CN113902182B - Robust charging optimization method for electric bus fleet considering energy consumption uncertainty - Google Patents
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Abstract
A robust charging optimization method for an electric bus fleet based on uncertain operation energy consumption is characterized by firstly utilizing a robust optimization technology to construct a budget uncertain set to depict uncertainty of operation energy consumption, ensuring the robustness of an obtained charging optimization result and avoiding the limitation that the traditional uncertain optimization method presupposes probability distribution of uncertain parameter clothes. Secondly, a robust charging optimization model of the electric bus fleet is built, the time-of-use electricity price, charging station resource constraints, uncertain energy consumption requirements of vehicles, the influence of multiple constraints such as operation time on the charging scheme of the electric bus fleet is fully considered, and the actual operation condition is fitted. And finally, solving according to a frame design column generation algorithm of the robust model, decomposing the robust optimization model into a main problem and a sub problem for iterative solution, and accelerating the solving efficiency. The invention provides an economic and reliable charging scheme for the charging management of the electric bus fleet, obviously reduces the charging cost, effectively copes with energy consumption fluctuation, and improves the economy and stability of the electric bus charging system.
Description
Technical Field
The invention belongs to the technical field of urban electric public system management and uncertain problem optimization, and relates to a robust optimization method for orderly charging of an electric bus fleet facing to energy consumption fluctuation influence, in particular to a robust charging optimization method for the electric bus fleet considering energy consumption uncertainty.
Background
The electric public transport system is used as the backbone of green traffic, and provides a good solution for solving the problems of environmental pollution, energy shortage and the like. Compared with the traditional fuel vehicle, the electric vehicle has the difficulties of mileage limitation, battery technology and the like, so that the charging management of the electric vehicle becomes a key link for ensuring the safe and stable operation of a bus fleet. However, the operation power consumption of the electric bus is greatly uncertain due to the influence of weather, road environment, traffic conditions, passenger capacity and other factors, and the power consumption of the same shift may have a deviation of about 5% -20% or more, thereby making the actual charging operation very challenging.
Through literature and patent search, it can be found that in current research results, a charging optimization strategy of an electric bus generally performs charging management based on a determined power consumption parameter, and ignores energy consumption uncertainty in a driving process. Wang et al, in the literature [ Wang Y, Huang Y, Xu J, et al, optimal retrieval scheduling for urban electric buses [ J ]. Transportation Research Part electronics & Transportation Review,2017,100: 115-. The method is characterized in that a discrete binary particle swarm algorithm is adopted in a document (Wu Xiao Mei, Von Qijin, rigor, permissive garden, and Pan Jing Hui, an electric bus ordered charging strategy [ J ] based on double-layer optimization, a power grid and clean energy source, 2021,37(01): 119-. He et al propose a network modeling framework in a document [ He Y, Liu Z, Song Z. optimal charging scheduling and management for a fast-charging basic electric bus system [ J ]. Transportation Research Part E: logics and Transportation Review,2020, 142 ] based on given capacity of a power distribution network, the number of chargers and bus operation information, so as to optimize the charging scheduling and management of the electric bus rapid charging system and effectively reduce the total charging cost including both power demand cost and energy cost. The charging optimization based on the determined power consumption parameters can simplify the problem, but the reliability of the compiled fleet charging strategy is poor. This is because the power consumption of the electric bus directly determines the required charging duration. Ignoring the uncertainty of the electric bus energy consumption can therefore result in suboptimal or even infeasible solutions for the public transportation charging system. For example, when the power consumption of the electric bus increases during a shift, the electric bus is executed according to the original charging strategy, and various situations that the battery power is lower than the safety threshold of the battery in the operation process or the battery is insufficient to supplement the power before the electric bus is out of the station, so that the delayed departure of the electric bus and the like may occur, and the reliability and the safety of the public transportation system are damaged.
Disclosure of Invention
Based on the problem of neglecting the uncertainty of energy consumption in the charging optimization of the electric bus fleet, the invention provides the robust charging optimization method of the electric bus fleet in consideration of the uncertainty of the energy consumption. Firstly, a budget uncertain set is constructed by using a robust optimization technology to depict uncertainty of operation energy consumption, on one hand, the robustness of the obtained charging optimization result is guaranteed, namely, the obtained solutions are feasible as long as the parameters are disturbed in the given uncertain set, and on the other hand, the limitation that the traditional uncertain optimization method assumes the probability distribution of uncertain parameter clothes in advance is avoided. And secondly, establishing a robust optimization model for charging of the electric bus fleet under uncertain operation energy consumption, wherein time-of-use electricity price, charging station resource limitation and fluctuation of operation energy consumption are fully considered, so that the final optimization result meets the uncertain energy consumption for charging of the electric bus fleet and the charging cost is minimum. And finally, solving according to a frame design column generation algorithm of the robust model, decomposing the robust optimization model into a main problem and a sub problem for iterative solution, and accelerating the solving efficiency.
The technical scheme of the invention is as follows:
the robust charging optimization method of the electric bus fleet considering the uncertainty of energy consumption comprises the following steps:
step (1) of constructing an uncertainty set of operation energy consumption
And in consideration of the fluctuation of the energy consumption of the vehicle m in the operation process, a budget uncertain set is introduced to deal with the uncertainty of the operation energy consumption. Uncertain energy consumption per vehicle m required to execute shift k
In a single interval, i.e.
Wherein
Is a nominal value for the uncertain energy consumption,
the maximum deviation value representing the uncertain energy consumption is generally assumed to be p times the nominal value, i.e.
The budget uncertainty set definition is shown in equation (1). Wherein Z is shown as formula (2).
Constructed uncertainty set U of operating energy consumption buget Can be defined as:
wherein r mk The energy consumption uncertainty budget for executing shift k is a parameter reflecting the uncertainty level, and the optimality and the robustness of the solution can be weighed. It should be noted that 0 ≦ r mk K ≦ k, i.e. r mk The maximum value of (a) is the maximum number of shifts for which the energy consumption deviates from its nominal value at the same time. When the deviation of the energy consumption of the vehicle m from the value of k is maximum for all shifts, i.e. per r mk K, which means that the worst case occurs, the charging scheme is most conservative.
Step (2) establishing electric bus fleet charging robust optimization model under uncertain operation energy consumption
And based on the defined uncertain energy consumption set, converting the electric bus charging determined optimization model into a robust optimization model for orderly charging of the electric bus fleet under uncertain operation energy consumption. The method can ensure the feasibility of the charging optimization result without knowing the specific distribution of uncertain energy consumption, and obtains a charging scheme with higher robustness and optimality.
(2.1) parameter definition
Collection
M is the number set of electric buses served by the charging station, wherein M is {1,2, …, M }
T is a set of times divided in one day, T {1,2, …,1440/Δ T }
K m Class set of electric public transport vehicles m, K m ＝{1,2,…,A m In which A is m Is the total number of shifts of the vehicle m
parameter(s)
f t The price of electricity at time t (unit: yuan/kW. h)
Delta t is unit time, (unit: minute), the invention selects 5 minutes as one unit time
q mk The remaining battery power of the vehicle m returning to the charging station after the shift k is executed (unit: kW. h)
q max Battery capacity (unit: kW. h)
q min Minimum threshold allowed by battery capacity (unit: kW. h)
Δe t Power consumption of electric bus per minute (unit: kW. h)
P max Total power supplied by charging station (unit: kW)
Mu loss rate of power transmission process from charging station to charging gun
N total number of charging guns in charging station
p a Maximum allowable charging power for battery of electric bus (unit: kW)
p b Maximum charging power allowed by charging gun (unit: kW)
p c Upper limit of charging power (unit: kW)
d mt If the bus m supplements the electric quantity at the moment t, d mt 1 is ═ 1; otherwise d mt ＝0
(2.2) decision variables
p mt Required charging function at time t of m for vehicleRate, integer variable, (unit: kW)
U mt Auxiliary binary variable for charge continuity control, for determining whether a change in the state of charge occurs from time t to t +1, U mt ∈{0,1}
V mt Auxiliary binary variable for charge continuity control, for determining whether a change in the state of charge has occurred from time t to t-1, V mt ∈{0,1}
(2.3) establishing a determined charging optimization model taking the minimum total charging cost of the electric bus fleet as an objective function, wherein the model is as follows:
an objective function:
constraint conditions are as follows:
U mt ∈{0,1} (13)
V mt ∈{0,1} (14)
0≤p mt ≤p c (15)
in the formula, the objective function formula (3) is intended to minimize the total charging cost of the electric bus fleet, and is the sum of the electricity rates of the consumed electric quantities of all vehicles at all times. The constraint condition formula (4) indicates that the electric bus can not be supplemented with electric quantity during the operationTherefore, the charging power is 0. Ensuring that the charging power can not exceed the maximum power p of the charging gun through the constraint condition formula (5) b And the maximum power p acceptable for the battery a . The constraint condition formula (6) and the constraint condition formula (7) are constraint conditions of charging station resource limitation, and the constraint condition formula (6) requires that the total charging power sum of the electric buses which are simultaneously served is not more than the total power available for the charging stations; the constraint condition formula (7) requires that the number of the electric buses charged simultaneously is at most the total number of the charging guns. Constraint expressions (8.1) to (8.3) and constraint expressions (9.1) to (9.5) both indicate charging continuity control, in which once charging of the vehicle is started, the vehicle reaches a predetermined amount of electricity or stops for a predetermined time, and charging is not performed intermittently. Note that the form of the constraints (9.1) - (9.5) is relatively complex, since the charging period after the last shift has been performed includes two parts, namely
The charging continuity of the two-part join requires special handling. The constraint condition formula (10) indicates that the battery electric quantity value after each electricity supplementing should not exceed the battery capacity; the constraint equation (11) ensures that the amount of power replenished is not below the minimum threshold after completion of the normal operation of the next shift. The constraint (12) means that the amount of electricity supplied and the amount of electricity consumed are identical, i.e. the electric bus must be fully charged before the next day of departure. The constraint equations (13) to (15) give the types of variables.
(2.4) converting the determined charging optimization model into a charging robust optimization model under the condition of uncertain operation energy consumption
In order to convert the deterministic charging optimization model into a robust model that takes into account the uncertain energy consumption, it is necessary to replace the constraints (10) - (12) with the following constraints (16) - (18), when the model contains uncertain energy consumption parameters.
The three constraints of the constraint equations (16) - (18) are difficult to handle, and can be converted into min/max forms, such as equations (19) - (21).
(2.5) conversion of robust optimization model
To further linearize the constraints (19) - (21), the transformations were derived from Bertsimas' robust optimization theory and dual theory. The specific process is as follows:
equation (22) is first obtained by unifying equations (19) - (21). The equivalent formula of equation (22) is equation (23).
The right end of equation (23) is the maximization problem, as in equation (24), and its dual problem is as shown in equation (25).
Wherein v and w i Are respectively
And 0. ltoreq. z i A dual variable of ≦ 1. From the strong dual theorem of linear programming, it can be seen that the problem of equation (24) is bounded, and then equation (25) is bounded as its dual problem, and the two objective function values are consistent. It can therefore be deduced that equation (22) is equivalent to equation (26) and is also equivalent to equation (27).
Finally, a peer constraint of constraints (19) - (21) is obtained, namely the following three sets of equations (28.1) - (30.4), each set of equations containing 4 subformulae.
Thus, the final robust model is transformed to include: an objective function (3), constraint expressions (4) - (9), constraint expressions (28.1) - (30.4), and constraint expressions (13) - (15).
Step (3) solving a design column generation algorithm
And (3.1) decomposing the final charging robust optimization model in the step (2) into a main problem and a sub problem, and accelerating the solving process.
The main problems are as follows: resource limit constraints (6) - (7) of total charging power and charging power in the original robust model are covered, and the task is to satisfy the charging scheme set R of uncertain energy consumption per bus m m To select a generated charging schedule to minimize the overall charging cost of the fleet. At the same time, the selected strategies all need to meet the resource constraints of the charging station. The mathematical expression is as follows:
x mr ∈{0,1} (35)
in the main problem, c mr Refers to the cost of the charging schedule r for the electric bus m. x is the number of mr Is a binary variable, x if vehicle m selects charging schedule r mr 1, otherwise x mr ＝0。p mtr And d mtr And (2.1) p in the parameter definition mt And d mt The expressed meanings are consistent, and the lower subscript R is only used as a scheme set R for distinguishing the vehicles m m Different schemes r. Equation (31) aims to minimize the total charge cost of the final selected scheme of the electric bus, consistent with the original robust model objective function. Constraints (32) are constraints on buses that represent each bus only with set R m One charging scheme of (1) is matched; restraint (33)For each time T epsilon T, the accumulated power of all charging strategies at the time T is required to be less than or equal to p max μ, the total power available for the electric bus charging station. The constraint (34) ensures that the number of electric buses simultaneously served by all selected charging strategies at any one time T e T does not exceed the total number of charging guns. Equation (35) indicates the variable type.
The sub-problems are: each vehicle is treated as a separate subproblem, the purpose of which is to find a new charging scheme that improves the current optimal solution of the main problem on the basis of the dual variables provided by the main problem. Under strategy r, the sub-problem of vehicle M, M ∈ M can be expressed as:
constraint expressions (4) - (5), constraint expressions (8) - (9), constraint expressions (28.1) - (30.4), and constraint expressions (13) - (15).
The objective function of the sub-problem is equation (36), consisting of dual variables of charging cost and charging resource, where δ mr A dual variable being a constraint equation (32); for charging power at time t, w t A dual variable of constraint formula (33); for the charging gun at time t,. epsilon t Is a dual variable of the constraint equation (34). Other constraints are consistent with the original robust model, and the generated charging scheme still needs to guarantee operation time limitation, charging continuity, battery capacity limitation, uncertain energy consumption requirements and the like.
And (3.2) solving the main problem in the step (3.1) to obtain dual variables for providing the sub-problem.
(3.3) respectively solving the subproblems of different electric buses m, and if the objective function of the subproblems is less than 0, adding a charging scheme set R m Updating the scheme set of the main problem and returning to the step (3.2); and if the objective functions of the sub-problems are all larger than 0, stopping solving to obtain the lowest charging cost of the robust optimization model of the electric bus fleet under the consideration of the energy consumption uncertainty.
The invention has the beneficial effects that:
the invention provides an electric bus fleet robust charging optimization method based on uncertain operation energy consumption. Firstly, the fluctuation of the operation energy consumption is described by adopting a budget uncertain set, so that the defects of traditional uncertain optimization methods such as random planning and fuzzy planning are overcome, the optimization result is insensitive to parameter disturbance, and the method is very suitable for public transport systems with higher requirements on stability. Secondly, an electric bus fleet robust charging optimization model is established, the influence of multiple constraints such as time-of-use electricity price, charging station resource constraint, uncertain energy consumption requirements of vehicles, operation time and the like on a charging scheme of the electric bus fleet is fully considered, actual operation conditions are fitted, and the practicability is high. And finally, designing a column generation algorithm to solve the model. The column generation algorithm has the advantages of accurate solving result, high calculating speed and the like in the aspect of solving the large-scale mixed integer programming model. In conclusion, the invention provides an efficient and reliable charging scheme for the charging management of the electric bus fleet, and when the actual energy consumption is deviated from the nominal value, the robust charging scheme with uncertain energy consumption can still normally operate in a given battery range, so that the fluctuation of energy consumption is effectively dealt with, the safe operation of the bus system is ensured, the charging cost is reduced, and the economical efficiency of the management level is improved.
Drawings
FIG. 1 is a schematic flow diagram of the process of the present invention.
Fig. 2 is an analysis graph of the operation energy consumption per minute of the electric bus in the embodiment of the present invention.
Fig. 3 is a distribution diagram of energy consumption of the electric bus fleet in different electricity price periods in the embodiment of the invention.
Fig. 4 is a graph comparing battery power of the electric bus B1 determination model and robust model pair in the embodiment of the present invention.
Fig. 5 is a graph comparing battery power of the electric bus B2 determination model and robust model pair in the embodiment of the present invention.
Detailed Description
The following describes in detail embodiments of the present invention with reference to examples, and simulates the effects of the present invention.
Fig. 1 is a schematic flow chart of the method of the present invention, as shown in fig. 1, which mainly comprises three steps: (1) constructing an uncertainty set of operation energy consumption; (2) establishing a robust optimization model for charging of the electric bus fleet under uncertain operation energy consumption, wherein the robust optimization technology is mainly used for converting the determined charging optimization model of the electric bus fleet into the robust optimization model considering the charging under uncertain operation energy consumption, and further carrying out linearization process on the robust optimization model; (3) the design column generation algorithm decomposes an original robust model, and the model is quickly solved by utilizing the independence of main problems and sub problems, so that a reliable and economic management scheme is obtained for charging the electric bus fleet.
The specific flow of this embodiment is as follows:
1. method for constructing uncertainty set of energy consumption by adopting certain city operation energy consumption data
As described in step (1), the vehicles m each need to perform an indeterminate energy consumption for shift k
In a single interval, i.e.
Wherein
Is a nominal value of the uncertain energy consumption,
representing the maximum deviation value, generally assumed to be p times the nominal value, i.e.
Constructed uncertainty set U of energy consumption buget Can be defined as:
from the above equation, the key parameters for determining the set of uncertainty in energy consumption include
P and f mk 。
In this embodiment, the operation energy consumption historical data of the pure electric bus in a certain city is selected to perform numerical fitting, as shown in fig. 2. The power consumption per minute has fluctuation along with the change of time, the minimum value is 0.142 kW.h, and the maximum value can reach 0.310 kW.h. The power consumption is integrally higher per minute probably due to traffic jam or work rush hour and the like, and the average value is close to 0.28 kW.h; the energy consumption value of 17:00-19:30 is also higher and is close to 0.25 kW.h; the operation energy consumption is relatively stable in other periods, and is basically about 0.18 kW.h.
For convenience, the average energy consumption per minute values of the vehicles in different periods of time in table 1 are collated according to the analysis results of the operation energy consumption of the electric buses. And accumulating the power consumption per minute according to the time period of the operation shift of the vehicle to obtain the power consumption of the shift.
TABLE 1 average energy consumption per minute values for different periods
I.e. when vehicle m executes shift k, i.e. departure time c mk Arrival time s mk Nominal value of cumulative energy consumption
By
And calculating to obtain.
The residual parameter ρ directly determines the fluctuation range of the uncertainty interval. R mk Is an important parameter that affects the degree to which a solution is conserved, r mk The larger the resulting solution is, the more conservative. Note that the parameter r mk In the embodiment, the difference of the vehicle m is not distinguished, only the difference of the shift is distinguished, and f mk Is satisfied as an integer between 0-k. Since the number of vehicle operation shifts to which this example relates is at most 8, f mk The total number of values of (2) is 8. For vehicles with less than 8 operating shifts, the vehicles only need to adopt Gamma consistent with the number of shifts mk The number of (2). Such as 3 vehicles in a shift, a r actually used in a trial calculation mk The parameters comprising only f m1 ,Г m2 And r m3 . The present embodiment adopts p of 20%, r m1 ＝1，Г m2 ＝2，Г m3 ＝3，Г m4 ＝4，Г m5 ＝5，Г m6 ＝6，Г m7 ＝7，Г m8 These parameters verify the utility of the effectiveness of the electric bus charging robusta optimization model that accounts for energy consumption uncertainty.
2. Setting experimental parameters according to the electric bus fleet charging robust optimization model under uncertain operation energy consumption established in the step (2) and the step (3)
In the experimental example, data of 64 vehicles on 3 lines actually operated in a certain city are selected, including a plurality of parameters such as an operation schedule, charging station resources, battery capacity, time-of-use electricity price and the like, and are specifically shown in tables 2 and 3. The time-of-use electricity price means that the charging price is changed in different time periods. According to the three distribution states of the electricity price, the electricity price is divided into a low-valley electricity price, a flat-peak electricity price and a high-peak electricity price. The price is the lowest between 00:00 and 8:00 and is 0.3 yuan/kW.h.
TABLE 2 parameters of the model for determination of electric bus charging optimization
TABLE 3 TIME-SHARE ELECTRICITY VALUE METER
Wherein the operation schedule needs to be converted into time, that is, one time Δ t every 5 minutes, 1440 minutes a day is discretized into 288 times t. The first time in the morning, 6:00, is set to be t ═ 1. A sample trip table for an electric bus is shown in table 4.
TABLE 4 example journey table of electric bus
3. Optimization results of robust charging model
The method is adopted to carry out robust charging optimization of the electric bus fleet in consideration of energy consumption uncertainty, and the optimization result shows that the lowest charging cost of 64 fleets is 1539.4 yuan. The total energy consumption required by the electric bus fleet is 3840.2kW & h, wherein the energy consumption with low electricity price accounts for 75% and is about 2880kW & h. According to the fact that the available total power resource in the low-electricity-price time period from 0:00 to 8:00 is 3040 kW.h, the resource utilization rate of the low-electricity price can be calculated to be as high as 95%, and the result of the robust optimization model can fully utilize the resource of the low-electricity price, reduce charging cost and improve economy of charging management. Meanwhile, since the charging resource of low power rates is almost occupied, a part of the power consumption needs to be distributed in the flat rate period, about 24% (fig. 3).
In order to further verify the anti-interference performance and stability of the charging scheme obtained by the robust charging optimization method, the battery power distribution corresponding to the vehicle can be obtained by considering the condition of determining the energy consumption and the condition of deviation of the energy consumption (rho is 20%). Fig. 4 and fig. 5 respectively show the comparison of the charging scheme of the optimization model and the charging scheme of the robust optimization model on the electric bus battery power distribution when the energy consumption has a deviation of 20%.
Fig. 4 shows that in the worst case, the electric bus B1 can operate normally in a given battery charge range with a minimum charge of 50kW · h above the charge safety threshold, in a solution to the robust model. Under the solution of the deterministic model, the battery use range of the electric bus exceeds a given range, the lowest electric quantity is only 7kW & h, and the safe operation of the electric bus is seriously influenced.
Fig. 5 illustrates that, even if the vehicle is relatively easy to operate, the battery capacities of the robust model and the deterministic model do not fall below the safety threshold of 20%, but at the time of the departure, the vehicle B2 cannot be fully charged according to the charging schedule of the deterministic model, the battery capacity is charged only to 84kW · h, the departure may be delayed, and the charging schedule of the next day may be affected even if the departure is not delayed. Therefore, when the worst case occurs, the electric bus charging system has two problems in determining the solution of the energy consumption model. First, in extreme cases where energy consumption deviates significantly from the nominal value, the electric bus may have its battery level below a safe threshold before returning to the charging station, which may reduce battery life beyond its given safe range. In severe cases, the battery power may be completely exhausted, thereby causing a safety hazard in the operation process. Second, the electric bus will need to spend longer charging time at the charging station due to the increase of energy consumption, delaying normal operation.
In conclusion, the robust charging optimization method for the electric bus fleet, which considers the uncertainty of energy consumption, can remarkably reduce the charging cost, effectively cope with the fluctuation of energy consumption, and improve the economy and stability of the electric bus charging system.
Claims (1)
1. The robust charging optimization method of the electric bus fleet considering the uncertainty of energy consumption is characterized by comprising the following steps of:
step (1) of constructing an uncertainty set of operation energy consumption
Considering the fluctuation of the energy consumption of the vehicle m in the operation process, a budget uncertain set is introduced to deal with the uncertainty of the operation energy consumption; uncertain energy consumption of each vehicle m required to execute shift k
In a single interval, i.e.
Wherein
Is a nominal value of the uncertain energy consumption,
maximum deviation value representing uncertain energy consumption, p times nominal value, i.e.
The budget uncertainty set is defined as shown in formula (1); wherein Z is shown as formula (2);
constructed uncertainty set U of operating energy consumption buget Can be defined as:
where M is the total number of vehicles, r mk The energy consumption uncertainty budget for executing shift k is a parameter reflecting the uncertainty level, and the optimality and the robustness of the solution can be balanced; and 0 is less than or equal to r mk K is less than or equal to k, i.e. r mk The maximum value of (a) is the maximum number of shifts of which the energy consumption deviates from the nominal value simultaneously; when the deviation value of the energy consumption of the vehicle m in all shifts k is maximum, namely, each shift r mk K, the worst case occurs, with the most conservative charging scheme;
step (2), establishing a robust optimization model for orderly charging of electric bus fleets under uncertain operation energy consumption
Converting the electric bus charging determination optimization model into a robust optimization model for orderly charging of the electric bus fleet under uncertain operation energy consumption based on the uncertainty set of the operation energy consumption defined in the step (1);
(2.1) parameter definition
Collection of
M is the number set of electric buses served by the charging station, wherein M is {1,2, …, M }
T is a set of times divided a day, {1,2, …,1440/Δ T }
K m Shift set of vehicle m, K m ＝P1,2,…,A m In which A is m Is the total number of shifts of the vehicle m
parameter(s)
f t : electricity price corresponding to time t, unit: Yuan/kW.h
Δ t: unit time, unit: minute (min)
q mk : the remaining battery power, unit, of the vehicle m returning to the charging station after performing shift k: kW. h
G mk : electric quantity value supplemented after vehicle m performed shift k, unit: kW.h
q max : battery capacity, unit: kW.h
q min : minimum threshold allowed for battery charge, unit: kW.h
Δe t : the power consumption of the electric bus per minute, unit: kW.h
E mk ：
P max : total power supplied by the charging station, in units: kW (power of kilowatt)
μ: loss rate from charging station to charging gun power transfer process
N: total number of charging guns in charging station
p a : maximum charging power allowed by the battery of the electric bus, unit: kW (kilo power)
p b : maximum charging power allowed by the charging gun, unit: kW (power of kilowatt)
p c : upper limit value of charging power, unit: kW (power of kilowatt)
d mt : if the vehicle m is replenished with electricity at time t, d mt 1; otherwise d mt ＝0
(2.2) decision variables
p mt : required charging power at time t of vehicle m, integer variable, unit: kW (power of kilowatt)
U mt : an auxiliary binary variable for charge continuity control, for determining whether a change, U, in the state of charge has occurred from time t to t +1 mt ∈{0，1}
V mt : auxiliary binary variable for charge continuity control, for determining whether a change in state of charge, V, has occurred from time t to t-1 mt ∈{0，1}
(2.3) establishing a determined charging optimization model taking the minimum total charging cost of the electric bus fleet as an objective function, wherein the model is as follows:
an objective function:
constraint conditions are as follows:
U mt ∈{0，1}(13)
V mt ∈{0，1} (14)
0≤p mt ≤p c (15)
in the formula, the objective function formula (3) aims to minimize the total charging cost of the electric bus fleet, and is the sum of the electricity prices of all vehicles at all times; the constraint condition formula (4) indicates that the electric bus cannot be supplemented with electric quantity during operation, so that the charging power is 0; ensuring that the charging power can not exceed the maximum power p of the charging gun through the constraint condition formula (5) b And the maximum power p acceptable for the battery a (ii) a The constraint condition formula (6) and the constraint condition formula (7) are constraint conditions of charging station resource limitation, and the constraint condition formula (6) requires that the total charging power sum of the electric buses which are served simultaneously is not more than the total power available for the charging station; the constraint condition formula (7) requires that the number of the electric buses charged simultaneously is at most the total number of the charging guns; the constraint expressions (8.1) - (8.3) and the constraint expressions (9.1) - (9.5) both represent charging continuity control, that is, once the vehicle starts charging, it reaches a predetermined amount of electricity or stops for a predetermined time, and charging is not performed intermittently; the constraint condition formula (10) indicates that the battery electric quantity value after each electricity supplementing should not exceed the battery capacity; the constraint condition formula (11) ensures that the battery capacity of the supplemented electric quantity is not lower than the lowest threshold value after the normal operation of the next shift is finished; the constraint equation (12) means that the amount of electricity supplied and the amount of electricity consumed are consistent, that is, the electric bus must be fully charged before the next day of departure; constraint equations (13) - (15) give the types of variables;
(2.4) converting the determined charging optimization model into a robust optimization model for orderly charging of electric bus fleets under uncertain operation energy consumption
In order to convert the determined charging optimization model into a robust model considering uncertain energy consumption, constraint conditional expressions (10) - (12) are replaced by constraint conditional expressions (16) - (18), and the model contains uncertain energy consumption parameters;
constraints (16) - (18) may transform the min/max form, as shown in equations (19) - (21):
(2.5) conversion of robust optimization model
To further linearize the constraint equations (19) - (21), the transformations are derived from Bertsimas' robust optimization theory and dual theory; the specific process is as follows:
firstly, unifying the formulas (19) - (21) to obtain a formula (22); the equivalent formula of the formula (22) is formula (23);
the right hand end of equation (23) is the maximization problem, as in equation (24), and its dual problem is as shown in equation (25):
wherein v and w i Are respectively
And 0. ltoreq. z i A dual variable of less than or equal to 1; according to the strong dual theorem of linear programming, the problem of the formula (24) is feasible and bounded, and then the problem of the formula (25) as the dual problem is also feasible and bounded, and the objective function values of the two are consistent; it can therefore be deduced that equation (22) is equivalent to equation (26), and in turn equivalent to equation (27):
finally, equivalent constraints of constraint conditions (19) - (21), namely equations (28.1) - (30.4), are obtained, each set of equations comprising 4 sub-equations:
thus, the final robust model is transformed to include: an objective function (3), constraint expressions (4) - (9), constraint expressions (28.1) - (30.4), and constraint expressions (13) - (15);
step (3) solving a design column generation algorithm
(3.1) decomposing a robust optimization model for orderly charging of the electric bus fleet under uncertain operation energy consumption in the step (2) into a main problem and a sub-problem, and accelerating the solving process;
the main problems are as follows: resource limit constraints (6) - (7) of the total charging power and the charging gun in the original robust model are covered, and the task of the resource limit constraints is to satisfy the charging scheme set R of the uncertain energy consumption per vehicle m m Selecting a generated charging scheme to minimize the overall charging cost of the fleet; meanwhile, the selected strategies all need to meet the resource constraint of the charging station; the mathematical expression is as follows:
x mr ∈{0，1}(35)
in the main problem, c mr Refers to the cost of the charging schedule r for the vehicle m; x is the number of mr Is a binary variable, x if vehicle m selects charging schedule r mr 1, otherwise x mr ＝0；p mtr The required charging power of the vehicle at the moment t of m in the charging scheme r is expressed as unit: kW; d is a radical of mtr Means that in the charging scheme r, the vehicle m supplements the electric quantity at the moment t, then d mtr 1, otherwise mtr 0; the lower corner mark R is only used as a scheme set R for distinguishing the vehicles m m Different charging schemes r; equation (31) aims to minimize the charging of the final selected scheme of the electric busThe total cost is consistent with the target function of the original robust model; the constraint expression (32) is a constraint condition for buses, and represents that each bus is only connected with the set R m One charging scheme of (1) is matched; in the constraint condition formula (33), for each time T epsilon T, the accumulated power of all charging strategies at the time T is required to be less than or equal to p max Mu, i.e. the total power available for electric bus-charging stations; the constraint condition formula (34) ensures that the number of the electric buses simultaneously served by all the selected charging strategies at any time te ∈ T does not exceed the total number of the charging guns; formula (35) indicates the variable type;
the sub-problems are: each vehicle is treated as a separate subproblem, with the aim of finding a new charging scheme that improves the current optimal solution of the main problem on the basis of the dual variables provided by the main problem; under the charging scheme r, the sub-problem of the vehicle M, M ∈ M can be expressed as:
constraint expressions (4) - (5), constraint expressions (8) - (9), constraint expressions (28.1) - (30.4), constraint expressions (13) - (15);
the objective function of the sub-problem is equation (36), consisting of dual variables of charging cost and charging resource, where δ mr A dual variable being a constraint equation (32); for charging power at time t, w t A dual variable of constraint formula (33); for the charging gun at time t,. epsilon t A dual variable being constraint (34); other constraints are consistent with the original robust model, and the generated charging scheme still needs to guarantee operation time limitation, charging continuity, battery capacity limitation, uncertain energy consumption requirements and the like;
(3.2) solving the main problem in the step (3.1) to obtain dual variables for providing to the sub-problems;
(3.3) respectively solving the sub-problems of different vehicles m, and if the objective function of the sub-problem is less than 0, adding a charging scheme set R m Updating the scheme set of the main problem and returning to the step (3.2); if the objective functions of the subproblems are all greater than 0, stoppingAnd solving to obtain the lowest charging cost of the robust optimization model for orderly charging of the electric bus fleet under the condition of considering the uncertain operation energy consumption of the energy consumption.
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