Research On Plane Based Linear Concaving Approximation And Its Application To Medium And Long Term Optimal Operation Of Cascaded Hydropower Reservoirs

The clean and low-carbon energy transformation has become a global consensus.The promotion of clean energy use can help improve environmental pollution problems and achieve economic development while protecting the ecological environment.As clean and renewable energy with mature operation technology,hydropower is an important force in promoting clean and low-carbon energy transformation.Hydropower is facing new challenges and opportunities in the new power system dominated by renewable energy.Among them,as the scale and complexity of cascaded hydropower reservoirs increase,nonlinearity has become a core factor affecting the efficiency and accuracy of its optimal scheduling and is a key challenge to further promote fine scheduling.This paper focuses on the most critical non-convex and nonlinear hydropower output function in the optimal operation of cascaded hydropower reservoirs.Supported by mathematical reasoning,we gradually delve into various innovative research methods for linear concaving approximation,including high efficiency,high precision,the balance between the two,and error elimination.Some valuable research results have been achieved,mainly in the following aspects:(1)A high-efficiency three-triangle based linear concaving approximation method is proposed: by restricting the plane intersection to the triangulation vertices,the introduction of integer variables can be avoided,thereby improving the solution efficiency.Meanwhile,it is mathematically proven that this method only requires local comparison but is equivalent to the all-triangle based linear concaving approximation method,ensuring that any convex hull plane specified applies within its corresponding triangular grid.Both methods achieve fitting errors that converge to zero on approximate the test concave function.In addition,in tests on the power output functions of four cascaded hydropower plants in the Lancang River,the root-mean-square error can be approximated to 2.05% of the installed capacity,but there is still room for improvement in the fitting accuracy.The proposed method has an average solution time of fewer than 0.097 seconds,which is 80 times faster than the global method(which takes at least 7.837 seconds),and this speed advantage becomes even more significant when frequent linearization processing is required.(2)A high-precision all-rectangle linear concaving approximation method is proposed.It avoids the drawback of existing rectangular grid linear approximation methods which introduce a large number of integer variables and reduce solution efficiency by avoiding the accurate expression of fitting error at the corner points.It is mathematically proved that the method based on this rectangular subdivision can converge to any convex function with arbitrary precision as the grid resolution increases.However,because global comparison is needed to ensure that the convex hull plane works within its corresponding rectangular grid,there is still room for further improvement in solution efficiency.The approximated results of the output functions of the four cascaded hydropower plants in the Lancang River show that both the proposed method and the existing method can reduce the average fitting error from 2.16% of installed capacity to 1.49% compared to the high-efficiency method.Although the proposed method is slower in solving speed than the high-efficiency method,it is significantly better than the unstable existing method.(3)A balanced accuracy and efficiency plane screening linear concaving approximation method is proposed.The proposed method ensures that each convex hull plane has at least one point that is effective in its corresponding rectangular grid by screening out ineffective planes,thereby improving the solution efficiency while maintaining accuracy.In the approximated results of the output function of the four cascaded hydropower plants in Lancang River,the proposed method can reduce the average calculation efficiency from 538.76 seconds of the high-precision method to 4.35 seconds while maintaining an average fitting error of 1.49%.Moreover,the simulation results of the historical long sequence optimal operation of the four cascaded hydropower reservoirs in Lancang River show that the linear programming model based on the plane based linear concaving approximation method can efficiently obtain the approximate optimal solution of the original problem.The linearization error of the multi-year simulation optimal operation can achieve 0.51% of the total installed capacity of the cascade,which is basically consistent with the fitting error of the hydropower output function,but is also affected by the distribution of fitting errors and cannot be eliminated by refining grid points.(4)Two(dynamic/static)corridor optimization iterative linear error elimination on a period-by-period basis method are proposed.The static method trades storage space for calculation time,offline determination of the convex hull plane for each area is carried out by dividing the region,and the convex hull plane in the optimal operation model is directly updated according to the current solution’s corresponding region during the optimization iteration.The dynamic method dynamically calibrates and updates the convex hull plane within the current solution’s neighborhood corridor.Simulation scheduling of the four cascaded hydropower reservoirs in the Lancang River shows that both corridor optimization methods converge,and even with the use of an efficient but poorly approximated method,the linearization error can be reduced to within 0.02% of the total installed capacity of the cascade from 1.69%,and the static method has an advantage in solving time,at least 217 times faster than the piecewise linear approximation method at the same grid resolution.Additionally,for the optimization scheduling problem considering forced spilling constraints,a new hydropower output relationship considering spillage factors is constructed based on the analysis of the relationship between the outflow of the hydropower plants,the maximum total turbined outflow,and spillage.Then,by introducing integer variable linearization for forced spilling constraints,the reservoir is forced to spill only when there is no space for water storage.Simulation results of the improved mixed-integer linear programming model in the single-year optimization scheduling of the four cascaded hydropower reservoirs in the Lancang River show that even spillage occurred in the early flood season,the target can be maximized while spilling only occurs when the reservoir level reaches the restricted level,which is more in line with actual operating situations.