CN109409569B - Discrete gradient stepwise optimization algorithm for long-term scheduling in reservoir group considering direct-current transmission constraint - Google Patents

Discrete gradient stepwise optimization algorithm for long-term scheduling in reservoir group considering direct-current transmission constraint Download PDF

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CN109409569B
CN109409569B CN201811100653.1A CN201811100653A CN109409569B CN 109409569 B CN109409569 B CN 109409569B CN 201811100653 A CN201811100653 A CN 201811100653A CN 109409569 B CN109409569 B CN 109409569B
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李崇浩
李树山
吴慧军
唐红兵
王凌梓
刘映尚
唐卓尧
程春田
廖胜利
赵志鹏
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Dalian University of Technology
China Southern Power Grid Co Ltd
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Abstract

The invention relates to the field of hydropower dispatching operation, relates to a discrete gradient stepwise optimization algorithm for medium-term and long-term dispatching of a reservoir group in consideration of direct-current transmission constraints, and is a novel optimization algorithm for solving the problem of dimension disaster in reservoir dispatching. The invention provides a discrete gradient calculation method and a corresponding discrete gradient descent method on the basis of the traditional gradient and the corresponding gradient descent method, combines the discrete gradient calculation method and the corresponding discrete gradient descent method with a two-stage sub-optimization problem of POA, and provides a discrete gradient stepwise optimization algorithm DGPOA. The method can determine the optimal search direction under a given discrete step length by using the information of the discrete gradient without direct derivation, and carries out iterative solution by using a line search iterative equation, thereby remarkably reducing the calculation scale and quickly obtaining the optimization result. The rule has important guiding significance for the power generation regulation of the cascade reservoir group.

Description

Discrete gradient stepwise optimization algorithm for long-term scheduling in reservoir group considering direct-current transmission constraint
Technical Field
The invention relates to the field of hydropower dispatching operation, relates to a discrete gradient stepwise optimization algorithm for medium-term and long-term dispatching of a reservoir group in consideration of direct-current transmission constraints, and is a novel optimization algorithm for solving the problem of dimension disaster in reservoir dispatching.
Background
By 2016, the installed capacity and annual energy generation of water and electricity in China break through 3 billion kilowatts and 1 trillion kilowatts, which respectively account for 20.9 percent and 19.4 percent of all energy sources, and the water and electricity become the largest renewable energy source in China. The water and electricity resources are fully utilized, and reservoir group optimization scheduling is an important measure for constructing a clean, low-carbon, safe and efficient modern energy system. The optimal scheduling of the cascade hydropower station group running in the modern alternating current and direct current large power grid is a typical high-dimensional, multi-stage, nonlinear and non-convex optimization problem, and the solving difficulty of the cascade hydropower station group is sharply increased along with the introduction of various mutually restricted constraint conditions such as the increase of the number of power stations, the safe running of the power grid, the comprehensive utilization of a reservoir and the like and the coupling of complex power and hydraulic connection. The conventional solving methods comprise linear programming, nonlinear programming, network flow, a large-system decomposition and coordination method, dynamic programming and an improved algorithm thereof, and emerging intelligent algorithms represented by genetic algorithms, particle swarm algorithms, differential evolution algorithms and the like. Methods such as linear programming, decomposition and coordination of a large system, network flow and the like all need to carry out certain approximate processing on the system, and deviation is easy to generate; the nonlinear programming has certain requirements on constraint conditions and the functional form of an objective function, and is difficult to be applied to practice; the intelligent algorithm has poor solution stability due to the existence of random factors. The dynamic programming is one of the most widely applied algorithms in the reservoir dispatching field due to the characteristics of less restriction on the function forms of the constraint conditions and the objective function, stable result and the like. But face this serious "dimensionality hazard" problem as the computational scale increases.
In solving the reservoir group optimization scheduling dimension disaster problem, a Progressive Optimization Algorithm (POA) is used as an improved Algorithm of dynamic programming, a multi-stage problem is converted into a plurality of two-stage problems, the number of state combinations between the stages is reduced, and the method is widely applied to hydropower optimization scheduling. When the two-stage subproblems are solved, the POA adopts a grid search method with given discrete step length, so that the global convergence on solving the subproblems is ensured, but the problem of serious 'dimension disaster' still exists along with the increase of the scale.
The achievement of the invention depends on the national science fund (91547201) and the national science fund-Yajiangjiang united fund funding project (U1765103), the method of the invention can realize the goal of maximum long-term scheduling generated energy in the reservoir group considering the direct current transmission constraint, and has strong practicability and wide popularization value.
Disclosure of Invention
Aiming at the problem of dimension disaster existing in the POA solving sub-optimization problem, the invention provides a discrete gradient stepwise optimization algorithm aiming at medium and long term cascade reservoir optimization scheduling.
The technical scheme of the invention is as follows:
a discrete gradient stepwise optimization algorithm for medium-term and long-term scheduling in a reservoir group considering direct-current transmission constraints comprises the following specific steps:
(1) setting algorithm parameters including a water level discrete step length h, a total number T of scheduling stages and a generated energy convergence precision epsilon;
(2) determining an initial water level process, and calculating the total step power generation amount E;
(3) setting a time interval variable T-1;
(4) obtaining water level state vector of time interval t
Figure BDA0001806598410000021
(5) The two-stage problem in the time period t is that a constrained optimization problem is converted into an unconstrained optimization problem;
minf(Z1,t,Z2,t,…ZN,t)
(6) setting the variable n of the iteration number to be 0 and setting the initial solution of the discrete gradient descent method
Figure BDA0001806598410000022
(7) Solving for discrete gradients
Figure BDA0001806598410000023
Calculating the search step size gammanSolving for a according to an iterative formulan+1(ii) a Judging whether the stop criterion is met, if so, enabling Zt＝anIf t is t +1, jumping to step (8), otherwise, making n be n +1, and repeating step (7);
(8) judging that t is less than or equal to 0, jumping to the step (9) if t is less than or equal to 0, and returning to the step (4) if t is less than or equal to 0;
(9) and (4) calculating the power generation amount E 'according to the optimized water level, judging that | E' -E | is less than or equal to epsilon, outputting a result if the power generation amount E 'is less than or equal to epsilon, ending the calculation, and returning to the step (4) if the power generation amount E' is not less than or equal to epsilon.
The invention has the beneficial effects that: the method can determine the optimal search direction under the given discrete step length by using the information of the discrete gradient without direct derivation, and carries out iterative solution by using a line search iterative equation, thereby obviously reducing the calculation scale and quickly obtaining the optimization result. The rule has important guiding significance for the power generation regulation of the cascade reservoir group.
Drawings
FIG. 1 is a flow chart of DGPOA calculation according to the present invention;
FIG. 2 is a graph of variation of calculated power generation amount with discrete accuracy of each algorithm in the scheme 1;
FIG. 3 is a graph of variation of calculated power generation amount with discrete accuracy of each algorithm in scheme 2;
fig. 4 is a schematic diagram of an optimized scheduling result of the DGPOA algorithm in the horizontal year.
Detailed Description
The following further describes a specific embodiment of the present invention with reference to the drawings and technical solutions.
The Progressive Optimization Algorithm (POA) is used as an improved Algorithm of dynamic programming, converts a multi-stage problem into a plurality of two-stage problems, reduces the number of state combinations between the stages, and is widely applied to hydropower optimization scheduling. When the two-stage subproblems are solved, the POA adopts a grid search method with given discrete step length, so that the global convergence on solving the subproblems is ensured, but the problem of serious 'dimension disaster' still exists along with the increase of the scale. The two-stage sub-optimization problem in the POA algorithm is essentially an optimization problem of a multivariable single-valued function. The invention provides a concept of Discrete Gradient (Discrete Gradient), and solves a sub-optimization problem by using a Discrete Gradient descent method, thereby providing a Discrete Gradient stepwise optimization algorithm (DGPOA). The method can determine the optimal search direction under a given discrete step length by using the information of the discrete gradient without direct derivation, and carries out iterative solution by using a line search iterative equation, thereby remarkably reducing the calculation scale and quickly obtaining the optimization result.
The specific operation method of each step is realized according to the following ideas (a) to (d):
(a) calculation of discrete gradients
For multivariable univocal functions f (x)1,x2,..xn) When the discrete step length h of the given water level is more than 0, order
Figure BDA0001806598410000031
Figure BDA0001806598410000032
Figure BDA0001806598410000033
In the formula:
Figure BDA0001806598410000034
respectively representing the position of a point x, the discrete step length h of the water level and xiPositive differential, negative differential, and self-differential of direction. Wherein x is (x)1,…xi,xi+1,…xn)T。
The approximate partial derivative under the water level discrete step length h is calculated according to the following formula:
Figure BDA0001806598410000035
Figure BDA0001806598410000036
in the formula:
Figure BDA0001806598410000037
the discrete partial derivative of the i dimension at point a is represented, s' the falling direction and s the differential direction.
The discrete gradient under the given water level discrete step length h is calculated according to the following formula:
Figure BDA0001806598410000038
(b) search step size and stop criteria
For reservoir scheduling, the optimized state variables (such as water level) and targets (such as power generation) often have different dimensions, so that the step length is easy to be too large or too small, and then the algorithm convergence and the search speed are influenced. The searching step length is dynamically set according to the water level discrete step length h and the discrete gradient value.
Setting:
Figure BDA0001806598410000039
then set the search step length to
Figure BDA00018065984100000310
The line search iteration equation is
Figure BDA00018065984100000311
The stopping criterion is as follows:
Figure BDA00018065984100000312
or
f(an)＜f(an+1)
For the discrete case, under a certain precision, it may happen that the search function value becomes large along the negative direction of the discrete gradient, and the optimal value is generally located at an,an+1In between, the search may be stopped, considering that the accuracy requirement has been met.
(c) Two-stage problem transformation in time t
The two-stage problem for time period t is a constrained optimization problem:
min-P(Z1,t,Z2,t,..Zn,t)
s.t.Zi,t∈C,i＝1,2,…n
in the formula: multivariate scalar function
Figure BDA0001806598410000041
C is a constraint set as a function of the power generation amount in the time period t.
Defining functions
Figure BDA0001806598410000042
Figure BDA0001806598410000043
In the formula: m is a sufficiently large constant.
It is converted into an unconstrained optimization problem:
minf(x1,x2,…xn)
(d) optimizing scheduling model settings
Objective function
The medium-and-long-term optimal scheduling mainly based on power generation generally adopts a model with the maximum power generation capacity. The calculation step length of the method is selected to be 1 month, and the scheduling period is 1 year.
Figure BDA0001806598410000044
In the formula: i and j are reservoir and time interval sequence numbers respectively; n is the total number of reservoirs participating in calculation; t is the total number of periods, and this embodiment is set to 12; e is the step total power generation amount, and the unit kW.h; a. theiThe output coefficient of the reservoir i is; hi,jIs the average net head of reservoir i time period j, unit m; Δ tjHours for time period j, in units of h; qi,jIs the generating flow of the reservoir i in the time interval j, and the unit m3/s。
Constraint conditions
(1) Top and bottom water level restriction
Figure BDA0001806598410000045
In the formula:
Figure BDA0001806598410000046
the initial and expected end levels, in m, are given for the reservoir i, respectively.
(2) Upper and lower limits of water level
Figure BDA0001806598410000047
In the formula: zi,jIs the water level of the reservoir i period j,Z i,j，
Figure BDA0001806598410000048
respectively the lower limit and the upper limit of the water level of the reservoir i time period j in the unit of m.
(3) Upper and lower limits of output
Figure BDA0001806598410000049
In the formula:N i,j，
Figure BDA00018065984100000410
the lower limit and the upper limit of the output of the reservoir i time interval j are respectively expressed in KW unit.
(4) Equation of water balance
Vi,j+1＝Vi,j+3600×(Ii,j-Qi,j-Si,j)Δtj
Wherein
Figure BDA0001806598410000051
In the formula, Vi,jThe last storage capacity of the reservoir i time period j, unit m3；Ii,j,qi,j,Si,jThe warehousing flow, the interval flow and the abandoning flow of the reservoir i time period j respectively are unit m3S, wherein qi,jIs known; omegaiSet of reservoirs directly upstream of reservoir i, for leading reservoir ΩiIs an empty set;
(5) power generation flow restriction
Figure BDA0001806598410000052
In the formula: qi,jThe generated flow of the reservoir i in the time period j,Q i,j，
Figure BDA0001806598410000053
the lower limit and the upper limit of the generating flow of the reservoir i time period j, unit m3/s。
(6) Warehouse-out flow limitation
Figure BDA0001806598410000054
In the formula:O i,j，
Figure BDA0001806598410000055
the lower limit and the upper limit of the delivery flow of the reservoir i time period j, unit m3/s。
The specific application of the invention is as follows:
(1) engineering background and parameter selection
The model algorithm is verified by taking 5 main power stations in the lan-lansanjiang dry flow of Yunnan province as an example. The water energy resources of the Lancang river basin are rich, and the Lancang river basin is one of thirteen hydropower bases which are mainly developed in China. 5 power stations comprise a bay, a large mountains, a glutinous rice ferry and a scenic flood from upstream to downstream in sequence, the total installed capacity reaches 14700MW, and the power station is an important peak-load-adjusting frequency-modulation power supply and a west-east power supply point in Yunnan. Except for the adjustment of the small bay and the glutinous rice ferry for years, the rest are season adjustment reservoirs, and the basic conditions and the initial and final water level setting values of all the reservoirs are shown in table 1. The power matching method is characterized in that power matching requirements are strictly met between the maximum power generation output of the power station for the small gulf and the glutinous rice ferry and the transmission power of the direct current line in order to ensure the safety and stability of a power grid.
The algorithm is compiled by adopting Python language, the initial solution is generated by an equal flow method, the power generation capacity of the power station and the corresponding power generation flow in each time period meet the constraint requirement of direct current power matching, and the convergence precision epsilon of the power generation capacity is set to be 0.001 hundred million kWh.
(2) Calculation results at different precisions
POA, POA-DPSA and DGPOA are respectively used for optimizing and scheduling the lancang river basin by utilizing the perennial data. Scheme 1 simplifies the lansanjiang basin into a bay and glutinous ferry two-library system, and the calculation results are shown in table 2. In the scheme 2, the calculation is carried out on the five-reservoir system of the minibay, the diffuse bay, the Dachaoshan mountain, the glutinous rice ferry and the Jinghong, and the calculation result is shown in a table 3. The calculated power generation amount of the three algorithms of the scheme 1 and the scheme 2 is changed along with the discrete accuracy as shown in fig. 2 and fig. 3 respectively. Defining POA calculation power generation amount e0POA-DPSA calculation power generation amount e1DGPOA calculation power generation amount e2. Relative deviation between POA-DPSA and POA calculation power generation amount
Figure BDA0001806598410000056
DGPOA and POA calculate the relative deviation of the generated energy
Figure BDA0001806598410000057
Relative deviation of DGPOA and POA-DPSA
Figure BDA0001806598410000058
(3) Calculation results under different incoming waters
And 5 main reservoirs in the lancang river basin are optimally scheduled by runoff data and POA-DPSA and DGPOA algorithms in the dry year (75%), the open year (50%), the rich year (25%) and the DGPOA algorithms. Since POA already presents a serious "dimensional disaster" problem in 5-reservoir systems, it is not calculated here. The discrete step length h is chosen to be 0.1 meter. The optimization results are shown in Table 6. The calculated results of the DGPOA reservoirs in the open water year are shown in figure 4. As shown in fig. 4, the water level cut is increased in both the small bay and the glutinous ferry of the two multi-year regulation reservoirs before the flood, the regulation reservoir capacity is fully exerted, and compensation scheduling is performed on other reservoirs. After entering the flood season, the miniinlets and the glutinous rice ferry start to store water so as to ensure the high water level operation in the later flood season. The reservoir is adjusted in three seasons of the bay, the great mountains and the flood, and is always in a high-water-level running state under the regulation and storage of the small bay glutinous ferry, so that the integral power generation benefit of the cascade is ensured.
The calculation results of multiple schemes of the reservoir system of the Langchan river basin 5 show that under the conventional engineering precision, the calculation speed of the DGPOA can reach 8-12 times of that of the POA-DPSA algorithm and 50-250 times of that of the POA algorithm under the condition that the global search capability is not remarkably reduced. And as the number of reservoirs is increased and the calculation precision is improved, the speed of the reservoirs is improved more obviously relative to the POA-DPSA and the POA.
TABLE 1
Figure BDA0001806598410000061
TABLE 2
Figure BDA0001806598410000062
TABLE 3
Figure BDA0001806598410000063
Figure BDA0001806598410000071
TABLE 4
Figure BDA0001806598410000072
TABLE 5
Figure BDA0001806598410000073
TABLE 6
Figure BDA0001806598410000074

Claims (1)

1. A discrete gradient gradual optimization method for medium-term and long-term scheduling of a reservoir group considering direct-current transmission constraint is characterized by comprising the following specific steps:
(1) setting method parameters including a water level discrete step length h, a total number T of scheduling stages and a generated energy convergence precision epsilon;
(2) determining an initial water level process, and calculating the total step power generation amount E;
(3) setting a time interval variable T-1;
(4) obtaining water level state vector of time interval t
Figure FDA0003241184570000011
(5) The two-stage problem in the time period t is that a constrained optimization problem is converted into an unconstrained optimization problem;
minf(Z1,t,Z2,t,…ZN,t)
(6) setting the variable n of the iteration number to be 0 and setting the initial solution of the discrete gradient descent method
Figure FDA0003241184570000012
(7) Solving for discrete gradients
Figure FDA0003241184570000013
Calculating the search step size gammanSolving for a according to an iterative formulan+1(ii) a Judging whether the stop criterion is met, if so, enabling Zt＝anIf t is t +1, jumping to step (8), otherwise, making n be n +1, and repeating step (7);
(8) judging that t is less than or equal to 0, jumping to the step (9) if t is less than or equal to 0, and returning to the step (4) if t is less than or equal to 0;
(9) calculating the generated energy E ' according to the optimized water level, judging that | E ' -E | is less than or equal to epsilon, if so, outputting a result, ending the calculation, and otherwise, enabling E to be E ', and returning to the step (4);
the specific operation of each step is as follows:
(a) calculation of discrete gradients
For multivariable univocal functions f (x)1,x2,..xn) Given discrete step length h of water level>In the case of 0, let
Figure FDA0003241184570000014
Figure FDA0003241184570000015
Figure FDA0003241184570000016
In the formula:
Figure FDA0003241184570000017
respectively representing the position of a point x, the discrete step length h of the water level and xiPositive, negative, and self-differencing of the directions; wherein x is (x)1,…xi,xi+1,…xn)T；
The partial derivative under the water level discrete step length h is calculated according to the following formula:
Figure FDA0003241184570000018
Figure FDA0003241184570000019
in the formula:
Figure FDA00032411845700000110
the discrete partial derivative of i dimension at the point a is shown, s' represents the descending direction, and s represents the differential direction;
the discrete gradient under the given water level discrete step length h is calculated according to the following formula:
Figure FDA00032411845700000111
(b) search step size and stop criteria
The searching step length is dynamically set according to the water level discrete step length h and the discrete gradient value;
setting:
Figure FDA0003241184570000021
then set the search step length to
Figure FDA0003241184570000022
The line search iteration equation is
Figure FDA0003241184570000023
The stopping criterion is as follows:
Figure FDA0003241184570000024
or
f(an)<f(an+1)
For the discrete case, the search function value in the negative direction of the discrete gradient becomes larger, and the optimal value is located at an,an+1Considering that the requirement of the generated energy convergence precision epsilon is met, and stopping searching;
(c) two-stage problem transformation in time t
The two-stage problem for time period t is a constrained optimization problem:
min-P(Z1,t,Z2,t,..Zn,t)
s.t.Zi,t∈C,i＝1,2,…n
in the formula: multivariate scalar function
Figure FDA0003241184570000025
C is a function of the generated energy in the time period t, and is a constraint set;
defining functions
Figure FDA0003241184570000026
Figure FDA0003241184570000027
In the formula: m is a constant;
it is converted into an unconstrained optimization problem:
minf(x1,x2,…xn)
(d) optimizing scheduling model settings
Objective function
The medium-long term optimized dispatching mainly based on power generation adopts a model with the maximum power generation capacity;
Figure FDA0003241184570000028
in the formula: i and j are reservoir and time interval sequence numbers respectively; n is the total number of reservoirs participating in calculation; t is the total time period number; e is the step total power generation amount, and the unit kW.h; a. theiThe output coefficient of the reservoir i is; hi,jIs the average net head of reservoir i time period j, unit m; Δ tjHours for time period j, in units of h; qi,jIs the generating flow of the reservoir i in the time interval j, and the unit m3/s；
Constraint conditions
(1) Top and bottom water level restriction
Figure FDA0003241184570000031
In the formula:
Figure FDA0003241184570000032
respectively setting an initial water level and an expected final water level of the reservoir i in a unit of m;
(2) upper and lower limits of water level
Figure FDA0003241184570000033
In the formula: zi,jIs the water level of the reservoir i period j,Z i,j，
Figure FDA0003241184570000034
respectively the lower limit and the upper limit of the water level of the reservoir i time period j, and the unit m;
(3) upper and lower limits of output
Figure FDA0003241184570000035
In the formula:N i,j，
Figure FDA0003241184570000036
respectively representing the lower limit and the upper limit of output of the reservoir i time period j in KW unit;
(4) equation of water balance
Vi,j+1＝Vi,j+3600×(Ii,j-Qi,j-Si,j)Δtj
Wherein
Figure FDA0003241184570000037
In the formula, Vi,jThe last storage capacity of the reservoir i time period j, unit m3；Ii,j,qi,j,Si,jThe warehousing flow, the interval flow and the abandoning flow of the reservoir i time period j respectively are unit m3S, wherein qi,jIs known; omegaiSet of reservoirs directly upstream of reservoir i, for leading reservoir ΩiIs an empty set;
(5) power generation flow restriction
Figure FDA0003241184570000038
In the formula: qi,jThe generated flow of the reservoir i in the time period j,Q i,j，
Figure FDA0003241184570000039
the lower limit and the upper limit of the generating flow of the reservoir i time period j, unit m3/s；
(6) Warehouse-out flow limitation
Figure FDA00032411845700000310
In the formula:O i,j，
Figure FDA00032411845700000311
the lower limit and the upper limit of the delivery flow of the reservoir i time period j, unit m3/s。
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