CN111342450A - Robust energy management method considering uncertain photovoltaic and load for traction power supply system - Google Patents

Robust energy management method considering uncertain photovoltaic and load for traction power supply system Download PDF

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CN111342450A
CN111342450A CN202010154115.1A CN202010154115A CN111342450A CN 111342450 A CN111342450 A CN 111342450A CN 202010154115 A CN202010154115 A CN 202010154115A CN 111342450 A CN111342450 A CN 111342450A
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power
traction
photovoltaic
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power supply
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CN111342450B (en
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陈民武
刘元立
程哲
付浩纯
陈迎涛
魏铭池
王梦
寇哲超
陈垠宇
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Southwest Jiaotong University
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    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
    • H02J3/00—Circuit arrangements for ac mains or ac distribution networks
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
    • H02J3/00—Circuit arrangements for ac mains or ac distribution networks
    • H02J3/008—Circuit arrangements for ac mains or ac distribution networks involving trading of energy or energy transmission rights
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
    • H02J3/00—Circuit arrangements for ac mains or ac distribution networks
    • H02J3/26—Arrangements for eliminating or reducing asymmetry in polyphase networks
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
    • H02J3/00—Circuit arrangements for ac mains or ac distribution networks
    • H02J3/28—Arrangements for balancing of the load in a network by storage of energy
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
    • H02J3/00—Circuit arrangements for ac mains or ac distribution networks
    • H02J3/28—Arrangements for balancing of the load in a network by storage of energy
    • H02J3/32—Arrangements for balancing of the load in a network by storage of energy using batteries with converting means
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
    • H02J3/00—Circuit arrangements for ac mains or ac distribution networks
    • H02J3/38—Arrangements for parallely feeding a single network by two or more generators, converters or transformers
    • H02J3/46—Controlling of the sharing of output between the generators, converters, or transformers
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
    • H02J7/00—Circuit arrangements for charging or depolarising batteries or for supplying loads from batteries
    • H02J7/34—Parallel operation in networks using both storage and other dc sources, e.g. providing buffering
    • H02J7/345—Parallel operation in networks using both storage and other dc sources, e.g. providing buffering using capacitors as storage or buffering devices
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
    • H02J7/00—Circuit arrangements for charging or depolarising batteries or for supplying loads from batteries
    • H02J7/34—Parallel operation in networks using both storage and other dc sources, e.g. providing buffering
    • H02J7/35—Parallel operation in networks using both storage and other dc sources, e.g. providing buffering with light sensitive cells
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02E—REDUCTION OF GREENHOUSE GAS [GHG] EMISSIONS, RELATED TO ENERGY GENERATION, TRANSMISSION OR DISTRIBUTION
    • Y02E40/00—Technologies for an efficient electrical power generation, transmission or distribution
    • Y02E40/50—Arrangements for eliminating or reducing asymmetry in polyphase networks

Abstract

The invention discloses a robust energy management method for a traction power supply system considering uncertain photovoltaic and load, which comprises the following steps of: 1. acquiring photovoltaic output data in a load process of a traction substation, and constructing an uncertain set; 2. establishing an objective function of a robust optimization model; 3. establishing a constraint condition of a robust optimization model, and linearizing the constraint condition of the optimization model; 4. establishing a two-stage robust optimization model according to the objective function obtained in the step (2) and the constraint condition obtained in the step (3); 5. solving the model obtained in the step 4 by using a column and constraint generation algorithm to obtain a traction substation power flow controller, a hybrid energy storage device and a photovoltaic optimal operation plan in the worst scene, namely completing robust energy management of a traction power supply system; the method overcomes the influence of uncertainty of photovoltaic and traction load on the operation of the traction power supply system, improves the operation economy of the traction substation in severe operation environment, and is closer to the reality.

Description

Robust energy management method considering uncertain photovoltaic and load for traction power supply system
Technical Field
The invention belongs to the field of energy management optimization of a traction power supply system, and particularly relates to a robust energy management method of the traction power supply system considering uncertain photovoltaic and load.
Background
Renewable energy sources in China develop rapidly in recent years, and photovoltaic power generation has the advantages of no pollution, no noise, small regional limitation and the like. Meanwhile, electrified railways represented by high-speed railways and heavy haul railways are constructed and operated in large scale in China, and the operation mileage of high-speed railways in China exceeds 3.5 kilometers by 2019. The electrified railway has the following features: 1) the regional distribution is wide, and the traffic network and the renewable energy network have more geographical intersections, such as Lanzhou New (Wulumangoqi) high-speed rails and Sichuan (Chengdu) Tibetan (Lasa) railways crossing southwest and northwest areas with rich wind and light resources; 2) the traction load is high in demand and has high consumption potential, and particularly, in weak areas of an external power supply, the access of renewable energy sources can also play an important supporting role. Therefore, the photovoltaic-based renewable energy is connected into the traction power supply system to promote local consumption and improve the permeability of the renewable energy.
On the other hand, the electric power department in China charges the electric charges for the electrified railway by adopting two large-scale industrial electricity-consumption power rates, wherein the basic electric charges are charged according to the basic capacity or the maximum demand. Maximum demand charging also sets a "threshold" for minimum installed capacity. From the statistics of railway operation departments, the electricity fee has become one of the main operation payment fees. If the hybrid energy storage system is accessed, peak clipping and valley filling of the traction load can be effectively realized, the required electric charge and the electric charge are greatly reduced, the utilization of the regenerative braking energy of the traction power supply system is promoted, and the system operation efficiency is improved. In addition, the requirement of the system on the capacity of the power supply equipment can be reduced. Therefore, the photovoltaic access and hybrid energy storage system in the traction power supply system of the electrified railway are increasingly concerned.
However, the random ripple characteristics of the traction load and the distributed power output represented by photovoltaic power present challenges to the operation of the traction power supply system. How to effectively deal with uncertain factors in a traction power supply system and realize safe, high-quality and high-efficiency operation is the key for solving the problem of energy management of the traction power supply system.
Disclosure of Invention
In order to overcome the influence of photovoltaic and load uncertainty on a traction power supply system, the electricity charge expenditure cost of a railway department is reduced in the worst scene, and the energy management optimization method is closer to reality. The invention provides a robust energy management method of a traction power supply system considering uncertain photovoltaic and load.
The invention discloses a robust energy management method of a traction power supply system considering uncertain photovoltaic and load, which comprises the following steps:
step 1: acquiring predicted values and fluctuation intervals of traction substation load process data and photovoltaic output data, and constructing uncertain sets of traction loads and photovoltaic outputs;
step 2: establishing a target function of a robust optimization model according to the electric charge parameters and the intervals of the load process data and the photovoltaic output data of the traction substation obtained in the step 1;
and step 3: according to power capacity parameters and three-phase voltage unbalance international parameters of the hybrid energy storage system and the photovoltaic system, establishing constraint conditions of a robust optimization model based on the traction substation load process data and the photovoltaic output data interval obtained in the step 1, and linearizing the constraint conditions of the robust optimization model;
and 4, step 4: establishing a robust energy management method of the traction power supply system based on a two-stage robust optimization model according to the objective function obtained in the step 2 and the constraint condition obtained in the step 3;
and 5: and (4) solving the model obtained in the step (4) by using a column and constraint generation algorithm to obtain the optimal charging and discharging power of the hybrid energy storage device, the optimal photovoltaic grid-connected power and the optimal power flow power of a back-to-back converter in the power flow controller under the worst scene, namely completing the robust energy management optimization of the traction power supply system.
Further, in the step 1, the load process data of the traction substation in the existing line can be obtained from the historical load data, and the traction substation in the newly-built line can be obtained by calculating load process simulation software, such as elbase/WEBANET, according to the high-speed railway line, the train and the schedule.
Further, the uncertain set of photovoltaic output and load in step 1 is:
Figure BDA0002403459970000021
Figure BDA0002403459970000022
Figure BDA0002403459970000023
in the formula: p, A and R are the uncertain set of photovoltaic power output, active load and reactive load, p, respectivelyt、atAnd rtFor photovoltaic power output, active load and reactiveUncertainty variable of load, Δ pt、ΔatAnd Δ rtRespectively represent predicted values pt f、at fAnd rt fDeviation value of, parameter Γp、ΓaAnd ΓrRepresenting uncertainty margin, i.e. the total relative deviation of the uncertainty variable from the predicted value, which ranges from 0 to the total time period N of the dayTIn the meantime.
Further, the objective function in step 2 is:
Figure BDA0002403459970000024
Figure BDA0002403459970000025
in the formula: f is an objective function and represents daily operation cost of the traction substation, t is a time period and P ist grid,buyElectric power purchased from the grid for traction substations, Pt grid,fedElectric power P fed back to the grid by the traction substationt PVFor photovoltaic output, Pt b,disFor discharging power of the battery, Pt b,chCharging power for batteries, Pt u,disDischarging power, P, for the super-capacitort u,chCharging power to the super capacitor, Pt demIn order to demand the power, the power supply is,
Figure BDA0002403459970000031
in order to achieve the cost price of photovoltaic operation and maintenance,
Figure BDA0002403459970000032
in order to maintain the cost price of the battery operation,
Figure BDA0002403459970000033
cost price for operating and maintaining super capacitor, cdemThe price of the required electricity charge is calculated,
Figure BDA0002403459970000034
the price of the electricity purchased by the traction substation is,
Figure BDA0002403459970000035
charge price levied for feedback back to the grid power, NdayFor the number of days of operation per month,
Figure BDA0002403459970000036
and
Figure BDA0002403459970000037
are all binary variables;
wherein:
Figure BDA0002403459970000038
further, the constraint conditions in the step 3 comprise power balance constraint, hybrid energy storage system constraint, public power grid power constraint, photovoltaic power generation constraint, back-to-back converter constraint and three-phase voltage unbalance constraint; the method specifically comprises the following steps:
power balance constraint conditions:
Pt grid,buy-Pt grid,fed＝Pt T+Pt α(6)
Pt α+Pt b,dis+Pt u,dis+Pt PV＝Pt β+Pt b,ch+Pt u,ch(7)
Pt T+Pt β＝at(8)
Figure BDA0002403459970000039
Figure BDA00024034599700000310
Figure BDA00024034599700000311
in the formula: pt TFor active power of traction transformers, Pt αFor α phase active power, P, of the back-to-back convertert βFor β phase active power, Q, of back-to-back convertert ββ phase reactive power P for back-to-back convertert PVThe photovoltaic output is obtained;
Figure BDA00024034599700000312
is the maximum limit value of the interaction power between the traction substation and the power grid,
Figure BDA00024034599700000313
is a binary variable representing the direction of power interaction between the traction substation and the grid,
Figure BDA00024034599700000314
indicating that the interactive power flows from the grid to the traction substation,
Figure BDA00024034599700000315
and the representation interactive power is fed back to the power grid by the traction substation.
Hybrid energy storage system constraint conditions:
Figure BDA00024034599700000316
Figure BDA00024034599700000317
Figure BDA00024034599700000318
Figure BDA00024034599700000319
Figure BDA00024034599700000320
Figure BDA0002403459970000041
Figure BDA0002403459970000042
in the formula: epsilonbIs the self-discharge rate of the battery, epsilonbIs the self-discharge rate of the super capacitor, ηb,disFor the discharge efficiency of the battery, ηb,chFor the charging efficiency of the battery, ηu,disFor the discharge efficiency of the supercapacitor, ηu,chTo the charging efficiency of the super capacitor, Δ t is the unit time period,
Figure BDA0002403459970000043
for the electric energy stored in the battery during the time period t +1, Et bStoring the electric energy for the battery in the time period t;
Figure BDA0002403459970000044
the electric energy stored by the super capacitor in the time period of t +1,
Figure BDA0002403459970000045
the electric energy stored for the super capacitor in the time period t;
Figure BDA0002403459970000046
is the rated power of the battery and is,
Figure BDA0002403459970000047
the power of the super capacitor is rated,
Figure BDA0002403459970000048
is the minimum state of charge of the battery,
Figure BDA0002403459970000049
the maximum state of charge of the battery is,
Figure BDA00024034599700000410
for rating the capacity of the battery，
Figure BDA00024034599700000411
The capacity of the super capacitor is rated,
Figure BDA00024034599700000412
the electrical energy stored by the battery for the time period t-1,
Figure BDA00024034599700000413
the electric energy stored by the super capacitor in the period of t-1,
Figure BDA00024034599700000414
the minimum state of charge of the super capacitor is obtained,
Figure BDA00024034599700000415
the maximum charge state of the super capacitor;
Figure BDA00024034599700000416
the electrical energy stored in the battery for the initial period of time each day,
Figure BDA00024034599700000417
the electrical energy stored in the battery for the last period of the day,
Figure BDA00024034599700000418
for the purpose of the initial state of charge per day,
Figure BDA00024034599700000419
the super capacitor stores electric energy for the initial time period every day,
Figure BDA00024034599700000420
the electric energy stored by the super capacitor for the last time period of each day,
Figure BDA00024034599700000421
the initial charge state of the super capacitor every day;
Figure BDA00024034599700000422
and
Figure BDA00024034599700000423
are all binary variables.
Photovoltaic power generation constraint:
0≤Pt PV≤pt(19)
in the formula: p is a radical oftThe photovoltaic output uncertain variable is the solar photovoltaic output upper limit value.
Back-to-back converter constraint:
Figure BDA00024034599700000424
Figure BDA00024034599700000425
in the formula:
Figure BDA00024034599700000426
for the capacity of the back-to-back inverter α phases,
Figure BDA00024034599700000427
the capacity of the back-to-back inverter β phases.
And (3) three-phase voltage unbalance degree constraint:
Figure BDA00024034599700000428
Figure BDA00024034599700000429
Figure BDA00024034599700000430
in the formula: epsilonUFor traction substation power grid side three-phase voltage unbalance degree, USFor the grid side line voltage, ScapFor the short-circuit capacity of the side line of the power grid,
Figure BDA0002403459970000051
is the upper limit value of the unbalance degree of the three-phase voltage in the national standard,
Figure BDA0002403459970000052
for grid side negative sequence current, UTFor the voltage at the outlet of the traction transformer, UαIs the voltage at the α phase outlet of the back-to-back converter, N1For single-phase traction transformer transformation ratio, N2For high voltage matching transformer transformation ratio, a is complex operator ej120°，
Figure BDA0002403459970000053
Is the voltage-current phase angle difference of the single-phase traction transformer,
Figure BDA0002403459970000054
is the voltage-current phase angle difference, I, of α phases of the back-to-back current transformerTFor drawing transformer currents, IαIs the current of the back-to-back current transformer α phases.
Further, the constraint condition linearization method in step 3 is as follows:
the max function in equation (5) is linearized by the following equation:
max(Pt dem)＝Pdem，max(25)
Figure BDA0002403459970000055
in the formula: pdem,maxIs an auxiliary variable representing the maximum demand value during the day.
The formula (21) after linearization is given by:
Figure BDA0002403459970000056
Figure BDA0002403459970000057
Figure BDA0002403459970000058
Figure BDA0002403459970000059
in the formula: n is a radical ofpThe number of the fan-shaped PQ semicircles is equal to that of the fan-shaped PQ semicircles, and the Q is more than or equal to 0; delta theta is the sector angle, (P)k,Qk) Is the division point coordinate of the fan shape and the PQ semicircle.
Equation (24) is linearized as follows:
Figure BDA00024034599700000510
Figure BDA00024034599700000511
Figure BDA00024034599700000512
in the formula:
Figure BDA00024034599700000513
and
Figure BDA00024034599700000514
are all auxiliary variables, and are all the auxiliary variables,
Figure BDA00024034599700000515
is a binary variable.
Further, the two-stage robust optimization model for robust energy management of the traction power supply system established in step 4 is as follows:
Figure BDA00024034599700000516
Figure BDA00024034599700000517
in the formula: x represents the binary decision variable vector of the first stage,
Figure BDA0002403459970000061
y denotes a second stage continuous decision variable vector,
Figure BDA0002403459970000062
Figure BDA0002403459970000063
c. b, D, D, E, E, F, F, G, H and I are parameter matrixes or parameter vectors.
Further, in step 5, based on the robust energy management model of the traction power supply system obtained in step 4, a main model and a sub model of the robust energy management model of the traction power supply system are formed through a column and constraint generation algorithm, and finally the main model and the sub model are solved in an iterative loop manner until a convergence standard is met.
Wherein, the expression of the main model is as follows:
Figure BDA0002403459970000064
s.t.x∈{0,1} (37)
Figure BDA0002403459970000065
Figure BDA0002403459970000066
Figure BDA0002403459970000067
Figure BDA0002403459970000068
Figure BDA0002403459970000069
Figure BDA00024034599700000610
Figure BDA00024034599700000611
in the formula: k is the number of iterative solution times, ylThe decision variables added to the main model at the first loop,
Figure BDA00024034599700000612
to solve for the worst photovoltaic output obtained by the submodel,
Figure BDA00024034599700000613
for solving the worst active load scenario, r, obtained for the submodell *And the worst reactive load scene obtained by solving the submodel is shown.
The sub-model expression is:
Figure BDA00024034599700000614
s.t.By≤d,(γ1) (46)
Dy＝e,(γ2) (47)
Fy≤f-Ex*,(γ3) (48)
Gy≤p,(γ4) (49)
Hy＝a,(γ5) (50)
Iy＝r,(γ6) (51)
in the formula: x is the number of*For the optimal solution of the main model, { gamma {1,γ2,γ3,γ4,γ5,γ6Is a dual variable of the constraint.
The submodel equivalent representation method is as follows:
worst scenario p in sub-model*、a*And r*To not determine the extreme values in sets P, A and R, equations (1) - (3) are therefore equivalent to:
Figure BDA0002403459970000071
Figure BDA0002403459970000072
Figure BDA0002403459970000073
in the formula: gt、mt、
Figure BDA0002403459970000074
And
Figure BDA0002403459970000075
are all binary variables.
Based on strong dual theory, submodels (45) - (51) are equivalent to:
Figure BDA0002403459970000076
-BTγ1+DTγ2-FTγ3-GTγ4+HTγ5+ITγ6＝cT(56)
γ1≥0,γ3≥0,γ4≥0,γ2,γ5,γ6is a free variable (57)
Figure BDA0002403459970000077
Figure BDA0002403459970000078
Figure BDA0002403459970000081
Figure BDA0002403459970000082
In the formula: lambda, mu, omega1And ω2Are all auxiliary variables, and are all the auxiliary variables,
Figure BDA0002403459970000083
the invention has the beneficial effects that:
1. the robust energy management optimization method for the traction power supply system can overcome the influence of uncertainty of photovoltaic and traction load on safe and efficient operation of the traction power supply system, effectively reduces the electric charge expenditure of railway departments, and accords with the development trend of green and intelligent traffic.
2. The robust energy management optimization model of the traction power supply system is established based on the uncertain set of photovoltaic output and traction load, the robustness of the energy management model of the traditional traction power supply system is improved, and the optimality of the traction power supply system in the worst scene is ensured;
3. the method establishes the mixed integer linear programming model by carrying out linearization treatment on the nonlinear elements in the objective function and the constraint condition, is convenient for directly solving by utilizing an optimization solver, and avoids the complexity of solving the mixed integer nonlinear model.
Drawings
Fig. 1 is a schematic diagram of a robust energy management structure of a traction power supply system in the invention.
FIG. 2 is a schematic flow chart of the present invention.
Detailed Description
The invention is further described with reference to the following figures and specific embodiments.
The structure of a traction power supply system aimed at by the invention is shown in fig. 1, and the flow of the robust energy management method of the traction power supply system considering uncertain photovoltaic and load is shown in fig. 2, specifically:
step 1: based on the actual measurement historical data, obtaining the predicted values and fluctuation intervals of the load process data and the photovoltaic output data of the traction substation, and establishing an uncertain set of traction load and photovoltaic output:
Figure BDA0002403459970000084
Figure BDA0002403459970000085
Figure BDA0002403459970000091
in the formula: p, A and R are the uncertain set of photovoltaic power output, active load and reactive load, p, respectivelyt、atAnd rtFor uncertain variables of photovoltaic contribution, active load and reactive load, Δ pt、ΔatAnd Δ rtRespectively represent predicted values
Figure BDA0002403459970000092
And
Figure BDA00024034599700000913
deviation value of, parameter Γp、ΓaAnd ΓrRepresenting uncertainty margin, i.e. the total relative deviation of the uncertainty variable from the predicted value, which ranges from 0 to the total time period N of the dayTIn the meantime.
Step 2: establishing an objective function of an optimization model according to the electric charge parameters and the intervals of the load process data and the photovoltaic output data of the traction substation obtained in the step 1;
the target function expression is:
Figure BDA0002403459970000093
Figure BDA0002403459970000094
in the formula: f is an objective function and represents daily operation cost of the traction substation, t is a time period and P ist grid,buyElectric power purchased from the grid for traction substations, Pt grid,fedWith traction substation feeding back to the gridElectric power, Pt PVFor photovoltaic output, Pt b,disFor discharging power of the battery, Pt b,chCharging power for batteries, Pt u,disDischarging power, P, for the super-capacitort u,chCharging power to the super capacitor, Pt demIn order to demand the power, the power supply is,
Figure BDA0002403459970000095
in order to achieve the cost price of photovoltaic operation and maintenance,
Figure BDA0002403459970000096
in order to maintain the cost price of the battery operation,
Figure BDA0002403459970000097
cost price for operating and maintaining super capacitor, cdemThe price of the required electricity charge is calculated,
Figure BDA0002403459970000098
the price of the electricity purchased by the traction substation is,
Figure BDA0002403459970000099
charge price levied for feedback back to the grid power, NdayFor the number of days of operation per month,
Figure BDA00024034599700000910
and
Figure BDA00024034599700000911
are all binary variables;
wherein:
Figure BDA00024034599700000912
and step 3: establishing constraint conditions of an optimization model based on the traction substation load process data and the photovoltaic output data interval obtained in the step 1 according to power capacity parameters and three-phase voltage unbalance international parameters of the hybrid energy storage system and the photovoltaic system, and linearizing the constraint conditions of the optimization model;
the constraint conditions comprise power balance constraint, hybrid energy storage system constraint, public power grid power constraint, photovoltaic power generation constraint, back-to-back converter constraint in a power flow controller and three-phase unbalance constraint.
The constraints are as follows:
power balance constraint conditions:
Pt grid,buy-Pt grid,fed＝Pt T+Pt α(6)
Pt α+Pt b,dis+Pt u,dis+Pt PV＝Pt β+Pt b,ch+Pt u,ch(7)
Pt T+Pt β＝at(8)
Figure BDA0002403459970000101
Figure BDA0002403459970000102
Figure BDA0002403459970000103
in the formula: pt TFor active power of traction transformers, Pt αFor α phase active power, P, of the back-to-back convertert βFor the back-to-back converter β phase active power,
Figure BDA0002403459970000104
β phase reactive power P for back-to-back convertert PVThe photovoltaic output is obtained;
Figure BDA0002403459970000105
is the maximum limit value of the interaction power between the traction substation and the power grid,
Figure BDA0002403459970000106
is a binary variable representing the direction of power interaction between the traction substation and the grid,
Figure BDA0002403459970000107
indicating that the interactive power flows from the grid to the traction substation,
Figure BDA0002403459970000108
and the representation interactive power is fed back to the power grid by the traction substation.
Hybrid energy storage system constraint conditions:
Figure BDA0002403459970000109
Figure BDA00024034599700001010
Figure BDA00024034599700001011
Figure BDA00024034599700001012
Figure BDA00024034599700001013
Figure BDA00024034599700001014
Figure BDA00024034599700001015
in the formula: epsilonbIs the self-discharge rate of the battery, epsilonbIs the self-discharge rate of the super capacitor, ηb,disFor the discharge efficiency of the battery, ηb,chFor the charging efficiency of the battery, ηu,disFor the discharge efficiency of the supercapacitor, ηu,chTo the charging efficiency of the super capacitor, Δ t is the unit time period,
Figure BDA00024034599700001016
for the electrical energy stored by the battery during the time period t +1,
Figure BDA00024034599700001017
storing the electric energy for the battery in the time period t;
Figure BDA00024034599700001018
the electric energy stored by the super capacitor in the time period of t +1,
Figure BDA00024034599700001019
the electric energy stored for the super capacitor in the time period t;
Figure BDA00024034599700001020
is the rated power of the battery and is,
Figure BDA00024034599700001021
the power of the super capacitor is rated,
Figure BDA00024034599700001022
is the minimum state of charge of the battery,
Figure BDA00024034599700001023
the maximum state of charge of the battery is,
Figure BDA00024034599700001024
is the rated capacity of the battery,
Figure BDA00024034599700001025
the capacity of the super capacitor is rated,
Figure BDA00024034599700001026
the electrical energy stored by the battery for the time period t-1,
Figure BDA00024034599700001027
electric energy stored for super capacitor in t-1 period，
Figure BDA00024034599700001028
The minimum state of charge of the super capacitor is obtained,
Figure BDA00024034599700001029
the maximum charge state of the super capacitor;
Figure BDA00024034599700001030
the electrical energy stored in the battery for the initial period of time each day,
Figure BDA0002403459970000111
the electrical energy stored in the battery for the last period of the day,
Figure BDA0002403459970000112
for the purpose of the initial state of charge per day,
Figure BDA0002403459970000113
the super capacitor stores electric energy for the initial time period every day,
Figure BDA0002403459970000114
the electric energy stored by the super capacitor for the last time period of each day,
Figure BDA0002403459970000115
the initial charge state of the super capacitor every day;
Figure BDA0002403459970000116
and
Figure BDA0002403459970000117
are all binary variables.
Photovoltaic power generation constraint:
0≤Pt PV≤pt(19)
in the formula: p is a radical oftThe photovoltaic output uncertain variable is the solar photovoltaic output upper limit value.
Back-to-back converter constraint:
Figure BDA0002403459970000118
Figure BDA0002403459970000119
in the formula:
Figure BDA00024034599700001110
for the capacity of the back-to-back inverter α phases,
Figure BDA00024034599700001111
the capacity of the back-to-back inverter β phases.
And (3) three-phase voltage unbalance degree constraint:
Figure BDA00024034599700001112
Figure BDA00024034599700001113
Figure BDA00024034599700001114
in the formula: epsilonUFor traction substation power grid side three-phase voltage unbalance degree, USFor the grid side line voltage, ScapFor the short-circuit capacity of the side line of the power grid,
Figure BDA00024034599700001115
is the upper limit value of the unbalance degree of the three-phase voltage in the national standard,
Figure BDA00024034599700001116
for grid side negative sequence current, UTFor the voltage at the outlet of the traction transformer, UαIs the voltage at the α phase outlet of the back-to-back converter, N1For single-phase traction transformer transformation ratio, N2For high voltage matching transformer transformation ratio, a is complex operator ej120°，
Figure BDA00024034599700001117
Is the voltage-current phase angle difference of the single-phase traction transformer,
Figure BDA00024034599700001118
is the voltage-current phase angle difference, I, of α phases of the back-to-back current transformerTFor drawing transformer currents, IαIs the current of the back-to-back current transformer α phases.
The constraint linearization method is as follows:
the max function in equation (5) is linearized by the following equation:
max(Pt dem)＝Pdem，max(25)
Figure BDA00024034599700001119
in the formula: pdem,maxIs an auxiliary variable representing the maximum demand value during the day.
The formula (21) after linearization is given by:
Figure BDA0002403459970000121
Figure BDA0002403459970000122
Figure BDA0002403459970000123
Figure BDA0002403459970000124
in the formula: n is a radical ofpThe number of the fan-shaped PQ semicircles is equal to that of the fan-shaped PQ semicircles, and the Q is more than or equal to 0; delta theta is the sector angle, (P)k,Qk) Is the division point coordinate of the fan shape and the PQ semicircle.
Equation (24) is linearized as follows:
Figure BDA0002403459970000125
Figure BDA0002403459970000126
Figure BDA0002403459970000127
in the formula:
Figure BDA0002403459970000128
and
Figure BDA0002403459970000129
are all auxiliary variables, and are all the auxiliary variables,
Figure BDA00024034599700001210
is a binary variable.
And 4, step 4: establishing a two-stage robust optimization model of robust energy management of the traction power supply system according to the objective function obtained in the step (2) and the constraint condition obtained in the step (3);
the two-stage robust optimization model for robust energy management of the traction power supply system comprises the following steps:
Figure BDA00024034599700001211
Figure BDA00024034599700001212
in the formula: x represents the binary decision variable vector of the first stage,
Figure BDA00024034599700001213
y denotes a second stage continuous decision variable vector,
Figure BDA00024034599700001214
Figure BDA00024034599700001215
c. b, D, D, E, E, F, F, G, H and I are parameter matrixes or parameter vectors.
And 5: and forming a main model and a sub model of the robust energy management model of the traction power supply system through a column and constraint generation algorithm, and finally, circularly and iteratively solving the main model and the sub model by using optimization software (such as a hybrid integer optimization solver CPLEX in a Matlab environment) to obtain the optimal charging and discharging power of the hybrid energy storage device, the optimal photovoltaic grid-connected power and the optimal power flow power of a back-to-back converter in a power flow controller under the worst scene, namely completing the robust energy management optimization of the traction power supply system.
The main model expression is:
Figure BDA0002403459970000131
s.t.x∈{0,1} (37)
Figure BDA0002403459970000132
Figure BDA0002403459970000133
Figure BDA0002403459970000134
Figure BDA0002403459970000135
Figure BDA0002403459970000136
Figure BDA0002403459970000137
Figure BDA0002403459970000138
in the formula: k is the number of iterative solution times, ylThe decision variables added to the main model at the first loop,
Figure BDA0002403459970000139
to solve for the worst photovoltaic output obtained by the submodel,
Figure BDA00024034599700001310
for solving the worst active load scenario, r, obtained for the submodell *Representing the worst reactive load scene obtained by solving the submodel;
the sub-model expression is:
Figure BDA00024034599700001311
s.t.By≤d,(γ1) (46)
Dy＝e,(γ2) (47)
Fy≤f-Ex*,(γ3) (48)
Gy≤p,(γ4) (49)
Hy＝a,(γ5) (50)
Iy＝r,(γ6) (51)
in the formula: x is the number of*For the optimal solution of the main model, { gamma {1,γ2,γ3,γ4,γ5,γ6Is a constraint dual variable;
the submodel equivalent representation method is as follows:
worst scenario p in sub-model*、a*And r*To not determine the extreme values in sets P, A and R, equations (1) - (3) are therefore equivalent to:
Figure BDA0002403459970000141
Figure BDA0002403459970000142
Figure BDA0002403459970000143
in the formula: gt、mt、
Figure BDA0002403459970000144
And
Figure BDA0002403459970000145
are all binary variables;
based on strong dual theory, submodels (45) - (51) are equivalent to:
Figure BDA0002403459970000146
-BTγ1+DTγ2-FTγ3-GTγ4+HTγ5+ITγ6＝cT(56)
γ1≥0,γ3≥0,γ4≥0,γ2,γ5,γ6is a free variable (57)
Figure BDA0002403459970000147
Figure BDA0002403459970000148
Figure BDA0002403459970000149
Figure BDA00024034599700001410
In the formula: lambda, mu, omega1And ω2Are all auxiliary variables, and are all the auxiliary variables,
Figure BDA00024034599700001411
examples
The robust energy management structure of the electrified railway traction power supply system considering uncertain photovoltaic and load is shown in figure 1, the parameters of the energy storage system are shown in table 1, the parameters of a traction substation are shown in table 2, and the power price parameters of a power grid are shown in table 3.
TABLE 1 hybrid energy storage System parameters
Figure BDA0002403459970000151
TABLE 2 traction substation parameters
Figure BDA0002403459970000152
TABLE 3 Electricity price parameter
Figure BDA0002403459970000153
The operating and maintenance costs of other parameters, such as photovoltaic, battery and super capacitor are all 0.1 ￥/kWh, and the charging of feedback electric energy is cfed＝-0.8cbuyIn linearization of PFC converter, the fan angle Δ θ is 30 °, C&The CG algorithm iteration convergence precision epsilon is 0.01. The uncertain interval of the photovoltaic output and the traction load is expressed as [ 1-lambda ]p,1+λp]pf、[1-λa,1+λa]afAnd [ 1-lambda ]r,1+λr]rfWherein λ isp、λaAnd λrFor predicting the deviation coefficients, 0.1 is taken.
The robust energy management optimization method of the traction power supply system is compared with the traditional deterministic energy management method, and the results are shown in tables 4 and 5 after simulation calculation.
TABLE 4 comparison of results of deterministic method and robust method under different uncertainty margins
Figure BDA0002403459970000154
TABLE 5 comparison of deterministic and robust method results under different prediction biases
Figure BDA0002403459970000161
As can be seen from tables 4 and 5, with the increase of the uncertainty margin and the prediction deviation coefficient, compared with the conventional deterministic energy management method, the robust energy management optimization method for the traction power supply system has the advantages that the total operating cost is lower in the worst scenario, and the saving rate is continuously increased. Particularly, when the prediction deviation coefficient reaches 0.6, the traditional deterministic energy management method has the situation of being unsolvable, so the robustness is low, and the robust energy management optimization method can be solvated in all scenes, so the robustness is high.
The invention provides a double-stage robust model-based energy management robust optimization method for a traction power supply system, which aims at the traction power supply system connected with a photovoltaic power generation system and an energy storage device, takes photovoltaic output and traction load uncertainty into consideration, and provides the double-stage robust model-based energy management robust optimization method for the traction power supply system. According to the method, in the first stage, based on photovoltaic output and traction load prediction information, a charging and discharging strategy of an energy storage device and an electric quantity trading scheme with a power grid are formulated, and in the second stage, the worst scene of the photovoltaic output and traction load and the corresponding optimal power flow are found. The column and constraint generation algorithm is adopted to solve the two-stage robust optimization model, and the result shows that compared with the traditional deterministic energy management method, the robust energy management optimization method has the advantages of optimality and robustness, especially under the condition that the uncertainty of the operating environment is increased. The method makes the energy management method of the traction power supply system more practical and provides a foundation for the access and engineering application of renewable energy sources and energy storage systems in future electrified railways.

Claims (7)

1. The method for managing the robust energy of the traction power supply system in consideration of uncertain photovoltaic and load is characterized by comprising the following steps of:
step 1: acquiring predicted values and fluctuation intervals of traction substation load process data and photovoltaic output data, and constructing uncertain sets of traction loads and photovoltaic outputs;
step 2: establishing a target function of a robust optimization model according to the electric charge parameters and the intervals of the load process data and the photovoltaic output data of the traction substation obtained in the step 1;
and step 3: according to power capacity parameters of a hybrid energy storage system and a photovoltaic system and national standard limit values of three-phase voltage unbalance, establishing constraint conditions of a robust optimization model based on the traction substation load process data and the photovoltaic output data interval obtained in the step 1, and linearizing the constraint conditions of the robust optimization model;
and 4, step 4: establishing a robust energy management method of the traction power supply system based on a two-stage robust optimization model according to the objective function obtained in the step 2 and the constraint condition obtained in the step 3;
and 5: and (4) solving the model obtained in the step (4) by using a column and constraint generation algorithm to obtain the optimal charging and discharging power of the hybrid energy storage device, the optimal photovoltaic grid-connected power and the optimal power flow power of a back-to-back converter in the power flow controller under the worst scene, namely completing the robust energy management optimization of the traction power supply system.
2. The method for robust energy management of a traction power supply system considering uncertain photovoltaic and load according to claim 1, wherein the uncertain set of photovoltaic output and load in step 1 is:
Figure FDA0002403459960000011
Figure FDA0002403459960000012
Figure FDA0002403459960000013
in the formula: p, A and R are the uncertain set of photovoltaic power output, active load and reactive load, p, respectivelyt、atAnd rtIs light ofUncertain variable, Δ p, of the outages, active and reactive loadst、ΔatAnd Δ rtRespectively represent predicted values
Figure FDA0002403459960000014
And rt fDeviation value of, parameter Γp、ΓaAnd ΓrRepresenting uncertainty margin, i.e. the total relative deviation of the uncertainty variable from the predicted value, which ranges from 0 to the total time period N of the dayTIn the meantime.
3. The method for robust energy management of a traction power supply system considering uncertain photovoltaic and load according to claim 1, wherein the objective function in the step 2 is as follows:
Figure FDA0002403459960000015
Figure FDA0002403459960000021
in the formula: f is an objective function and represents daily operation cost of the traction substation, t is a time period and P ist grid,buyElectric power purchased from the grid for traction substations, Pt grid,fedElectric power P fed back to the grid by the traction substationt PVFor photovoltaic output, Pt b,disFor discharging power of the battery, Pt b,chCharging power for batteries, Pt u,disDischarging power, P, for the super-capacitort u,chCharging power to the super capacitor, Pt demIn order to demand the power, the power supply is,
Figure FDA0002403459960000022
in order to achieve the cost price of photovoltaic operation and maintenance,
Figure FDA0002403459960000023
for transporting batteriesThe cost price of the maintenance of the line,
Figure FDA0002403459960000024
cost price for operating and maintaining super capacitor, cdemThe price of the required electricity charge is calculated,
Figure FDA0002403459960000025
the price of the electricity purchased by the traction substation is,
Figure FDA0002403459960000026
charge price levied for feedback back to the grid power, NdayFor the number of days of operation per month,
Figure FDA0002403459960000027
and
Figure FDA0002403459960000028
are all binary variables;
wherein:
Figure FDA0002403459960000029
4. the method for robust energy management of a traction power supply system considering uncertain photovoltaic and load according to claim 1, wherein the constraint conditions in the step 3 are as follows:
power balance constraint conditions:
Pt grid,buy-Pt grid,fed＝Pt T+Pt α(6)
Pt α+Pt b,dis+Pt u,dis+Pt PV＝Pt β+Pt b,ch+Pt u,ch(7)
Pt T+Pt β＝at(8)
Figure FDA00024034599600000210
Figure FDA00024034599600000211
Figure FDA00024034599600000212
in the formula: pt TFor active power of traction transformers, Pt αFor α phase active power, P, of the back-to-back convertert βFor the back-to-back converter β phase active power,
Figure FDA00024034599600000213
β phase reactive power P for back-to-back convertert PVThe photovoltaic output is obtained;
Figure FDA00024034599600000214
is the maximum limit value of the interaction power between the traction substation and the power grid,
Figure FDA00024034599600000215
is a binary variable representing the direction of power interaction between the traction substation and the grid,
Figure FDA00024034599600000216
indicating that the interactive power flows from the grid to the traction substation,
Figure FDA00024034599600000217
representing that the interactive power is fed back to the power grid by the traction substation;
hybrid energy storage system constraint conditions:
Figure FDA00024034599600000218
Figure FDA00024034599600000219
Figure FDA0002403459960000031
Figure FDA0002403459960000032
Figure FDA0002403459960000033
Figure FDA0002403459960000034
Figure FDA0002403459960000035
in the formula: epsilonbIs the self-discharge rate of the battery, epsilonbIs the self-discharge rate of the super capacitor, ηb,disFor the discharge efficiency of the battery, ηb,chFor the charging efficiency of the battery, ηu,disFor the discharge efficiency of the supercapacitor, ηu,chTo the charging efficiency of the super capacitor, Δ t is the unit time period,
Figure FDA0002403459960000036
for the electrical energy stored by the battery during the time period t +1,
Figure FDA0002403459960000037
storing the electric energy for the battery in the time period t;
Figure FDA0002403459960000038
the electric energy stored by the super capacitor in the time period of t +1,
Figure FDA0002403459960000039
the electric energy stored for the super capacitor in the time period t;
Figure FDA00024034599600000310
is the rated power of the battery and is,
Figure FDA00024034599600000311
the power of the super capacitor is rated, bSOCis the minimum state of charge of the battery,
Figure FDA00024034599600000312
the maximum state of charge of the battery is,
Figure FDA00024034599600000313
is the rated capacity of the battery,
Figure FDA00024034599600000314
the capacity of the super capacitor is rated,
Figure FDA00024034599600000315
the electrical energy stored by the battery for the time period t-1,
Figure FDA00024034599600000316
the electric energy stored by the super capacitor in the period of t-1, uSOCthe minimum state of charge of the super capacitor is obtained,
Figure FDA00024034599600000317
the maximum charge state of the super capacitor;
Figure FDA00024034599600000318
the electrical energy stored in the battery for the initial period of time each day,
Figure FDA00024034599600000319
the electrical energy stored in the battery for the last period of the day,
Figure FDA00024034599600000320
for the purpose of the initial state of charge per day,
Figure FDA00024034599600000321
the super capacitor stores electric energy for the initial time period every day,
Figure FDA00024034599600000322
the electric energy stored by the super capacitor for the last time period of each day,
Figure FDA00024034599600000323
the initial charge state of the super capacitor every day;
Figure FDA00024034599600000324
and
Figure FDA00024034599600000325
are all binary variables;
photovoltaic power generation constraint:
0≤Pt PV≤pt(19)
in the formula: p is a radical oftThe photovoltaic output uncertain variable is an intra-day photovoltaic output upper limit value;
back-to-back converter constraint:
Figure FDA00024034599600000326
Figure FDA00024034599600000327
in the formula:
Figure FDA00024034599600000328
for the capacity of the back-to-back inverter α phases,
Figure FDA00024034599600000329
capacity of the back-to-back inverter β phases;
and (3) three-phase voltage unbalance degree constraint:
Figure FDA00024034599600000330
Figure FDA0002403459960000041
Figure FDA0002403459960000042
in the formula: epsilonUFor traction substation power grid side three-phase voltage unbalance degree, USFor the grid side line voltage, ScapFor the short-circuit capacity of the side line of the power grid,
Figure FDA0002403459960000043
is the upper limit value of the unbalance degree of the three-phase voltage in the national standard,
Figure FDA0002403459960000044
for grid side negative sequence current, UTFor the voltage at the outlet of the traction transformer, UαIs the voltage at the α phase outlet of the back-to-back converter, N1For single-phase traction transformer transformation ratio, N2For high voltage matching transformer transformation ratio, a is complex operator ej120°，
Figure FDA0002403459960000045
Is the voltage-current phase angle difference of the single-phase traction transformer,
Figure FDA0002403459960000046
is the voltage-current phase angle difference, I, of α phases of the back-to-back current transformerTFor drawing transformer currents, IαIs the current of the back-to-back current transformer α phases.
5. The method for robust energy management of a traction power supply system considering uncertain photovoltaic and load according to claim 1, wherein the constraint condition linearization method in the step 3 is as follows:
the max function in equation (5) is linearized by the following equation:
max(Pt dem)＝Pdem,max(25)
Figure FDA0002403459960000047
in the formula: pdem,maxIs an auxiliary variable representing the maximum demand value during the day;
the formula (21) after linearization is given by:
Figure FDA0002403459960000048
Figure FDA0002403459960000049
Figure FDA00024034599600000410
Figure FDA00024034599600000411
in the formula: n is a radical ofpThe number of the fan-shaped PQ semicircles is equal to that of the fan-shaped PQ semicircles, and the Q is more than or equal to 0; delta theta is the sector angle, (P)k,Qk) The coordinates of the division points of the fan shape and the PQ semicircle;
equation (24) is linearized as follows:
Figure FDA00024034599600000412
Figure FDA00024034599600000413
Figure FDA0002403459960000051
in the formula:
Figure FDA0002403459960000052
and
Figure FDA0002403459960000053
are all auxiliary variables, and are all the auxiliary variables,
Figure FDA0002403459960000054
is a binary variable.
6. The method for robust energy management of a traction power supply system considering uncertain photovoltaic and load according to claim 1, wherein the two-stage robust optimization model for robust energy management of the traction power supply system established in the step 4 is as follows:
Figure FDA0002403459960000055
Figure FDA0002403459960000056
in the formula: x represents the binary decision variable vector of the first stage,
Figure FDA0002403459960000057
y denotes a second stage continuous decision variable vector,
Figure FDA0002403459960000058
Figure FDA0002403459960000059
c. b, D, D, E, E, F, F, G, H and I are parameter matrixes or parameter vectors.
7. The method for managing the robust energy of the traction power supply system in consideration of uncertain photovoltaic and load according to claim 1, wherein a main model and a sub model of the robust energy management model of the traction power supply system are formed through a column and constraint generation algorithm in the step 5, the main model and the sub model are solved in a circulating iteration mode to obtain the optimal charging and discharging power of the hybrid energy storage device in the worst scene, the optimal photovoltaic grid-connected power and the optimal power flow power of a back-to-back converter in a power flow controller, and therefore the robust energy management optimization of the traction power supply system is completed;
wherein, the expression of the main model is as follows:
Figure FDA00024034599600000510
s.t.x∈{0,1} (37)
Figure FDA00024034599600000511
Figure FDA00024034599600000512
Figure FDA00024034599600000513
Figure FDA00024034599600000514
Figure FDA00024034599600000515
Figure FDA00024034599600000516
Figure FDA00024034599600000517
in the formula: k is the number of iterative solution times, ylThe decision variables added to the main model at the first loop,
Figure FDA0002403459960000061
to solve for the worst photovoltaic output obtained by the submodel,
Figure FDA0002403459960000062
for solving the worst active load scenario, r, obtained for the submodell *Representing the worst reactive load scene obtained by solving the submodel;
the sub-model expression is:
Figure FDA0002403459960000063
s.t.By≤d,(γ1) (46)
Dy＝e,(γ2) (47)
Fy≤f-Ex*,(γ3) (48)
Gy≤p,(γ4) (49)
Hy＝a,(γ5) (50)
Iy＝r,(γ6) (51)
in the formula: x is the number of*For the optimal solution of the main model, { gamma {1,γ2,γ3,γ4,γ5,γ6Is a constraint dual variable;
the submodel equivalent representation method is as follows:
worst scenario p in sub-model*、a*And r*To not determine the extreme values in sets P, A and R, equations (1) - (3) are therefore equivalent to:
Figure FDA0002403459960000064
Figure FDA0002403459960000065
Figure FDA0002403459960000066
in the formula: gt、mt、
Figure FDA0002403459960000067
And
Figure FDA0002403459960000068
are all binary variables;
based on strong dual theory, submodels (45) - (51) are equivalent to:
Figure FDA0002403459960000071
-BTγ1+DTγ2-FTγ3-GTγ4+HTγ5+ITγ6＝cT(56)
γ1≥0,γ3≥0,γ4≥0,γ2,γ5,γ6is a free variable (57)
Figure FDA0002403459960000072
Figure FDA0002403459960000073
Figure FDA0002403459960000074
Figure FDA0002403459960000075
In the formula: lambda, mu, omega1And ω2Are all auxiliary variables, and are all the auxiliary variables,
Figure FDA0002403459960000076
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