Xi is the i-th individual of the population.

[0016] Step S3, introduce the reverse elite learning strategy into the main loop of the golden eagle optimization algorithm, and use the exponential decay function to obtain the attack coefficient p a and the cruising coefficient p c ;

[0017] Step S4, the economic load dispatch model of the power system performs optimization based on the improved golden eagle optimization algorithm under the constraint conditions, and calculates the minimum generation cost of the generator set using the optimal solution.

[0018] Furthermore, the ELD objective function in Step S1 is expressed as:

[0019]

[0020] where: F cost is the total generation cost of the system; T is the total number of dispatch periods, and when dealing with static optimization problems, T is 1; M is the number of units in the system; P i is the active power output value of unit i; F i (P i ) represents the generation cost of the unit.

[0021] Furthermore, the mathematical models of the target cost function are divided into two categories: without valve point effect and considering valve point effect:

[0022] When there is no valve point effect, the generation cost of thermal power unit i is represented by its consumption characteristic function, and the mathematical expression is:

[0023]

[0024] In the formula: a i , b i , c i are the consumption characteristic coefficients of unit i;

[0025] When considering the valve point effect, the objective function can be expressed as:

[0026]

[0027] In the formula: e i , f i are the valve point effect coefficients of unit i; P imin is the lower limit of the active power output of unit i.

[0028] Furthermore, the constraint conditions for economic load distribution include power balance constraint, unit output constraint, ramp rate constraint, and unit operation prohibited zone constraint.

[0029] (1) The system power balance constraint consists of the active power output of the units, the system power loss, and the total system load, and its expression is as follows:

[0030]

[0031] Where: P loss is the system power loss; P load is the total system load.

[0032] (2) The expression of the unit output constraint is as follows:

[0033] P imin ≤P i ≤P imax (6)

[0034] Where: P imin , P imax are the lower limit and upper limit of the active power output of unit i, respectively.

[0035] (3) The expression of the unit ramp constraint is as follows:

[0036] -DR i ≤P ti -P (t-1)i ≤UR i (7)

[0037] Where: DR i , UR i are the extreme value of the output deceleration and the extreme value of the output acceleration of unit i, respectively; P 0i is the active power output of unit i at the previous moment.

[0038] (4) The unit operation forbidden zone constraint is expressed as:

[0039]

[0040] Where: P imin is the lower limit of the output of unit i; P i is the active power output value of unit i; is the lower limit of the minimum embargo zone in the total number of operation forbidden zones of unit i; is the upper limit of the (j - 1)th operation forbidden zone of unit i; is the lower limit of the jth operation forbidden zone of unit i; N g is the total number of operation forbidden zones of unit i; is the upper limit of the maximum embargo zone in the total number of operation forbidden zones of unit i; P max is the upper limit of the output of the ith unit.

[0041] Furthermore, the system power loss is obtained by the B - coefficient method P loss ,

[0042]

[0043] Where: B ij , B oi , B oo is the line loss coefficient.

[0044] Furthermore, the value of β is 0.4.

[0045] Furthermore, step S3 specifically includes the following steps:

[0046] Step S3-1, sort the current golden eagle population individuals in ascending order of fitness value, and then select the top N golden eagle individuals with better fitness values (N represents the number of elite individuals) as the elite group;

[0047] Furthermore, the number of elite individuals is generally taken as N = 0.1 × population size P.

[0048] Step S3-2, obtain the reverse solutions of the elite group. The calculation formula for each reverse individual is:

[0049]

[0050] Where is the reverse solution of the elite group, x Em (t) is the elite group, and l(t) and u(t) respectively represent the minimum and maximum value vectors of each dimension parameter in the current golden eagle population at different times;

[0051] Step S3-3, generate N uniformly distributed new individuals using each elite solution and its reverse solution. The calculation formula is:

[0052]

[0053] Where s Em (t) represents the mth individual;

[0054] Step S3-4, obtain the current attack vector of the golden eagle. The calculation formula is as follows:

[0055]

[0056] Where: A i is the attack vector of golden eagle i, is the best position (prey) visited by golden eagle f so far, is the current position of golden eagle i;

[0057] Step S3-5, obtain the target point on the cruise hyperplane. The expression is as follows:

[0058]

[0059] where: c k is the k-th element of the target point C, a j is the j-th element of the attack vector A i , d is on the right side of Equation (10), is the k-th element of the attack vector A i , and k is the index of the fixed variable;

[0060] Step S3-6, perform a non-linear adjustment and change on the attack coefficient p a and the cruise coefficient p c , and the specific formula is as follows:

[0061]

[0062]

[0063] where: and are the maximum and minimum values of the attack coefficient p a respectively; and are the maximum and minimum values of the cruise coefficient p c respectively; α is the adjustment coefficient; t and T are the current iteration number and the maximum iteration number respectively.

[0064] Furthermore, the specific steps of Step S3-5 are as follows:

[0065] Step S3-5-1, calculate the hyperplane of the cruise vector of the golden eagle i in iteration t, and the calculation formula is as follows:

[0066]

[0067] where: a j is the j-th element of the attack vector A i , t is the current iteration number, is the j-th element of the attack vector A i corresponding to the current iteration number, A i ={a1, a2,..., a n} is the attack vector of the golden eagle i; x j is the j-th element of the decision variable vector X i , X i ={x1, x2,..., x n} is the decision variable vector of the golden eagle i, that is, the current position of the golden eagle; is the j-th element of the position X * of the prey selected by the golden eagle, is the position of the prey selected by the golden eagle;

[0068] Step S3-5-2: Obtain the cruising vector c of the golden eagle i in the cruising hyperplane of the eagle in the iteration k , the cruising vector c k is defined by the following formula:

[0069]

[0070] In the formula: c k is the k-th element of the target point C, a j is the j-th element of the attack vector A i , d is the right side of formula (10), is the k-th element of the attack vector A i , and k is the index of the fixed variable;

[0071] Step S3-5-3: Obtain a random target point on the cruising hyperplane. The expression of the target point on the cruising hyperplane is as follows:

[0072]

[0073] Furthermore, in step S3-6, c max takes the value of 1, c min takes the value of 0.5, and α takes the value of 4.

[0074] Furthermore, step S4 specifically includes the following steps:

[0075] Step S4-1: Use the improved golden eagle optimization algorithm to perform elite reverse learning processing on the current population to obtain a new population;

[0076] Step S4-2: Update the attack coefficient and the cruising coefficient according to the current iteration number; the attack coefficient p a and the cruising coefficient p c

[0077] Step S4-3: Each golden eagle selects a target prey from the memory of the entire flock based on the best solution found so far;

[0078] Step S4-4: Calculate the attack vector and the cruising vector of each golden eagle and calculate Δx;

[0079]

[0080] The step vector of the golden eagle i in the iteration is defined by formula (14).

[0081]

[0082] In the formula: is the attack coefficient in the iteration t, For the cruise coefficient in iteration t, the influence of the golden eagle being attacked and cruising is adjusted. and is a random vector with elements in the interval [0, 1];

[0083] The position of the golden eagle in iteration t + 1 can be calculated by adding the step vector in iteration t to the position in iteration t.

[0084]

[0085] Step S4 - 5: Determine whether all golden eagle individuals meet the embargo area constraint, slow - change area constraint, and power and power balance constraint of the ELD problem; when all meet, continue to loop and execute step S4 - 6; when there are those that do not meet the constraint conditions, re - randomize the non - compliant golden eagle individuals; continue to loop until all individuals meet the constraint conditions.

[0086] Step S4 - 6: Substitute the current position of the golden eagle into the objective function, calculate the current fitness values of all golden eagles and compare them with the memory position, and determine whether the current fitness value is better than the saved optimal fitness value; if so, use the current fitness value as the optimal fitness value; otherwise, keep the current optimal fitness value unchanged.

[0087] Step S4 - 7: Take the iterative population as the new current population, and determine whether the iteration end condition (reaching the maximum number of iterations) is met; if so, output the optimal individual as the optimal solution; otherwise, return to execute step S4 - 1.

[0088] Furthermore, in step S4 - 4 For and The intermediate value of is calculated by linear transition.

[0089]

[0090] In the formula: t represents the current iteration number, t represents the maximum iteration number, and are respectively the initial value and the final value of the attack coefficient p a and and are respectively the initial value and the final value of the cruise coefficient p c respectively.

[0091] The present invention adopts the above technical solutions to improve the GEO algorithm. By utilizing the characteristics of Tent chaotic mapping, which has uniformity and orderliness, the diversity of the initial population is enhanced. The elite reverse group strategy is used to make full use of the characteristics of excellent individuals to expand the search space of the population. The linear attack and cruise weight of the original GEO algorithm are non-linearly improved to increase the convergence speed of the algorithm. Finally, the improved GEO algorithm is applied to solve the economic load dispatch optimization model. The simulation results of the test functions and the IEEE 15-machine and IEEE 40-machine test systems show that the proposed scheme is reasonable and effective. Compared with the multi-machine power system under the action of the existing swarm intelligence optimization algorithms, the multi-machine multi-node power system with economic load dispatch optimized by the improved Golden Eagle Optimization algorithm (IGEO) has superior system dynamic performance. Brief Description of the Drawings

[0092] The following further elaborates on the present invention in detail in conjunction with the drawings and specific embodiments;

[0093] Figure 1 It is a schematic diagram of the main step flow of the power system economic load dispatch optimization method based on the improved Golden Eagle Optimization algorithm of the present invention;

[0094] Figure 2 It is a flow chart of the improved Golden Eagle Optimization algorithm;

[0095] Figure 3 It is a flow chart of the ELD application of the improved Golden Eagle Optimization algorithm;

[0096] Figure 4 It is a Matyas iteration convergence graph;

[0097] Figure 5 It is a Sphere iteration convergence graph;

[0098] Figure 6 It is a Three-hump camel iteration convergence graph;

[0099] Figure 7 It is an Ackley 1 iteration convergence graph;

[0100] Figure 8 It is a Penalized 1 iteration convergence graph;

[0101] Figure 9 It is a Periodic iteration convergence graph;

[0102] Figure 10 It is an IEEE 15 test unit iteration graph;

[0103] Figure 11 It is an IEEE 40 test unit iteration graph. Specific Embodiments

[0104] To make the objectives, technical solutions, and advantages of the embodiments of this application clearer, the technical solutions in the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings in the embodiments of this application.

[0105] As Figure 1 shown, the present invention discloses an economic load dispatch optimization method for a power system based on an improved golden eagle optimization algorithm, which includes the following steps:

[0106] Step S1: Establish a mathematical model for economic load dispatch of the power system with the minimum total generation cost of the generator set as the objective function; the mathematical model for economic load dispatch of the power system includes generation cost and constraint conditions;

[0107] Step S2: Introduce the Tent chaos mapping strategy into the golden eagle optimization algorithm and initialize the positions of the golden eagles and relevant initial data;

[0108] The iterative formula of the Tent mapping is:

[0109]

[0110] In the formula: In the present invention, β is taken as 0.4, and a chaotic sequence [z1, z2,..., z p is generated through iteration, and z t is the t-th value of the chaotic sequence generated through iteration. Finally, an initial population with a uniform distribution is generated, and the expression is:

[0111] X i = l s + z i (u s - l s )

[0112] In the formula: l s and u s respectively represent the lower and upper limits of the search space; z i is the i-th value in the Tent chaotic sequence.

[0113] Step S3: Introduce the reverse elite learning strategy into the main loop of the golden eagle optimization algorithm and use the exponential decay function to obtain the attack p a vector and the cruise vector p c ;

[0114] Step S4: The economic load dispatch model of the power system performs optimization based on the improved golden eagle optimization algorithm under constraint conditions, and uses the optimal solution to calculate the minimum generation cost of the generator set.

[0115] The following is a detailed description of the specific principle of the present invention:

[0116] To determine the economic distribution of load among operating units, the operating cost of a unit is generally expressed as a function of the unit's output. If the cost function of the power system is regarded as the cumulative sum of a series of quadratic polynomials, the ELD objective function can be expressed as:

[0117]

[0118] Where: F cost is the total power generation cost of the system; T is the total number of dispatching periods. When dealing with static optimization problems, T is 1; M is the number of units in the system; P i is the active power output value of unit i; F i (P i ) represents the power generation cost of the unit.

[0119] For the static economic dispatch problem of a power system with single - energy power generation adopted in the present invention, the mathematical models of the objective cost function are divided into two categories: without valve - point effect and considering valve - point effect, which can be represented by Equation (2) and Equation (3) respectively.

[0120] Generally, the power generation cost of thermal power unit i can be represented by its consumption characteristic function, and the mathematical expression is:

[0121]

[0122] In the formula: a i , b i , c i are the consumption characteristic coefficients of unit i.

[0123] If the valve - point effect is considered, the objective function can be expressed as:

[0124]

[0125] In the formula: e i , f i are the valve - point effect coefficients of unit i; P imin is the lower limit of the active power output of unit i.

[0126] Finally, the constraint conditions of the economic load dispatch of the power system are introduced. The economic load distribution problem includes multiple groups of constraint conditions, including power balance constraint, unit output constraint, ramp - rate constraint, and unit operation forbidden - zone constraint.

[0127] (1) System power balance constraint

[0128] The system power balance constraint consists of the active power output of the units, the system network loss, and the system total load.

[0129]

[0130] In the formula: P lossis the system network loss; P load is the total system load. The system network loss is obtained by the B - coefficient method as shown below.

[0131]

[0132] In the formula: B ij , B oi , B oo are network loss coefficients.

[0133] (2) Generator output constraint

[0134] P imin ≤P i ≤P imax (6)

[0135] In the formula: P imin , P imax are the lower and upper limits of the active power output of generator set i respectively.

[0136] (3) Generator ramp rate constraint

[0137] -DR i ≤P ti -P (t-1)i ≤UR i (7)

[0138] In the formula: DR i , UR i are the extreme values of the output deceleration and output acceleration of generator set i respectively; P 0i is the active power output of generator set i at the previous moment.

[0139] (4) Generator operation forbidden zone constraint

[0140] When a thermal power generator is in operation, there will be some sub - intervals within its operating range. When the thermal power generator operates in these sub - intervals, it will cause the vibration amplitude of the generator bearings to be too large. Therefore, it is necessary to set an operation forbidden zone within the operating range to avoid these sub - spaces and prevent excessive vibration of the generator bearings. The operating range with an operation forbidden zone set can be expressed as:

[0141]

[0142] In the formula: P imin is the lower limit of the output of generator set i; P i is the active power output value of generator set i; is the lower limit of the smallest embargo zone among the total number of operation forbidden zones of generator set i; is the upper limit of the (j - 1) - th operation forbidden zone of generator set i; is the lower limit of the j - th operation forbidden zone of generator set i; N gis the total number of operating restricted areas of unit i; is the upper limit of the largest embargo area among the total number of operating restricted areas of unit i; P max is the upper limit of the output of the i-th unit.

[0143] Basic principle of the Golden Eagle Optimization Algorithm: In 2021, Mohammadi - Balani Abdolkarim et al. proposed the Golden Eagle Optimization Algorithm (GEO). In the GEO algorithm, golden eagles and target prey are important groups. The individual golden eagle represents a candidate solution to the optimization problem, and the prey is the target around which the golden eagle chooses to cruise until the current iteration. Therefore, the prey can serve as a "vane" for the golden eagle to cruise in the feasible search space. Each golden eagle cruises around a prey. When the cruising intention is stronger in the early stage of iteration, when a better solution is found, it updates its memory of the optimal prey.

[0144] Obtain the current attack vector of the golden eagle:

[0145]

[0146] In the formula: A i is the attack vector of golden eagle i, is the best position (prey) visited by golden eagle f so far, is the current position of golden eagle i.

[0147] Before calculating the cruising vector, the equation of the tangent hyperplane must be calculated first. Formula (10) is the scalar form of the hyperplane equation in n-dimensional space.

[0148]

[0149] In the formula: is the normal vector, is the variable vector, is an arbitrary point on the hyperplane,

[0150] If we consider (the position of the golden eagle) as an arbitrary point on the hyperplane and consider (the attack vector) as the normal of the hyperplane, then the hyperplane of (the cruising vector of golden eagle i in iteration t) can be obtained according to formula (11).

[0151]

[0152] In the formula: A i ={a1, a2,..., a n} is the attack vector of golden eagle i, X i ={x1, x2,..., xn} is the decision variable vector of Golden Eagle i (i.e., the current position of the Golden Eagle), the position of the prey selected by the Golden Eagle.

[0153] After calculating the cruising hyperplane of the eagle in the iteration, the cruising vector of Golden Eagle i can be found in this hyperplane, as shown in Equation (12).

[0154]

[0155] In the formula: c k is the k-th element of the target point C, a j is the j-th element of the attack vector A i , d is the right side of Equation (10), is the k-th element of the attack vector A i , and k is the index of the fixed variable.

[0156] Find a random target point on the cruising hyperplane. Equation (5) shows the general representation of the target point on the cruising hyperplane.

[0157]

[0158] The displacement of the Golden Eagle includes attack and vector. Define the step vector of Golden Eagle i in the iteration as Equation (14).

[0159]

[0160] In the formula: is the attack coefficient in iteration t, is the cruising coefficient in iteration t, which adjusts the influence of the Golden Eagle being attacked and cruising. and are random vectors whose elements are in the interval [0, 1].

[0161]

[0162] The position of the Golden Eagle in iteration t + 1 can be calculated by simply adding the step vector in iteration t to the position in iteration t.

[0163]

[0164] If the fitness of the new position of the Golden Eagle is better than its position in memory, update the memory of this eagle to the new position. Otherwise, the memory position remains unchanged, but the eagle will stay at the new position. In the new iteration, each Golden Eagle randomly selects a Golden Eagle from the population to circle around its memory-optimal position, calculates the attack vector, calculates the cruising vector, and finally calculates the step vector and the new position for the next iteration. Execute this loop until any termination condition is met.

[0165] GEO uses p a and p c to switch from cruise exploration to development. The algorithm starts from low p a and high p c As the iteration progresses, p a gradually increases and p c gradually decreases. The initial and final values of both parameters are defined by the user. The intermediate values can be calculated by the linear transition shown in Equation (17).

[0166]

[0167] In the formula: t represents the current iteration number, t represents the maximum iteration number, and are the initial value and the final value of the attack tendency p a respectively, and are the initial value and the final value of the cruise tendency p c respectively.

[0168] Take and This means that p a is set to 0.5 in the first iteration and linearly decreases to 2 in the last iteration. The same is true for p c which starts with 1 in the first iteration and linearly decreases to 0.5 in the last iteration.

[0169] Improved Golden Eagle Optimization Algorithm: Improved measures are added to the three links of population initialization, loop phase, and attack and cruise vector update in the Golden Eagle Optimization Algorithm, which is beneficial to improving the optimization efficiency of the algorithm and the ability to jump out of the local optimal solution, forming an improved Golden Eagle Optimization Algorithm.

[0170] Tent Chaotic Initialization of Population: The Golden Eagle Optimization Algorithm uses the method of randomly initializing the population, which is difficult to retain the diversity of the population and has poor optimization effect. The Tent chaotic mapping has the characteristics of randomness, uniformity, and orderliness. Using the Tent chaotic mapping can improve the population diversity of the initial population in the search space, making the dragonfly individuals closer to the optimal solution to improve the convergence efficiency of the algorithm. The iterative formula of the Tent mapping is:

[0171]

[0172] In the formula: In the present invention, β takes 0.4, and after iteration, a chaotic sequence [z1, z2,..., z p is generated, and z t is the t-th value of the chaotic sequence generated after iteration. Finally, an initial population with a uniform distribution is generated, and the expression is

[0173] X i = l s + z i (u s - l s )

[0174] where: l s and u s represent the lower and upper limits of the search space respectively; z i is the i-th value in the Tent chaotic sequence, and X i is the i-th individual of the population.

[0175] Elitist Opposition-based Learning Strategy: The elitist opposition-based learning strategy is introduced into the main loop of the Golden Eagle Optimization Algorithm. During the loop stage of the algorithm, the elitist opposition strategy is introduced to improve the diversity of the population and reduce the possibility of the algorithm falling into local optimal solutions. The opposition-based learning (OBL) strategy was proposed by Tizhoosh, which evaluates both the feasible solution and its opposite solution simultaneously and selects the optimal one to enter the next generation. The elitist opposition group strategy combines the idea of the elitist algorithm on the basis of OBL, generates a group of uniformly distributed elitist opposition groups between each elite and its opposite solution, and selects the B groups of solutions with better fitness to form a new population. The specific steps are as follows:

[0176] Step 1: Arrange the individuals of the current golden eagle population in ascending order of fitness values, and then select the top N golden eagle individuals with better fitness values (N represents the number of elite individuals) as the elite group. Generally, the number of elite individuals is taken as N = 0.1 × population size P

[0177] Step 2: Obtain the opposite solutions of the elite group. The calculation formula for each opposite individual is

[0178]

[0179] where is the opposite solution of the elite group, x Em (t) is the elite group, and l(t) and u(t) represent the minimum and maximum value vectors of each dimension parameter in the current golden eagle population at different times respectively.

[0180] Step 3: Generate N uniformly distributed new individuals using each elite solution and its opposite solution. The calculation formula is:

[0181]

[0182] where s Em (t) represents the m-th individual.

[0183] Improving the Attack and Cruise Vector Updates with an Exponentially Decreasing Function: Use the exponentially decreasing function to obtain the attack coefficient p a and the cruise coefficient pc It is an improvement on the calculation methods of the attack coefficient p Figure 2 as shown and the cruise coefficient p a to accelerate the convergence speed of the algorithm. In the original Golden Eagle algorithm, the attack coefficient p c and the cruise coefficient p a are both linearly varying values. In actual situations, as the number of algorithm iterations increases, the convergence speed of the algorithm will gradually decrease, showing a curve decline. In the algorithm, the convergence speeds of the attack coefficient p c and the cruise coefficient p a are inconsistent with the overall convergence speed of the algorithm, which will reduce the convergence speed of the algorithm and thus affect the overall efficiency of the algorithm. In the present invention, an exponential decay strategy is introduced, and the attack coefficient p c and the cruise coefficient p a are non-linearly adjusted and changed through the following formula: c In the formula:

[0184]

[0185]

[0186] Where: and are respectively the maximum and minimum values of the attack coefficient p a ; and are respectively the maximum and minimum values of the cruise coefficient p c ; α is the adjustment coefficient; t and T are respectively the current iteration number and the maximum iteration number.

[0187] Experimental simulation verification

[0188] Test functions: To verify the performance of the proposed improved Golden Eagle optimization algorithm (IGEO), IGEO is compared with GEO. Six standard test functions are shown in Table 1, where f1 to f3 are unimodal functions and f4 to f6 are multimodal functions. The simulation parameters of each algorithm are set as follows: In GEO and IGEO the maximum iteration number of each algorithm is 600, and the population size is 30. The four optimization algorithms are optimized 30 times on six standard test functions, and the test results are shown in Table 2, and the iteration curves are as Figures 4 to 9 shown.

[0189] Table 1: Standard test functions

[0190]

[0191] To avoid the contingency of the experiment, each function is independently run 30 times, and the average value and error results of the final optimization of each algorithm are shown in Table 2.

[0192] Table 2: Comparison Results of Function Tests

[0193]

[0194]

[0195] It can be clearly seen from the data in the table that compared with the other original Golden Eagle optimization algorithms, the optimal fitness value of the IGEO algorithm is closest to the target value, and its standard deviation is the smallest. It can be clearly seen from the convergence graphs of each test function that both the convergence speed and the ability to avoid falling into local optimal solutions of IGEO are better than those of GEO.

[0196] To demonstrate the superiority of the present invention, in this embodiment, simulations are respectively carried out using the IEEE15-unit and 40-unit generator test systems. The economic dispatch statistical results obtained by using the Particle Swarm Optimization algorithm (PSO), Grey Wolf Optimization algorithm (GWO), Golden Eagle Optimization algorithm (GEO), and the improved Golden Eagle Optimization algorithm (IGEO) based on the present invention are compared respectively. At the same time, the valve point effect is introduced. Due to the excessive number of units, the grid loss is not considered for the time being, and the basic parameters of the system are set to the conventional settings.

[0197] IEEE15-unit test system: During the simulation, the maximum number of iterations of the IEEE15-unit test system is set to T = 700 times, and the population size is set to 30. The 15-unit test system also considers the ramp rate constraint, that is, the ramp constraint, on the basis of the static system. Its total load demand is 2630 MW. The upper and lower limits of the output of each unit are considered, the embargo area constraint is considered, and the network loss is considered. To avoid the randomness problem of the algorithm, each of the four algorithms is independently executed 50 times and the average value is obtained for comparison. The setting parameters of the IEEE15-unit test system, such as the generator set system restricted area and coefficient B: [B ij , B oi , B oo , and the power loss factor are shown in Table 3, and the parameters of the 15-unit test system are shown in Table 4.

[0198] The results of the four algorithms in the 15-unit test system are shown in Table 5, and the iterative comparison graph is shown in Figure 10 .

[0199] Table 3: Settings of Generator Set Restricted Areas in the 15-Unit Test System

[0200]

[0201] The power loss factor B is expressed as:

[0202]

[0203] B 0i= [-0.0001 -0.0002 0.0028 -0.0001 0.0001 -0.0003 -0.0002 -0.0002 0.0006 0.0039 -00017 0.0000 -0.0032 0.0067 -0.0064]

[0204] B oo = 0.0055

[0205] Table 4: Parameters of the 15 - unit test system

[0206] unit <![CDATA[α ($ / MW 2 )]]> β ($ / MW) γ ($) UR (MW / h) LR (MW / h) <![CDATA[P0]]> <![CDATA[P min (MW)]]> <![CDATA[P max (MW)]]> 1 0.000299 10.1 671 80 120 400 150 455 2 0.000183 10.2 574 80 120 300 150 455 3 0.001126 8.8 374 130 130 105 20 130 4 0.001126 8.8 374 130 130 100 20 130 5 0.000205 10.4 461 80 120 90 150 470 6 0.000301 10.1 630 80 120 400 135 460 7 0.000364 9.8 548 80 120 350 135 465 8 0.000338 11.2 227 65 100 95 60 300 9 0.000807 11.2 173 60 100 105 25 162 10 0.001203 10.7 175 60 100 110 25 160 11 0.003586 10.2 186 80 80 60 20 80 12 0.005513 9.9 230 80 80 40 20 80 13 0.000371 13.1 225 80 80 30 25 85 14 0.001929 12.1 309 55 55 20 15 55 15 0.004447 12.4 323 55 55 20 15 55

[0207] Table 5: Comparison of results of 15 units

[0208]

[0209]

[0210] IEEE 40 - machine test system: During simulation, the maximum number of iterations of the IEEE 40 - unit test system is set to T = 1000 times, and the population size is set to 50. The 40 - unit test system is a static system. The ramp - rate constraints of the units are not considered. Its total load demand is 10500 MW. The embargo - zone constraints and network losses are not considered. The 40 - machine test system is a larger system with more local optimal solutions than the 15 - machine test system. Therefore, the robustness of the algorithm can be better demonstrated during the calculation of the 40 - unit system. In the present invention, considering the practicality of the model, the "valve - point effect" is considered when calculating the 40 - machine test system. To avoid the randomness problem of the algorithm, each of the four algorithms is independently executed 50 times and the average value is obtained. The parameters of the 40 - unit test system are shown in Table 6.

[0211] The results of the four algorithms in the 40 - unit test system are shown in Table 7, and the iteration comparison graph is shown in Figure 11 .

[0212] Table 6: Parameters of the 40 - unit test system

[0213]

[0214]

[0215] Table 7: Comparison of results of 40 units

[0216] algorithm average value maximum value minimum value standard deviation PSO 142942.5 153461.3 135543.6 4042.266 GWO 130614.1 133261.3 128039.9 1235.554 GEO 131311.2 134812.8 128427.5 1391.697 IGEO 130423.9 132697.8 128037.1 959.5933

[0217] From Figure 10 、 Figure 11From the curves in and the data in Table 5 and Table 7, it can be seen that in high-dimensional cases, the IGEO algorithm has significantly stronger optimization and configuration capabilities compared to other algorithms. The running results of the simulation experiments show that the average, maximum, and minimum values of the optimal values obtained by the IGEO algorithm are the lowest among the listed algorithms, its convergence performance is excellent, and its standard deviation is also lower than that of other algorithms, indicating that the IGEO algorithm can still maintain good robustness in high dimensions.

[0218] In summary, the test results verify the superiority and effectiveness of using the improved golden eagle optimization algorithm to solve the economic dispatch problem of power systems. The obtained optimal allocation solution has a higher convergence accuracy and a lower required generation cost.

[0219] The present invention adopts the above technical solutions to improve the GEO algorithm. By utilizing the characteristics of uniformness and orderliness of the Tent chaotic mapping, the diversity of the initial population is improved; by utilizing the characteristics of the elite reverse group strategy that can make full use of excellent individuals, the search space of the population is expanded; the linear attack and cruise weight of the original GEO algorithm are non-linearly improved to increase the convergence speed of the algorithm. Finally, the improved GEO algorithm is applied to solve the economic load dispatch optimization model. The simulation results of the test function and the IEEE 15-machine and IEEE 40-machine test systems show that the proposed scheme is reasonable and effective. Compared with the multi-machine power system under the action of the existing swarm intelligence optimization algorithm, the multi-machine multi-node power system with economic load dispatch optimized by the improved golden eagle optimization algorithm (IGEO) has more excellent system dynamic performance.

[0220] Obviously, the described embodiments are part of the embodiments of the present application, rather than all of them. Without conflict, the embodiments and features in the present application can be combined with each other. Usually, the components of the embodiments of the present application described and illustrated herein can be arranged and designed in various different configurations. Therefore, the detailed description of the embodiments of the present application is not intended to limit the scope of the present application claimed, but merely represents the selected embodiments of the present application. Based on the embodiments in the present application, all other embodiments obtained by those of ordinary skill in the art without creative efforts belong to the scope of protection of the present application.

Claims

1. An economic load dispatch optimization method for power systems based on an improved golden eagle optimization algorithm, characterized in that: It includes the following steps: Step S1: Establish a mathematical model for economic load dispatch of the power system with the minimum total generation cost of the generating set as the objective function. The mathematical model for economic load dispatch of the power system includes generation cost and constraint conditions. The objective function is expressed as: Among them: F cost is the total power generation cost of the system; T is the total number of dispatching periods. When dealing with static optimization problems, T is 1; M is the number of units in the system; P i is the active power output value of unit i; F i (P i ) represents the power generation cost of the unit; The mathematical model of the objective function is divided into two categories: without valve point effect and considering valve point effect: When there is no valve point effect, the generation cost of thermal power unit i is represented by its consumption characteristic function, and the mathematical expression is: where: a i , b i , c i are the consumption characteristic coefficients of unit i; When considering the valve point effect, the objective function is expressed as: Where: e i , f i are the valve point effect coefficients of unit i; P imin is the lower limit of the active power output of unit i; The constraint conditions for economic load distribution include power balance constraint, unit output constraint, ramp constraint, and unit operation embargo area constraint; (1) The system power balance constraint consists of the active power output of the units, the system power loss, and the total system load, and its expression is as follows: Where: P loss is the system network loss; P load is the total system load; (2) The expression of the unit output constraint is as follows: P imin ≤P i ≤P imax (6) where: P imin , P imax are respectively the lower limit and upper limit of the active power output of unit i; (3) The expression of the unit ramp constraint is as follows: -DR i ≤P ti -P (t-1)i ≤UR i (7) where: DR i , UR i are respectively the extreme value of output speed reduction and the extreme value of output speed increase of unit i; P (t-1)i is the active power output of unit i at the previous moment; (4) The unit operation no-go zone constraint is expressed as: Where: P imin is the lower limit of the output of unit i; P i is the active power output value of unit i; is the lower limit of the minimum embargo zone among the total number of operating embargo zones of unit i; is the upper limit of the (j - 1)-th operating embargo zone of unit i; is the lower limit of the j-th operating embargo zone of unit i; N g is the total number of operating embargo zones of unit i; is the upper limit of the maximum embargo zone among the total number of operating embargo zones of unit i; P max is the upper limit of the output of the i-th unit; Step S2: Introduce the Tent chaos mapping strategy into the Golden Eagle optimization algorithm, and initialize the positions of the golden eagles and related initial data; The iterative formula of the Tent mapping is: where: β = 0.4, to iteratively generate a chaotic sequence [z1, z2, …, z p , z t is the t-th value of the chaotic sequence generated through iteration, and finally a uniformly distributed initial population is generated, with the expression: X i = l s + z i (u s - l s ) where: l s and u s represent the lower and upper bounds of the search space, respectively; z i is the i-th value in the Tent chaotic sequence; X i is the i-th individual of the population; Step S3: Introduce the reverse elite learning strategy into the main loop of the golden eagle optimization algorithm, and use the exponential decay function to obtain the attack coefficient p a and the cruise coefficient p c ; Step S3 specifically includes the following steps: Step S3-1: Arrange the individuals of the current golden eagle population in ascending order of fitness value, and then select the top N golden eagle individuals with better fitness values as the elite group, where N represents the number of elite individuals; Step S3-2: Obtain the reverse solutions of the elite group. The calculation formula for each reverse individual is: where is the reverse solution of the elite group, and x Em (t) is the elite group, and l(t) and u(t) respectively represent the minimum and maximum value vectors of each dimension parameter in the current golden eagle population at different times; Step S3-3: Generate N uniformly distributed new individuals using each elite solution and its reverse solution. The calculation formula is: where s Em (t) represents the m-th individual; Step S3-4: Obtain the current attack vector of the golden eagle. The calculation formula is as follows: Where: A i is the attack vector of Golden Eagle i, is the best location (prey) visited by Golden Eagle f so far, is the current location of Golden Eagle i; Step S3-5: Obtain the target point on the cruising hyperplane. The expression is as follows: where: c k is the k-th element of the target point C, a j is the j-th element of the attack vector A i ; is obtained from the hyperplane equation in the n-dimensional space, h j is the j-th element of the normal vector , x j is the j-th element of the variable vector , p j is the j-th element of any point on the hyperplane; a k is the k-th element of the attack vector A i , and k is the index of the fixed variable; Step S3-6, perform a non-linear adjustment and change to the attack coefficient p a and the cruise coefficient p c The specific formula is as follows: Where: and are the maximum and minimum values of the attack coefficient p a respectively; and are the maximum and minimum values of the cruise coefficient p c respectively; α is the adjustment coefficient, and the value of α is 4; t and T are the current iteration number and the maximum iteration number respectively; The specific steps of Step S3-5 are as follows: Step S3-5-1, calculate the cruise vector of Golden Eagle i in iteration t of the hyperplane, and the calculation formula is as follows: Where: a j is the j-th element of the attack vector A i at the current iteration t, is the j-th element of the attack vector A i corresponding to the current iteration t, and A i = {a1, a2, …, a n} is the attack vector of the golden eagle i; x j is the j-th element of the decision variable vector X i of the golden eagle i, and X i = {x1, x2, …, x n} is the decision variable vector of the golden eagle i, i.e., the current position of the golden eagle; is the j-th element of the position X * of the prey selected by the golden eagle, and is the position of the prey selected by the golden eagle; Step S3-5-2, obtain the cruising vector c of the golden eagle i in the cruising hyperplane of the middle-aged eagle in the iteration k , the cruising vector c k is formulated as follows: where: c k is the k-th element of the target point C, a j is the j-th element of the attack vector A i is the k-th element of the attack vector A k is the k-th element of the attack vector A i and k is the index of the fixed variable; Step S3-5-3, obtaining a random target point on the cruising hyperplane. The expression of the target point on the cruising hyperplane is as follows: Step S4: The economic load dispatch model of the power system performs optimization based on the improved Golden Eagle optimization algorithm under constraint conditions, and calculates the minimum generation cost of the generating set using the optimal solution. Step S4 specifically includes the following steps: Step S4-1: Use the improved Golden Eagle optimization algorithm to perform elite reverse learning processing on the current population to obtain a new population; Step S4-2, update the attack coefficient p according to the current iteration number a and the cruise coefficient p c ; Step S4-3: Each golden eagle selects a target prey from the memory of the entire flock based on the best solution found so far; Step S4-4, calculate the attack vector of each golden eagle and the cruise vector and obtain Δx; The step vector of golden eagle i in the iteration is defined as Equation (14): where: p a is the attack coefficient in iteration t, and p c is the cruise coefficient in iteration t, which adjusts the influence of the golden eagle being attacked and cruising; and are random vectors whose elements are in the interval [0, 1]; Then add the step vector in iteration t to the position in iteration t to calculate the position of the golden eagle in iteration t + 1, that is: Step S4-5: Determine whether all golden eagle individuals meet the embargo area constraint, slow change area constraint, and power balance constraint of the ELD problem; when all meet, continue to loop and execute Step S4-6; when there are individuals that do not meet the constraint conditions, randomly reprocess the non-compliant golden eagle individuals; continue to loop until all individuals meet the constraint conditions; Step S4-6: Substitute the current position of the golden eagle into the objective function, calculate the current fitness values of all golden eagles and compare them with the memory position, and determine whether the current fitness value is less than the saved optimal fitness value; if so, use the current fitness value as the optimal fitness value and update the memory position to the current position; otherwise, keep the current optimal fitness value and memory position unchanged; Step S4-7: Take the iterative population as the new current population and determine whether the maximum number of iterations has been reached. If so, output the optimal individual as the optimal solution; otherwise, return to execute Step S4-1.

2. The economic load dispatch optimization method for power systems based on the improved golden eagle optimization algorithm according to claim 1, characterized in that: In step S3-6, the value is 2, the value is 0.5, the value is 1, the value is 0.

5.

3. The economic load dispatch optimization method for power systems based on the improved golden eagle optimization algorithm according to claim 1, characterized in that: In step S4-4, the intermediate value of p a and p c is calculated by linear transition: where: \(t\) represents the current iteration number, \(T\) represents the maximum iteration number, and are the initial value and the final value of the attack tendency \(p\) a respectively, and are the initial value and the final value of the cruising tendency \(p\) c respectively.

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