Power system active power dynamic optimization method considering valve point effect
By combining the improved Lagrangian relaxation algorithm with the alternating direction multiplier method, the non-convexity problem caused by the valve point effect in the active power dispatch of the power system is solved, and efficient and accurate generator output optimization and fuel consumption minimization are achieved.
Patent Information
- Application Number
- CN202510739047.8
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-06-04
- Publication Date
- 2025-09-23
AI Technical Summary
When considering the valve point effect, the traditional power system active scheduling model has non-convexity and non-smoothness, which causes the optimization results to deviate from the actual optimal value, the heuristic algorithm results to be unstable, and the mathematical optimization method has iterative oscillation problems, making it difficult to apply.
An improved Lagrangian relaxation algorithm is combined with the surrogate gradient method and the alternating direction multiplier method. By constructing a smooth surrogate function to approximate the valve point effect, the non-convex optimization problem is decomposed, and the Lagrangian multiplier is iteratively updated through the alternating direction multiplier method to achieve global coordination.
It provides accurate optimization scheduling results within seconds, significantly reduces computational complexity, improves computational efficiency, avoids iterative oscillations, and obtains more accurate generator output and fuel consumption minimization.
Smart Images
Figure CN120691513A_ABST
Abstract
Description
Technical Field
[0001] The present invention belongs to the technical field of power system operation control, and in particular relates to a method for dynamic optimization of active power of a power system taking valve point effects into consideration. Background Art
[0002] Active power dispatch is one of the core optimization tasks in power system operation. Its purpose is to achieve the optimal operating state of the power system by rationally allocating and adjusting the output of each generator unit while meeting load demand and system safety constraints, thereby minimizing total power generation consumption. In traditional models, the active power dispatch function of the generator is simply viewed as a smoothed quadratic polynomial of the active power output and is solved using classical optimization algorithms such as linear programming, quadratic programming, or the Lagrange multiplier method. However, the output of large steam turbines is affected by the steam inlet valve, which can produce multiple local minima in the active power dispatch function of the unit, resulting in non-convexity of the active power dispatch function, which in turn affects the active power balance of the system and causes the optimization results of traditional models to deviate from the actual optimal value.
[0003] When considering the valve point effect, the system's active power dispatch function exhibits nonlinearity and non-convexity. Therefore, targeted research on corresponding optimization methods is necessary. In the past, heuristic optimization methods mainly included particle swarm optimization, self-organizing migration, Q-learning, simulated annealing, and Taboo search. However, due to the random number generation problem during the solution process, there are problems with premature results, and the optimization results are often different in each solution, resulting in irreproducible results. Therefore, heuristic algorithms are still difficult to put into practical application at this stage. Mathematical programming optimization methods mainly include lambda iteration, Newton's method, dynamic programming, and Lagrangian relaxation. They can usually guarantee the convergence of the problem and only need to be run once. However, due to the non-smooth characteristics of the optimization function and the model indivisibility and iterative oscillation problems of the Lagrangian relaxation algorithm, mathematical optimization methods are generally not suitable for solving active power dispatch considering the valve point effect.
[0004] Therefore, a hybrid optimization strategy of alternating direction multiplier method and proxy gradient method is proposed to efficiently solve the complex economic load distribution problem considering valve point effect, solve the problem that heuristic algorithms are difficult to put into practical application, and solve the problems of splitting and iterative oscillation of traditional mathematical optimization methods. Summary of the Invention
[0005] In order to address the deficiencies of the above-mentioned prior art, the present invention provides a method for dynamic optimization of active power of an electric power system taking into account the valve point effect, determines the active power scheduling objective function when the overall operating state is optimal, and determines the constraints for system power balance, generator output limit, and line transmission capacity; in order to enhance the convexity of the optimization problem after piecewise linearization and accelerate the convergence speed, it is proposed to adopt an improved Lagrangian relaxation algorithm, relax the coupling constraints and form Lagrangian duality, use the power balance constraint as the penalty term, and the objective function exhibits good local convexity; in order to solve the problems of model inseparability and iterative oscillation existing in the augmented Lagrangian relaxation algorithm, a solution strategy combining a proxy gradient algorithm and an alternating direction multiplier method is proposed, so as to obtain the optimal fuel consumption and the corresponding generator output.
[0006] The present invention provides a method for dynamic optimization of active power of an electric power system taking into account valve point effect, which comprises the following steps:
[0007] S1. Establish a power system active power dispatch model considering valve point effect;
[0008] The optimization objective function of the active power dispatch model is to minimize the consumption of thermal power units. When considering the valve point effect, the objective function is expressed as a rectified sine curve. The constraints are given by the system power balance, generator output limit and line transmission capacity.
[0009] S2, improved Lagrangian relaxation algorithm;
[0010] When considering the valve point effect of the generator set, the power balance constraint is used as the penalty term, and the augmented Lagrangian relaxation algorithm is expressed as:
[0011]
[0012] k=1,...,m i ;
[0013] Where D is the system load demand forecast value, a i ,b i ,c i is the fuel consumption coefficient of thermal power unit i, p i is the active power output of the thermal power unit, v, w l are the Lagrangian relaxation factors corresponding to the coupled constraints of system power balance and line transmission capacity, α i,k and β i,k is the coefficient of the linear approximation polynomial of unit i in the kth segment interval, m i is the total number of segment intervals, k li is the power transfer allocation factor of unit i to line l, λ is the penalty term coefficient, and δ is the number of iterations;
[0014] S3, introduce the alternating direction multiplier method and the proxy gradient method to solve the problem;
[0015] The local optimal solution of expression (1) is:
[0016]
[0017] k=1,...,m i , combining the proxy gradient method with the alternating direction multiplier method to update the Lagrange multiplier;
[0018] S4, alternately iteratively update the unit active power output and Lagrange multiplier;
[0019] The solution obtained from expression (2) is used as the initial value of each segment optimization. By solving each segment separately and updating the multiplier alternately until the iteration converges, the optimal fuel consumption and the corresponding generator output are obtained.
[0020] Preferably, S1, establishing an active power dispatch model of the power system considering the valve point effect, specifically:
[0021] S11. Determine the objective function of active power scheduling;
[0022] When the valve point effect is considered, the active power dispatch function is expressed as a rectified sine curve:
[0023]
[0024] Among them, e i ,f i is a constant, p i The lower output limit of unit i.
[0025] S12. Determine the constraints of the objective function;
[0026] 1) Determine constraints for system power balance:
[0027]
[0028] 2) Determine constraints for generator output limits:
[0029]
[0030] 3) Determine constraints based on line transmission capacity limits:
[0031]
[0032] Furthermore, since expressions (4) and (6) are linear constraints, the expression of Lagrangian duality is:
[0033]
[0034] in, is a constant.
[0035] In a preferred solution, the power balance constraint is used as a penalty term, and the augmented Lagrangian relaxation algorithm of expression (7) is expressed as:
[0036]
[0037] k=1,...,m i , where the penalty coefficient λ is the penalty coefficient.
[0038] In a preferred embodiment, the expression of the augmented Lagrangian relaxation algorithm is:
[0039]
[0040] stconstraint(5).
[0041] Furthermore, at the δth iteration, for the generator set with a unit number less than i, the active output of the generator set is the result of the δth iteration; and for the generator set with a unit number greater than or equal to i, the active output of the generator set is p j Take the result of the δ-1th iteration.
[0042] Compared with the prior art, the present invention has the following beneficial effects:
[0043] (1) The method for dynamic optimization of active power of power system considering valve point effect of the present invention converts the original problem into a differentiable optimization form by constructing a smooth proxy function to approximate the non-smooth sinusoidal fluctuation term in the valve point effect, thus avoiding the iterative oscillation problem caused by non-smoothness in traditional methods.
[0044] (2) The method for dynamic optimization of active power of power system considering valve point effect of the present invention utilizes the variable splitting characteristic of the alternating direction multiplier method to decompose the non-convex optimization problem into multiple sub-problems, and realizes global coordination through alternating iterative update of Lagrange multipliers, which can provide more accurate optimization scheduling results within seconds, significantly reduce computational complexity and improve computational efficiency. BRIEF DESCRIPTION OF THE DRAWINGS
[0045] Figure 1 This is a schematic flow chart of the active power dynamic optimization method considering valve point effect of the present invention;
[0046] Figure 2 This is a typical unit active power output curve considering the valve point effect of the present invention;
[0047] Figure 3This is the fitting error curve of a typical unit considering the valve point effect of the present invention;
[0048] Figure 4 This is the convergence characteristic curve of the present invention applied to a 40-machine system. DETAILED DESCRIPTION
[0049] To fully describe the technical content, objectives and effects of the present invention, the following will be described in detail with reference to the accompanying drawings.
[0050] The present invention provides a method for dynamic optimization of active power of power system considering valve point effect, such as Figure 1 As shown, the method includes the steps of:
[0051] S1. Establish a power system active power dispatch model considering valve point effect;
[0052] Generally, the minimum consumption of thermal power units, that is, the optimal overall operating state, is taken as the optimization objective function of active power scheduling. When the valve point effect is considered, the active power scheduling function is usually represented by a rectified sine curve, and constraints are given for system power balance, generator output limit, and line transmission capacity.
[0053] S11. Establishing an objective function model for active power scheduling;
[0054] The objective function of active power dispatch is to optimize the overall operating state, which can generally be expressed as the following quadratic function:
[0055]
[0056] When considering the valve point effect of the unit, the opening and closing process of the steam turbine inlet valve will significantly affect the unit efficiency, and the unit fuel consumption curve will show non-convex and non-smooth characteristics. The objective function of active scheduling usually uses a rectified sinusoidal function to describe the valve point effect. The objective function expression is:
[0057]
[0058] S12. Determine the constraints of the objective function;
[0059] 1) Determine constraints for system power balance:
[0060]
[0061] 2) Determine constraints for generator output limits:
[0062]
[0063] 3) Determine constraints based on line transmission capacity limits:
[0064]
[0065] S2, improved Lagrangian relaxation scheduling algorithm;
[0066] For the active power dispatching model, the Lagrangian relaxation method is used to solve it, and the corresponding Lagrangian duality expression is:
[0067]
[0068] Where θ(·) is the Lagrange dual function, inf{·} is the infimum function, l is the transmission line number, L is the total number of transmission lines, S c is the generator output limit constraint.
[0069] Since the coupling constraints (9) and (11) are both linear constraints, expression (11) can be further rewritten as follows:
[0070]
[0071] in, is a constant.
[0072] As can be seen, Equation (12) is the sum of multiple single-unit constraint optimization subproblems. Therefore, the original optimization problem can be expressed as multiple parallel single-unit optimization subproblems, where the variables and constraints in each subproblem are specific to a single unit and independent of the others. The subproblem expressions are shown in Equation (13).
[0073]
[0074] For traditional active power dispatch, f(p i ) is a quadratic polynomial function, and the subproblem is a quadratic programming problem. It is relatively easy to obtain the optimal solution. After the subproblem is solved optimally, the Lagrange multiplier of the main problem is updated, and the main problem and subproblem are iterated continuously to gradually approach the optimal solution. However, when considering the valve point effect of the generator set, f(p i ) is transformed into a non-convex and non-smooth function form, such as Figure 2 As shown in the figure, an improved Lagrangian relaxation scheduling algorithm is proposed for this optimization problem.
[0075] The piecewise linear approximation algorithm is used to linearize the valve point effect, and the expression e i |sinf i (p i - p i )|Perform polynomial approximate fitting and obtain:
[0076] e i |sinf i (p i -p i )|≈α i,k p i +β i,k ,k=1,...,m i (14);
[0077] α i,k , m i and β i,k The calculation methods are as follows:
[0078]
[0079] Where K is the number of segments between two consecutive valve points, is the upper limit of active power output of segment k, and its calculation formula is:
[0080]
[0081] When calculating the piecewise linearization fitting error, the normalized calculation method is used: fitting error = |fitting value - true value| / fuel consumption under the rated output of the unit. The error can be converted into a dimensionless percentage form, making the errors of units of different capacities comparable. Figure 2 Taking the typical active power output curve of the unit considering the valve point effect as an example, the fitting error of the piecewise linear approximation is as follows: Figure 3 As shown, it can be seen that when the number of segments between two consecutive valve points increases, the maximum error of the fitting curve will decrease. When K>10, the maximum error of the fitting curve will be less than 0.1%. The accuracy of the current piecewise linearization meets the engineering requirements.
[0082] Substituting the fitted polynomial (14) into formula (3), the fuel consumption of the thermal power unit can be obtained as follows:
[0083]
[0084] k=1,2.....,m i , since f(p i ) is a piecewise quadratic polynomial function. Therefore, the present invention adopts the augmented Lagrangian relaxation method to convexify it to accelerate the convergence speed. With the power balance constraint as the penalty term, the augmented Lagrangian relaxation algorithm of problem (12) is:
[0085]
[0086] k=1,...,m i ;
[0087] stconstraint(5);
[0088] It can be seen that expression (20) is still a quadratic programming problem, since its quadratic term coefficient is a i +λ, therefore, when the penalty coefficient λ is large enough, the objective function can exhibit good local convexity.
[0089] S3, introduces a solution method combining the alternating direction multiplier method and the proxy gradient method;
[0090] Although the augmented Lagrangian relaxation method can accelerate the convergence of active power scheduling to a certain extent, the introduction of quadratic terms makes the optimization problem indecomposable, making direct decomposition difficult. Therefore, this paper proposes an alternating iterative solution strategy based on the alternating direction multiplier method.
[0091] The augmented Lagrangian relaxation problem (20) can be equivalently written as follows:
[0092]
[0093] k=1,...,m i ;
[0094] stconstraint(5);
[0095] Furthermore, formula (20) can be rewritten as a general expression:
[0096]
[0097] stconstraint(5);
[0098] From the above formula, we can get that at the δth iteration, for the generator set with a unit number less than i, its active output is the result of the δth iteration; and for the generator set with a unit number greater than i, its active output p j Then take the result of the δ-1th iteration, and the optimization problem is decomposed into multiple serial single-machine optimization sub-problems.
[0099] Its local optimal solution is:
[0100]
[0101] k=1,...,m i ;
[0102] At the same time, in order to solve the oscillation problem of the traditional Lagrangian relaxation algorithm, the present invention applies the proxy gradient method to the update process of the Lagrangian multiplier and combines it with the alternating direction multiplier method to form the following multiplier update strategy:
[0103]
[0104] S4, alternately iteratively update the unit active power output and Lagrange multiplier;
[0105] The solution obtained from expression (2) is used as the initial optimization value for each segment. By solving each segment separately and iteratively updating the multiplier until the iteration converges, the optimal fuel consumption and corresponding generator output are obtained. In a preferred embodiment, the local optimal solution obtained is used as the initial optimization value for each segment. The active output of each unit and the Lagrange multiplier are iteratively updated alternately until the maximum fitting error is less than 0.1%.
[0106] The following takes an actual system of 40 units as an example to specifically illustrate the method for dynamic optimization of active power of a power system taking valve point effects into consideration according to the present invention.
[0107] All units in this system are heating units, and their parameter values are shown in Table 1. The system load demand is 10,500 MW.
[0108] Table 1 Generator parameters of the 40-unit test system
[0109]
[0110]
[0111] When the valve point effect is considered, the simulation results of the proposed method are compared with other methods such as the evolutionary programming method, the evolutionary programming method considering Cauchy mutation, the hybrid method of differential evolution and sequential quadratic programming, the chaotic differential evolution and quadratic programming method, the Taguchi method, and the Lagrangian method considering Maclaurin series, as shown in Table 2.
[0112] Table 2 Comparison of the results of the method proposed in this invention with other methods
[0113]
[0114] As can be seen from Table 2, the proposed solution strategy combining the alternating direction multiplier method with the proxy gradient algorithm minimizes the active scheduling function and optimizes system operation. Furthermore, the runtime is only approximately 1.51 seconds, making it more efficient than the evolutionary programming method, the evolutionary programming method with Cauchy mutation, the hybrid method of differential evolution and sequential quadratic programming, the chaotic differential evolution and quadratic programming method, and the Taguchi method.
[0115] The convergence characteristic curve of the present invention applied to the actual system of 40 machines is as follows: Figure 4 As shown in the figure, the curve has converged after 50 iterations, which shows that this method has a faster convergence speed and good stability.
[0116] The embodiments described above are merely descriptions of preferred implementations of the present invention and are not intended to limit the scope of the present invention. Without departing from the spirit of the present invention, various modifications and improvements made to the technical solutions of the present invention by ordinary technicians in this field should fall within the scope of protection determined by the claims of the present invention.
Claims
1. A method for dynamic optimization of active power of a power system considering valve point effect, characterized in that: The method comprises the steps of: S1. Establish a power system active power dispatch model considering valve point effect; The optimization objective function of the active power dispatch model is to minimize the consumption of thermal power units. When considering the valve point effect, the objective function is expressed as a rectified sine curve. The constraints are given by the system power balance, generator output limit and line transmission capacity. S2, improved Lagrangian relaxation algorithm; When considering the valve point effect of the generator set, the power balance constraint is used as the penalty term, and the augmented Lagrangian relaxation algorithm is expressed as: Where D is the system load demand forecast value, a i ,b i ,c i is the fuel consumption coefficient of thermal power unit i, p i is the active power output of the thermal power unit, v, w l are the Lagrangian relaxation factors corresponding to the coupled constraints of system power balance and line transmission capacity, α i,k and β i,k is the coefficient of the linear approximation polynomial of unit i in the kth segment interval, m i is the total number of segment intervals, k li is the power transfer allocation factor of unit i to line l, λ is the penalty term coefficient, and δ is the number of iterations; S3, introduce the alternating direction multiplier method and the proxy gradient method to solve the problem; The local optimal solution of expression (1) is: k=1,...,m i , combining the proxy gradient method with the alternating direction multiplier method to update the Lagrange multiplier; S4, alternately iteratively update the unit active power output and Lagrange multiplier; The solution obtained from expression (2) is used as the initial value of each segment optimization. By solving each segment separately and updating the multiplier alternately until the iteration converges, the optimal fuel consumption and the corresponding generator output are obtained.
2. The method for dynamic optimization of active power of a power system considering valve point effect according to claim 1, characterized in that: S1. Establish a power system active power dispatch model considering the valve point effect, specifically: S11. Determine the objective function of active power scheduling; When the valve point effect is considered, the active power dispatch function is expressed as a rectified sine curve: Among them, e i ,f i is a constant, The lower output limit of unit i. S12. Determine the constraints of the objective function; 1) Determine constraints for system power balance: 2) Determine constraints for generator output limits: 3) Determine constraints based on line transmission capacity limits:
3. The method for dynamic optimization of active power of a power system considering valve point effect according to claim 1, characterized in that: Since expressions (4) and (6) are linear constraints, the expression of Lagrangian duality is: in, is a constant.
4. The method for dynamic optimization of active power of a power system considering valve point effect according to claim 1, characterized in that: Taking the power balance constraint as the penalty term, the augmented Lagrangian relaxation algorithm of expression (7) is expressed as: k=1,...,m i , where the penalty coefficient λ is the penalty coefficient.
5. The method for dynamic optimization of active power of a power system considering valve point effect according to claim 1, characterized in that: The expression of the augmented Lagrangian relaxation algorithm is: stconstraint(5).
6. The method for dynamic optimization of active power of a power system considering valve point effect according to claim 5, characterized in that: At the δth iteration, for the generator set with a unit number less than i, the active output of the generator set is the result of the δth iteration; and for the generator set with a unit number greater than or equal to i, the active output of the generator set is p j Take the result of the δ-1th iteration.