CN112685917B - Battery equalization modeling system and method based on nonlinear efficiency model - Google Patents
Battery equalization modeling system and method based on nonlinear efficiency model Download PDFInfo
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Abstract
The invention provides a battery equalization modeling method based on a nonlinear efficiency model, which comprises the following steps: s1, establishing a relational expression of the residual electric quantity of the battery and the balance current; s2, obtaining efficiency values under different equalizing currents, and establishing a nonlinear relation between the efficiency and the equalizing currents, wherein the energy consumed by the jth battery in an equalizing period in the equalizing period is obtained, and an equalizing loss target function of the bus type equalizing system in the equalizing process is obtained; and S3, combining the residual electricity quantity change equations of the n batteries together according to the structure of the bus type equalizing system to obtain a state space model of the bus type equalizing system, then controlling the state space model of the bus type equalizing system by utilizing model prediction control, and establishing an equalizing model of nonlinear efficiency based on a model prediction control strategy. The invention greatly reduces the solving difficulty by utilizing the fusion of the linear efficiency model and the nonlinear efficiency model, so that the model applied to the dynamic battery equalization is more practical.
Description
Technical Field
The invention relates to the field of automatic control, in particular to a battery equalization modeling system and method based on a nonlinear efficiency model.
Background
The model predictive control essentially belongs to the interdisciplinary discipline of optimization and control, and because the Model Predictive Control (MPC) is convenient to build a model, has low requirements on the model, adopts a rolling optimization strategy and has the characteristics of better dynamic control effect and the like, the application of the model predictive control on the dynamic battery balance can improve the balance performance;
however, most of the research in the battery equalization model is based on a linear model, and little research is on the nonlinear modeling of the battery equalization circuit; the efficiency of a practical equalizer is a non-linear function of the input current and the input-output voltage of the equalizer; under the equilibrium strategy of model predictive control, a large amount of nonlinear constraints exist especially when the number of batteries and the number of prediction steps are large, and the solution of an objective function is very troublesome; therefore, a trust domain reflection algorithm can be adopted, and the solving result of the objective function under the linear efficiency model is given to a trust domain reflection algorithm solver as an initial value, so that the difficulty of the optimization process is greatly reduced.
Disclosure of Invention
The invention aims to at least solve the technical problems in the prior art, and particularly provides a battery equalization modeling system and method based on a nonlinear efficiency model.
In order to achieve the above object of the present invention, the present invention provides a battery equalization modeling system based on a nonlinear efficiency model, comprising: the system comprises a charge-discharge module, n equalizers, n batteries and an energy bus;
the first end of the 1 st battery is connected with the first end of the charge-discharge module and the first end of the 1 st equalizer, the second end of the 1 st battery is connected with the second end of the 1 st equalizer, the first end of the 2 nd battery and the first end of the 2 nd equalizer, the third end of the 1 st equalizer is connected with the negative electrode of the energy bus, and the fourth end of the 1 st equalizer is connected with the positive electrode of the energy bus;
the first end of the 2 nd battery is connected with the second end of the 1 st battery, the second end of the 1 st equalizer and the first end of the 2 nd equalizer, the second end of the 2 nd battery is connected with the first end of the 3 rd battery, the first end of the 3 rd equalizer and the second end of the 2 nd equalizer, the third end of the 2 nd equalizer is connected with the negative pole of an energy bus, and the fourth end of the 2 nd equalizer is connected with the positive pole of the energy bus;
the first end of the 3 rd battery is connected with the second end of the 2 nd battery, the second end of the 2 nd equalizer and the first end of the 3 rd equalizer, the second end of the 3 rd battery is connected with the first end of the 4 th battery, the first end of the 4 th equalizer and the second end of the 3 rd equalizer, the third end of the 3 rd equalizer is connected with the negative electrode of the energy bus, and the fourth end of the 3 rd equalizer is connected with the positive electrode of the energy bus;
……；
the first end of the nth battery is connected with the second end of the (n-1) th battery, the second end of the (n-1) th equalizer and the first end of the nth equalizer, the second end of the nth battery is connected with the second end of the nth equalizer and the second end of the charge-discharge module, the third end of the nth equalizer is connected with the negative electrode of the energy bus, and the fourth end of the nth equalizer is connected with the positive electrode of the energy bus;
n is the total number of cells and is also the total number of equalizers corresponding to the cells.
Further, each equalizer includes:
the first end of battery links to each other with the first end of inductance L1, and the second end of inductance L1 links to each other with the first end of electric capacity C1, the first end of switch tube Q1, and the second end of battery links to each other with the first end of inductance L3, and the second end of inductance L3 links to each other with the first end of electric capacity C2, the second end of switch tube Q1, and the second end of electric capacity C1 links to each other with the first end of switch tube Q2, the first end of inductance L2The second end of the capacitor C2 is connected with the second end of the switch tube Q2 and the first end of the inductor L4, and the second end of the inductor L2 is connected with the capacitor C bus Is connected to the energy bus, and the second terminal of the inductor L4 is connected to the capacitor C bus And the second end of the energy bus is connected with the energy bus.
The invention also provides a battery equalization modeling method based on the nonlinear efficiency model, which comprises the following steps:
s1, obtaining the relation between the SOC of the battery and the balance current in a bus type balance system according to an ampere-hour integral method, and accordingly establishing a relation between the residual electric quantity of the battery and the balance current;
s2, according to the structure of the bus type equalizing system, efficiency values under different equalizing currents can be obtained through equipment, a nonlinear relation between the efficiency and the equalizing currents is established, energy consumed by the jth battery in an equalizing period in the equalizing period is established according to the nonlinear relation of the efficiency, and an equalizing loss target function of the bus type equalizing system in the equalizing process is obtained;
and S3, combining the residual electricity quantity change equations of the n batteries together according to the structure of the bus type equalization system to obtain a state space model of the bus type equalization system, then controlling the state space model of the bus type equalization system by utilizing model prediction control, and establishing a nonlinear efficiency equalization model based on a model prediction control strategy.
Further, the solving of the non-linear efficiency model, which is the equilibrium model of the non-linear efficiency based on the model predictive control strategy in S3, belongs to the non-linear optimization, and includes:
firstly, setting the transmission efficiency of an equalizer as a constant, establishing a linear model, namely a linear efficiency model, solving an optimal solution through multi-target linear programming, and assigning the optimal solution to a trust domain reflection algorithm as an initial value so as to solve the optimal solution of a nonlinear efficiency model; the problem that the calculated amount is too large to solve is solved;
most of the constraints are nonlinear constraints, so that the solution is very difficult; solving a nonlinear objective function problem with nonlinear constraints is generally called a nonlinear multi-objective programming problem; in multi-objective planning with a large number of nonlinear constraints, a search-class algorithm or a confidence domain algorithm is commonly used, and an optimal value is usually found by gradient descent or confidence radius update through a given initial value; however, for a large optimization function, the search algorithm and the trust domain algorithm have the disadvantage that if the initial value is not selected properly, the global optimum value is difficult to obtain, and the problem is well solved by the proposed method.
Further, the S1 includes:
s1-1, establishing a relation between the SOC (state of charge) of the jth battery and the balance current according to the relation between the SOC of the jth battery and the balance current of the jth battery in a bus type balance system, so as to obtain a relation between the residual electric quantity of the battery and the balance current;
s1-2, adding the self-loss of the battery in the equation of the relation between the residual capacity of the battery and the balance current, and setting the residual capacity of the j-th battery as x j ＝Q j *SOC j And obtaining a complete equation of the relation between the residual battery capacity and the balance current.
Further, the S1 includes:
in the bus-type equalization circuit, n represents the total number of cells,
represents the current between the jth cell and the equalizer, <' > or>
Representing the current between the energy bus and the jth equalizer, j ∈ {1,2, ..., n } is the battery number, and is also the equalizer number corresponding to the battery, v ∈ {1,2, \ 8230 } j Is the voltage of the jth cell; />
According to the relation between the residual electric quantity of the lithium ion battery and the balance current, writing out an SOC change equation of the j-th battery as follows:
wherein Q is j The capacity of the j-th battery;
SOC j (t) representing the SOC change equation of the jth battery at the moment of time t;
SOC j (t 0 ) Is represented at time t 0 At the moment, the SOC change equation of the j-th battery;
by differentiating equation (1), the following equation can be obtained:
wherein Q j The capacity of the j-th battery;
for completeness of the equation, adding the self-loss of the cell in equation (2) can result in the following equation:
wherein Q is j The capacity of the jth battery;
τ is used to describe the self-discharge rate of the cell;
Q j the capacity of the jth battery;
SOC j represents the percentage of the remaining capacity of the jth battery;
let the residual capacity x of the j-th battery j ＝Q j *SOC j And obtaining the relation between the residual battery capacity and the balance current as follows:
wherein the content of the first and second substances,
is the remaining capacity x of the battery j Differentiation of (1);
τ is used to describe the self-discharge rate of the cell;
x j the residual capacity of the jth battery is;
a relationship between the remaining capacity of the battery and the equalizing current has been established so far.
Further, the S2 includes:
for the jth equalizer, from the nonlinear efficiency, the following can be obtained:
wherein the content of the first and second substances,
is the equalization efficiency between the equalizer and the battery;
v j Is the voltage of the jth cell;
v bus is the bus voltage;
wherein the content of the first and second substances,
representing the current between the jth battery and the equalizer;
v j is the voltage of the jth cell;
v bus is the bus voltage;
when energy is transferred from the battery to the energy bus, the input current of the equalizer is
When both the input voltage and the output voltage are constant, a switch is activated>
Only and->
Related to; can be set>
Wherein it is present>
Represents an equalized current pick-up>
And a nonlinear equation of relationship for efficiency; when energy is transferred from the bus to the battery, the equalizer input current is &>
When both the input voltage and the output voltage are constant, a switch is activated>
Only and>
(ii) related; can be set>
Wherein +>
Expressed as equalizing current->
And a nonlinear equation of relationship for efficiency;
to facilitate the expression, two variables u are redefined j,1 、u j,2 Control signals used to describe the solution to be solved; wherein u is j,1 、u j,2 The specific expression of (A) is shown as follows:
wherein u is j,1 Represents the current flowing from the energy bus to the jth equalizer;
wherein u is j,2 Represents the current flowing from the jth battery to the jth equalizer;
the default direction of the current is that the current flows from the bus to the equalizer and the current flows to the battery by the equalizer is positive, otherwise, the current is added with negative;
the battery balancing strategy is to achieve the purpose of SOC balancing by controlling the current flowing direction and the current flowing size between the battery and the energy bus;
when the battery is charged, the current flows into the battery through the equalizer via the bus, and the following formula can be obtained from formula (6):
wherein, the first and the second end of the pipe are connected with each other,
representing the current between the jth battery and the equalizer;
v bus Is the bus voltage;
v j is the voltage of the jth cell;
h(u j,1 ) Indicating the efficiency between the energy bus and the jth equalizer;
u j,1 represents the current flowing from the energy bus to the jth equalizer;
when the battery is discharged, the current is directly discharged from the battery, and the following formula can be obtained:
wherein, the first and the second end of the pipe are connected with each other,
represents the current between the jth battery and the equalizer;
u j,2 represents the current flowing from the jth battery to the jth equalizer;
substituting equations (9) (10) into equation (4) yields the following equation:
wherein the content of the first and second substances,
indicates the remaining capacity x of the battery j Differentiation of (1);
τ is used to describe the self-discharge rate of the cell;
x j representing the residual capacity of the j-th battery;
h(u j,1 ) Represents the efficiency between the energy bus and the jth equalizer;
v bus is the bus voltage;
u j,1 represents the current flowing from the energy bus to the jth equalizer;
v j is the voltage of the jth cell;
u j,2 represents the current flowing from the jth battery to the jth equalizer;
s2-2, according to the battery equalization model with the nonlinear efficiency, the energy consumed by the jth battery in the process of an equalization period can be represented as the following formula:
wherein E represents the energy lost by the jth battery in the process of one balancing cycle;
t eq represents an equalization period;
h(u j,1 ) Represents the efficiency between the energy bus and the jth equalizer;
u j,1 represents the current flowing from the energy bus to the jth equalizer;
v bus is the bus voltage;
f(u j,2 ) Is the efficiency between the jth battery and the jth equalizer;
u j,2 represents the current flowing from the jth battery to the jth equalizer;
v j is the voltage of the jth cell;
therefore, an equalization loss objective function of the bus type equalization system in the equalization process is obtained.
Further, the S3 includes:
s3-1, combining the residual electricity quantity change equations of n batteries together to obtain a state space model of the bus type equalization system according to the structure of the bus type equalization system, discretizing the state space model to obtain a discrete state space model, and obtaining an equalization loss target function and constraint of the target function in the discretization model;
and S3-2, controlling the state space model of the bus type equalization system by using a model prediction control strategy, so that a target function and a constraint condition of the target function which minimize equalization loss in model prediction control can be obtained, and according to the rule that the battery charge state changes along with time in the equalization process, the equalization speed is related to the area enclosed by a charge state change curve and a longitudinal axis in the equalization process, so that the time efficiency is combined into the target function with the minimum equalization loss.
Further, the S3-1 comprises:
combining the remaining capacity change equations of the n batteries together to obtain a state space model of the bus type equalization system as shown in the following formula:
wherein the content of the first and second substances,
representing a state space model formed by combining equations of n batteries together;
A 0 ＝-τI n×n is a matrix of n × n; τ is the self-discharge rate of the cell, I n×n Representing an n × n identity matrix, n being the total number of cells;
x＝[x 1 ,x 2 ,…x n ] Τ is a state vector of n x 1 dimension, representing the residual capacity of all batteries in the equalizing system, x 1 Indicates the remaining capacity, x, of the 1 st cell 2 Indicates the remaining capacity of the 2 nd battery, x n Representing the residual capacity of the nth battery; a means of Τ Representing a transpose;
u＝[u 1,1 ,u 1,2 ,u 2,1 ,u 2,2 ,…u n,1 ,u n,2 ,] Τ is a control input of 2n × 1 dimension, representing the control action of the equalizer on the battery, u 1,1 Representing the current, u, of the energy bus flowing to the 1 st equalizer 1,2 Represents the current of the 1 st cell to the 1 st equalizer, u 2,1 Representing the current, u, of the energy bus flowing to the 2 nd equalizer 2,2 Representing the current of the 2 nd battery to the 2 nd equalizer, u n,1 Representing the current of the energy bus flowing to the nth equalizer, u n,2 Representing the current of the nth battery flowing to the nth equalizer;
B 0 the matrix for n × 2n is as follows:
wherein, h (u) 1,1 ) Represents the efficiency between the energy bus and the 1 st equalizer;
v bus is the bus voltage;
v 1 voltage of the 1 st cell;
h(u 2,1 ) Represents the efficiency between the energy bus and the 2 nd equalizer;
v 2 the voltage of the 2 nd battery;
h(u n,1 ) Representing the efficiency between the energy bus and the nth equalizer;
v n is the voltage of the nth cell;
discretizing the formula (13) with kt as the sampling time 0 -t 0 Substituting the sampling instant into equation (13) may result in the following equation:
wherein the content of the first and second substances,
representing a time of kt 0 -t 0 The state space model of the bus type equalization system;
x(kt 0 -t 0 ) Representing the battery at time kt 0 -t 0 The remaining capacity of electricity;
u(kt 0 -t 0 ) Representing the battery at time kt 0 -t 0 The magnitude of the balancing current;
t 0 the sampling time of the discretization of the continuous state space model, and the control signal is unchanged in a sampling period;
when t is 0 Sufficiently small, then equation (15) may become the following equation:
wherein x (k) represents the remaining capacity of the battery at the kth sampling moment;
x (k-1) represents the remaining capacity of the battery at the k-1 th sampling point;
t 0 the sampling time of the discretization of the continuous state space model, and the control signal is unchanged in a sampling period;
x (k-1) represents the remaining capacity of the battery at the k-1 th sampling point;
u (k-1) represents a control signal at the k-1 th sampling point;
the equation (16) can be simplified as shown in the following equation:
x(k)＝(I n×n +A 0 t 0 )x(k-1)+B 0 t 0 u(k-1) (17)
wherein x (k) represents the remaining capacity of the battery at the kth sampling moment;
I n×n an identity matrix of n × n;
t 0 the sampling time of the discretization of the continuous state space model, and the control signal is unchanged in a sampling period;
x (k-1) represents the remaining capacity of the battery at the k-1 th sampling point;
u (k-1) represents a control signal at the k-1 th sampling point;
so far, a discretization model of the formula (13) such as the formula (17) can be obtained; for convenience of expression, let A = I here n×n +A 0 t 0 、B＝B 0 t 0 Wherein A is 0 ＝-τI n×n Is a matrix of n × n, τ is the self-discharge rate of the cell, I n×n Representing an n × n identity matrix, B 0 Is a matrix of n × 2 n; for the discretized state space model, the time when equalization is completed is K, then the equalization loss objective function in the discretized state space model can be obtained according to equation (12) as shown in the following equation:
wherein E represents the energy lost by the jth battery in the process of one balancing cycle;
k represents the time when equalization is completed;
t 0 a sampling time representing a discretization of the continuous state space model;
F 0 (k) The F matrix at the 0 th step of the kth sampling moment is obtained;
u (k) represents the control action of the equalizer on the battery at the k sampling moment;
v bus is the bus voltage;
h[u 1,1 (k)]representing the efficiency between the energy bus and the 1 st equalizer at the kth sampling instant;
h[u n,1 (k)]representing the efficiency between the energy bus and the nth equalizer at the kth sampling instant;
v 1 voltage of the 1 st cell;
f[u 1,2 (k)]the efficiency between the 1 st battery and the 1 st equalizer at the kth sampling moment is shown;
v n is the voltage of the nth cell;
f[u 1,2 (k)]represents the 1 st battery and the 1 st at the kth sampling momentEfficiency between equalizers;
f[u n,2 (k)]the efficiency between the nth battery and the nth equalizer at the kth sampling moment is shown;
for optimization problems, it can be generally considered that the constraints on the state variables are soft constraints, the constraints on the control inputs belong to hard constraints, and the following constraints exist for equation (18): formula (19) is a soft constraint that the remaining capacity of each battery is equal at the time K of completing equalization, formula (20) is a hard constraint that the total current flowing into the bus and the total current flowing out of the bus are equal to ensure the bus voltage is stable during equalization, and formula (21) is a hard constraint on the control signal;
x 1 (K)＝x 2 (K)＝…＝x n (K) (19)
wherein x is 1 (K) Indicating the residual capacity of the 1 st battery at the moment K of finishing the balance;
x 2 (K) Indicating the residual capacity of the 2 nd battery at the moment K of finishing the balance;
x n (K) Indicating the residual capacity of the nth battery at the moment K of finishing the balance;
wherein u is j,1 (k) Is the current from the bus to the equalizer at the kth sampling instant;
u j,2 (k) Efficiency between the equalizer and the battery at the kth sampling instant;
u j,2 (k) Is the current from the battery to the equalizer at the kth sampling instant;
v j is the voltage of the jth cell;
v bus is the bus voltage;
u j,1 (k)×u j,2 (k)＝0 k＝0,1,...,K-1,j＝1…,n (21)
wherein u is j,1 (k) Is the current from the bus to the equalizer at the kth sampling instant;
u j,2 (k) Is the current from the battery to the equalizer at the kth sampling instant;
k is the time for completing the equalization;
n means n batteries;
and establishing the discrete state space model and corresponding constraint conditions of the discrete state space model.
Further, the S3-2 comprises:
controlling a state space model of the bus type balance system by using a model prediction control strategy; and (3) controlling the balance system by adopting model prediction control, and obtaining a prediction state in a prediction step length according to a formula (17) as follows:
x(k+1|k)＝Ax(k)+Bu(k) (22)
the state for the next time is as follows:
x(k+2|k)＝Ax(k+1)+Bu(k+1) (23)
substituting equation (22) into equation (23) yields the following equation:
x(k+2|k)＝A 2 x(k)+ABu(k)+Bu(k+1) (24)
wherein u (k) means the control signal at the kth sampling instant, u (k + 1) means the control signal at the k +1 th sampling instant, a = I n×n +A 0 t 0 ，B＝B 0 t 0 Wherein A is 0 ＝-τI n×n Is a matrix of n × n, τ is the self-discharge rate of the cell, I n×n Representing an n × n identity matrix, B 0 Is a matrix of n × 2n, t 0 Is the sampling time of the discretization of the continuous state space model; the step size of the model predictive control is actually the time interval t between adjacent moments 0 (ii) a x (k) and x (k + 1) respectively represent the residual capacity of the battery at the kth sampling moment and the residual capacity of the battery at the kth +1 sampling moment; x (k +1 k) represents the residual capacity of the battery predicted in the step 1 at the kth sampling moment, and x (k +2 k) represents the residual capacity of the battery predicted in the step 2 at the kth sampling moment; similar iterations can result in the following prediction matrices:
wherein x (k + 1) is the predicted state of step 1 at the kth sampling time; x (k +2 k) is the prediction state in the step 2 at the kth sampling moment; x (k + N | k) is the prediction state of the Nth step at the k sampling moment, namely the prediction step length is N; (. Cndot.) 2 Represents the square, (.) N Representing the power of N, x (k + i | k) being at time kt 0 Predicting the state of the ith step at the moment; u (k + i | k) is at time kt 0 Inputting the prediction of the ith step at the moment;
b (k | k), u (k | k), B (k + N-1 luminance k) are equations derived from the above equations:
from equation (13), u = [ u ] 1,1 ,u 1,2 ,u 2,1 ,u 2,2 ,…u n,1 ,u n,2 ,] Τ U (k | k) is the control input at the kth sampling moment, u (k +1 luminance k) is the k sampling moment, and the prediction input in the step 1 is carried out; u (k + N-1) is predicted input of the step N-1 at the k sampling moment;
b = B as shown in formula (17) 0 t 0 (ii) a From equation (14), the matrices B and h (u) 1,1 )、h(u 2,1 )…h(u n,1 ) (ii) related; so matrix B also sums u = [ u ] 1,1 ,u 1,2 ,u 2,1 ,u 2,2 ,…u n,1 ,u n,2 ,] Τ Therefore, B (k | k) is a matrix obtained by substituting u (k | k) into the B matrix at the k-th sampling time; b (k + i | k) is a matrix obtained by substituting u (k + i | k) into the B matrix when predicting i steps at the kth sampling moment; b (k + N-1) cunning is a matrix obtained by substituting u (k + N-1) cunning k into the B matrix at the k sampling moment when predicting the N-1 step;
wherein k represents the kth sampling instant; n is the prediction step size, expressed for simplicity as:
wherein the content of the first and second substances,
for convenience of the following description, the following redefinitions are made; wherein the input variable matrix U (k) is a matrix of dimension (2 × N, 1), the state variable X (k + 1) is a matrix of dimension (N × N, 1), and F (k) is a row vector of dimension (1, 2 × N);
U(k)＝[u(k|k),u(k+1|k),…,u(k+N-1|k)] Τ (27)
X(k+1)＝[x(k+1|k),x(k+2|k),…,x(k+N|k)] Τ (28)
F(k)＝[F 0 (k|k),F 0 (k+1|k),…,F 0 (k+N-1|k)] (29)
wherein, F 0 (k | k) denotes the matrix F, F at the prediction of step 0 at the k-th sampling instant 0 (k +1 purple k) represents the F matrix in the 1 st prediction at the kth sampling moment, F 0 (k + N-1 caldus) represents the F matrix during the N-1 st prediction at the kth sampling time · Τ Representing a transpose; in order to minimize the loss during equalization, i.e., minimize equation (18); in the model predictive control strategy, it is equivalent to minimizing the loss within the whole prediction step; the objective function for minimizing the equalization loss in model predictive control can be obtained as follows:
min t 0 F(k)U(k) (30)
t 0 is the time interval between adjacent sampling instants of the discretization of the continuous-state space model, F (k) is a row vector of dimension (1, 2 × N), and the variable matrix U (k) is a matrix of dimension (2 × N, 1);
meanwhile, the constraint of the discrete state space model under the model prediction control strategy can be obtained; as shown in equations (31) (32) (33) (34) (35); wherein equation (31) is the range limit for the input variables; equation (32) is the range limit for the state variables; formula (33) is a constraint that the current flowing into the bus is equal to the current flowing out of the bus in an equalization period; equation (34) is the limit for the battery charge to be equal after the equalization process is over; formula (35) is a constraint that the control signal has only one direction at a certain moment in the predicted step length;
0≤U(k)≤i m (31)
x l ≤X(k+1)≤x u (32)
x 1 (k+N|k)＝x 2 (k+N|k)＝…＝x n (k+N|k) (34)
u j,1 (k+h|k)u j,2 (h|k)＝0 h＝0,1,…N-1,j＝1,…,n (35)
wherein i m Is the maximum value of the input variable; x is the number of l 、x u Respectively the minimum residual capacity and the maximum residual capacity which can be allowed by the battery;
f[u j,2 (k+h|k)]the prediction efficiency of the h step between the jth battery and the jth equalizer at the kth sampling moment is shown; u. of j,1 (k + h | k) represents the prediction of the h step of the bus to equalizer current at the kth sample time; u. of j,2 (k + h | k) represents the current prediction of the h step between the jth battery and the jth equalizer at the kth sampling instant;
k represents the sampling time, j represents the j-th battery, N represents the total number of batteries, N represents the predicted step length, h represents the predicted step number at the k-th sampling time, v bus Is the bus voltage, v j Is the voltage of the jth cell;
x 1 (k + N | k) represents the remaining capacity of the 1 st battery at the k sampling moment when predicting N steps, x 2 (k + N | k) represents the remaining capacity of the 2 nd battery at the k sampling moment when predicting N steps, x n (k + N | k) represents the remaining capacity of the nth battery at the kth sampling moment when the nth battery predicts the N steps;
according to the rule that the state of charge of the battery changes along with time in the balancing process, the balancing speed is related to the area enclosed by a state of charge change curve and a longitudinal axis in the balancing process, and when the enclosed area is smaller, the balancing time is shorter, and the speed is faster, a target function related to time efficiency can be obtained as shown in the following formula:
wherein, beta is a weight coefficient, x (K + 1) is the residual capacity of the battery at the (K + 1) th sampling moment, K is the moment of completing equalization, K is the kth sampling moment, M 0 ＝[m 1 ,m 2 ,…,m n ]Wherein m is j (j =1,2 \ 8230n) is as follows:
M 0 is a defined coefficient matrix, m n The residual capacity of the nth battery is equal to the average residual capacity of the n batteries, and if the residual capacity of the nth battery is greater than the average residual capacity of the n batteries, m is n Is 1, otherwise is-1; m is j The residual capacity of the j-th battery is equal to the average residual capacity of the j-th battery, and if the residual capacity of the j-th battery is larger than the average residual capacity of the j-th battery, the m j Is 1, otherwise is-1; x is a radical of a fluorine atom j 、x eq Respectively the residual capacity of the j-th battery and the average residual capacity of the n batteries;
according to the prediction step size of N, the other coefficient matrix M = [ M = 0 ,…,M 0 ] (1,N×n) The objective function related to time efficiency can be obtained as shown in the following equation:
minβMX(k+1) (38)
wherein n is the total number of cells, β is a weight coefficient, and X (k + 1) ) Is a matrix of dimension (N × N, 1);
adding equation (30) to equation (38) results in a final objective function of time efficiency and loss efficiency, as shown in the following equation:
mint 0 F(k)U(k)+βMX(k+1) (39)
from the formulas (26) (39), the following formulaThe objective function is reduced to min (F (k) t) 0 + β M (k) S) U (k) + β MRx (k), where t0 is the interval duration between adjacent sampling points of the discretization of the continuous state space model, F (k) is a row vector of dimension (1, 2 × N), the variable matrix U (k) is a matrix of dimension (2 × N, 1), β is a weight coefficient, the state variable X (k + 1) is a matrix of dimension (N × N, 1), since β MRx (k) belongs to a constant independent of the input U (k), the objective function can be simplified as shown in the following equation:
thus, an objective function of the equilibrium model based on the nonlinear efficiency of the model predictive control strategy and a constraint condition thereof are obtained.
In summary, due to the adoption of the technical scheme, the invention has the beneficial effects that:
1. under a model prediction control strategy, a nonlinear relation between the efficiency of an equalizer and an equalizing current is considered to model a bus type equalizing circuit, so that the model of the bus type equalizing circuit is more practical;
2. and (3) solving the multi-target nonlinear optimization problem, and taking the result of the linear model as the initial value of the nonlinear model optimization by utilizing the fusion of the linear efficiency model and the nonlinear efficiency model. The solving difficulty is greatly reduced.
Additional aspects and advantages of the invention will be set forth in part in the description which follows and, in part, will be obvious from the description, or may be learned by practice of the invention;
drawings
The above and/or additional aspects and advantages of the present invention will become apparent and readily appreciated from the following description of the embodiments, taken in conjunction with the accompanying drawings of which:
FIG. 1 is a schematic diagram of a single improved bussed bi-directional Cuk circuit of the present invention;
FIG. 2 is a block diagram of a bus-based equalization system architecture of the present invention;
FIG. 3 is a graph of a non-linear function between equalization efficiency and equalization current of the present invention;
FIG. 4 is a graph of state of charge of a battery of the present invention over time;
FIG. 5 is a flowchart of the overall algorithm of the present invention;
FIG. 6 is a flow chart of the model predictive control strategy of the present invention;
FIG. 7 is a SOC curve under the model predictive control equalization strategy of the present invention.
Detailed Description
Reference will now be made in detail to the embodiments of the present invention, examples of which are illustrated in the accompanying drawings, wherein like reference numerals refer to the same or similar elements or elements having the same or similar functions throughout; the embodiments described below with reference to the drawings are illustrative only for the purpose of illustrating the invention and are not to be construed as limiting the invention;
in the description of the present invention, it is to be understood that the terms "longitudinal", "lateral", "upper", "lower", "front", "rear", "left", "right", "vertical", "horizontal", "top", "bottom", "inner", "outer", etc. indicate orientations or positional relationships based on those shown in the drawings, and are merely for convenience of description and simplicity of description, but do not indicate or imply that the device or element referred to must have a particular orientation, be constructed in a particular orientation, and be operated, and thus, are not to be construed as limiting the present invention.
In the description of the present invention, unless otherwise specified and limited, it is to be noted that the terms "mounted," "connected," and "connected" are to be construed broadly, and can be, for example, a mechanical or electrical connection, a communication between two elements, a direct connection, or an indirect connection via an intermediate medium, and the specific meaning of the terms can be understood by those skilled in the art according to specific situations.
FIG. 1 shows a single improved bussed bi-directional Cuk circuit diagram structure, namely the equalizer portion of FIG. 2: the first end of the battery is connected with the first end of the inductor L1, the second end of the inductor L1 is connected with the first end of the capacitor C1 and the first end of the switching tube Q1, and the second end of the battery is connected with the power supplyThe first end of inductance L3 links to each other, and the second end of inductance L3 links to each other with the first end of electric capacity C2, the second end of switch tube Q1, and the second end of electric capacity C1 links to each other with the first end of switch tube Q2, the first end of inductance L2, and the second end of electric capacity C2 links to each other with the second end of switch tube Q2, the first end of inductance L4, and the second end of inductance L2 and electric capacity C2 bus Is connected to the energy bus, and the second end of the inductor L4 is connected to the capacitor C bus And the second end of the energy bus is connected with the energy bus.
Fig. 2 shows a bus-based equalization system obtained by a cell-pack-cell method, in which a battery with a high state of charge is charged by an equalizer through a bus to a battery with a lower state of charge. The bus here mainly serves as an energy schedule. According to the relation between the residual battery capacity and the balance current, a relation expression between the residual battery capacity and the balance current can be obtained, as shown in the following formula, wherein Q j The capacity of the jth battery.
Therein, SOC j (t) represents the SOC variation equation of the j-th battery at the time t,
is SOC j A differential form of (t); />
Representing the current between the jth battery and the equalizer;
in the above equation, plus the self-loss of the cell, the following equation can be obtained, where τ is used to describe the self-discharge rate of the cell, and τ is a very small number.
Q SOC in the above formula j The remaining capacity of the battery. Let the residual capacity x of the jth battery j ＝Q j *SOC j In which，SOC j Represents the percentage of the jth battery remaining capacity; the relationship between the remaining battery capacity and the balance current can be obtained as shown in the following formula:
fig. 3 shows a non-linear function between equalization efficiency and equalization current. Can be measured using a dedicated device and then a function fit is used to obtain a non-linear function between the equalizer efficiency and the equalizer current. Wherein the relationship between equalizer to battery current and equalizer efficiency is shown as:
y 1 ＝-0.0002x 1 4 +0.0043x 1 3 -0.0317x 1 2 +0.0897x 1 +0.8113
the relationship between bus to equalizer current and equalizer efficiency is shown as follows:
y 2 ＝0.0003x 2 4 -0.0038x 2 3 -0.0019x 2 2 +0.0949x 2 +0.6878
fig. 4 shows a graph of the state of charge of a battery over time. The upper graph is a plot of the SOC of the cell over time in a standard cell balancing problem. In the same battery state, it is apparent that the equalization time t indicated by the broken line eq1 Less than the equalisation time t indicated by the solid line eq2 . The essence of the method is that the area enclosed by the dotted line and the coordinate axes is smaller than the area enclosed by the solid line and the coordinate axes. From the principle of minimum area and minimum time, the objective function related to time efficiency can be obtained as shown in the following formula:
wherein, beta is a weight coefficient, x (K + 1) is the residual capacity of the battery at the (K + 1) th sampling moment, K is the moment of completing equalization, K is the kth sampling moment, M 0 ＝[m 1 ,m 2 ,…,m n ]Wherein m is j (j =1,2 \ 8230n) is as follows:
since the prediction step is N, another coefficient matrix M = [ M = [ [ M ] 0 ,…,M 0 ] (1,N×n) The objective function related to time efficiency can be obtained as shown in the following equation:
minβMX(k+1)
where N is the total number of cells, β is a weight coefficient, and X (k + 1) is a matrix with dimension (N × N, 1).
Fig. 5 shows the overall algorithm flow diagram of the present invention. In order to solve the problem that the calculated amount is too large to solve, the invention firstly sets the transmission efficiency of the equalizer as a constant, thereby establishing a linear model and solving the optimal solution u through multi-objective linear programming 0 And then assigning the optimal solution to a trust domain reflection algorithm of a discrete state space model of the nonlinear equilibrium system as an initial value, thereby solving the optimal solution u of the nonlinear efficiency model 0 . The solving difficulty of the confidence domain reflection algorithm can be greatly reduced by utilizing the fusion of the linear model and the nonlinear model.
Fig. 6 shows a specific algorithm flow chart of the model-based prediction strategy. The key of the model predictive control is to use the first control action in the optimal solution as the input of the current control object. In fact, in the next sampling period, the measured value at the next moment is used again, and the measured value at the next moment is used for solving the control input at the next moment, so that the model predictive control has the function of feedback correction.
S-A, initializing se:Sup>A sampling time K to be 0, and setting the time K for completing the equalization to be K;
S-B, obtaining an optimal value u according to a linear model 0 ；，
S-C, reaction of u 0 As an initial value of the nonlinear multi-objective optimization;
S-D, solving the problem of multi-target nonlinear programming under constraint conditions to obtain an optimal solution U (k), wherein U (k) is an input variable matrix;
S-E, the value of k is equal to the last k value plus 1;
S-F, taking the first control action in the optimal solution U (k) as an input;
S-G, judging whether K is less than or equal to K, if so, skipping to execute S-D; if not, executing S-H;
and S-H, finishing.
FIG. 7 shows a simulation diagram of a bus-based equalization model solution for nonlinear efficiency under a model predictive control strategy. The selected simulation parameters are: number of cells n =5, sampling time of discretization of continuous state space model, i.e. interval time t between two adjacent sampling times 0 =30, prediction step N =100, battery self-discharge rate τ =10 -8 Capacity of battery Q = [20;20;20;20;20]X 3600, initial remaining capacity x of battery 0 ＝[0.45；0.51；0.55；0.57；0.65]X 20X 3600, lowest state of charge SOC of battery min =0.3, maximum state of charge SOC of battery max =0.7, voltage v of battery j ＝[3.0；3.1；3.3；3.4；3.2]Bus voltage v bus ＝8。
In the figure, the SOC equalization results when the upper and lower halves are equal to the equalization speed and the weighting factors are β =0.01 and β =0, respectively. As can be seen from the figure, when the equalization speed weight coefficient β =0, the equalization is reached at a time of 3000 s. When the weighting coefficient β =0.01 of the equalizing speed, that is, the equalizing speed efficiency is considered into the objective function, only about 1050s is required to achieve the equalization.
In the description herein, references to the description of the term "one embodiment," "some embodiments," "an example," "a specific example," or "some examples," etc., mean that a particular feature, structure, material, or characteristic described in connection with the embodiment or example is included in at least one embodiment or example of the invention. In this specification, the schematic representations of the terms used above do not necessarily refer to the same embodiment or example. Furthermore, the particular features, structures, materials, or characteristics described may be combined in any suitable manner in any one or more embodiments or examples.
While embodiments of the present invention have been shown and described, it will be understood by those of ordinary skill in the art that: various changes, modifications, substitutions and alterations can be made to the embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the claims and their equivalents.
Claims (7)
1. A battery equalization modeling system based on a nonlinear efficiency model is characterized by comprising: the system comprises a charge-discharge module, n equalizers, n batteries and an energy bus;
the first end of the 1 st battery is connected with the first end of the charge-discharge module and the first end of the 1 st equalizer, the second end of the 1 st battery is connected with the second end of the 1 st equalizer, the first end of the 2 nd battery and the first end of the 2 nd equalizer, the third end of the 1 st equalizer is connected with the negative electrode of the energy bus, and the fourth end of the 1 st equalizer is connected with the positive electrode of the energy bus;
the first end of the 2 nd battery is connected with the second end of the 1 st battery, the second end of the 1 st equalizer and the first end of the 2 nd equalizer, the second end of the 2 nd battery is connected with the first end of the 3 rd battery, the first end of the 3 rd equalizer and the second end of the 2 nd equalizer, the third end of the 2 nd equalizer is connected with the negative electrode of the energy bus, and the fourth end of the 2 nd equalizer is connected with the positive electrode of the energy bus;
the first end of the 3 rd battery is connected with the second end of the 2 nd battery, the second end of the 2 nd equalizer and the first end of the 3 rd equalizer, the second end of the 3 rd battery is connected with the first end of the 4 th battery, the first end of the 4 th equalizer and the second end of the 3 rd equalizer, the third end of the 3 rd equalizer is connected with the negative electrode of the energy bus, and the fourth end of the 3 rd equalizer is connected with the positive electrode of the energy bus;
……；
the first end of the nth battery is connected with the second end of the (n-1) th battery, the second end of the (n-1) th equalizer and the first end of the nth equalizer, the second end of the nth battery is connected with the second end of the nth equalizer and the second end of the charge-discharge module, the third end of the nth equalizer is connected with the negative electrode of the energy bus, and the fourth end of the nth equalizer is connected with the positive electrode of the energy bus;
n is the total number of the batteries and is also the total number of the equalizers corresponding to the batteries;
each equalizer includes:
the first end of battery links to each other with inductance L1's first end, inductance L1's second end and electric capacity C1's first end, switch tube Q1's first end links to each other, the second end of battery links to each other with inductance L3's first end, inductance L3's second end and electric capacity C2's first end, switch tube Q1's second end links to each other, electric capacity C1's second end and switch tube Q2's first end, inductance L2's first end links to each other, electric capacity C2's second end and switch tube Q2's second end, inductance L4's first end links to each other, inductance L2's second end and electric capacity C2's first end link to each other bus Is connected to the energy bus, and the second terminal of the inductor L4 is connected to the capacitor C bus The second end of the energy bus is connected with the energy bus;
the first end of battery links to each other with inductance L1's first end, inductance L1's second end and electric capacity C1's first end, switching tube Q1's first end links to each other, the second end of battery links to each other with inductance L3's first end, inductance L3's second end and electric capacity C2's first end, switching tube Q1's second end links to each other, electric capacity C1's second end and switching tube Q2's first end, inductance L2's first end links to each other, electric capacity C2's second end and switching tube Q2's second end, inductance L4's first end links to each other, inductance L2's second end and electric capacity C2's first end link to each other bus Is connected to the energy bus, and the second terminal of the inductor L4 is connected to the capacitor C bus The second end of the energy bus is connected with the energy bus.
2. A battery equalization modeling method based on a nonlinear efficiency model is characterized by comprising the following steps:
s1, obtaining the relation between the SOC of the battery and the balance current in a bus type balance system according to an ampere-hour integral method, and accordingly establishing a relation between the residual electric quantity of the battery and the balance current;
s2, according to the structure of the bus type equalizing system, efficiency values under different equalizing currents can be obtained through equipment, a nonlinear relation between the efficiency and the equalizing currents is established, energy consumed by the jth battery in an equalizing period in the equalizing period is established according to the nonlinear relation of the efficiency, and an equalizing loss target function of the bus type equalizing system in the equalizing process is obtained;
obtaining the equalization loss objective function comprises the following steps:
for the jth equalizer, from the nonlinear efficiency, the following can be obtained:
wherein, the first and the second end of the pipe are connected with each other,
is the equalization efficiency between the equalizer and the battery;
v j is the voltage of the jth cell;
v bus is the bus voltage;
wherein, the first and the second end of the pipe are connected with each other,
representing the current between the jth battery and the equalizer;
v j is the voltage of the jth cell;
v bus is the bus voltage;
when energy is transferred from the battery to the energy bus, the input current of the equalizer is
When both the input voltage and the output voltage are constant, a switch is activated>
Only and->
(ii) related; can be set>
Wherein it is present>
Representing equalizing current>
And a nonlinear equation of relationship for efficiency; when energy is transferred from the bus to the battery, the input current to the equalizer is ≥ r>
When both the input voltage and the output voltage are constant, a switch is activated>
Only and->
(ii) related; can be set>
Wherein it is present>
Expressed as equalizing current->
And a nonlinear equation of relationship for efficiency;
for convenience of expression, two variables u are redefined j,1 、u j,2 Control signals used to describe the solution to be solved; wherein u is j,1 、u j,2 The specific expression of (b) is shown by the following formula:
wherein u is j,1 Represents the current flowing from the energy bus to the jth equalizer;
wherein u is j,2 Represents the current flowing from the jth battery to the jth equalizer;
the default direction of the current is that the current flows from the bus to the equalizer and the current flows to the battery from the equalizer to be a positive direction, and the current is added with a negative sign on the contrary;
the battery balancing strategy is to achieve the purpose of SOC balancing by controlling the current flowing direction and magnitude between the battery and the energy bus;
when the battery is charged, the current flows into the battery through the equalizer via the bus, and the following formula can be obtained from formula (6):
wherein the content of the first and second substances,
representing the current between the jth battery and the equalizer; />
v bus is the bus voltage;
v j is the voltage of the jth cell;
h(u j,1 ) Represents the efficiency between the energy bus and the jth equalizer;
u j,1 represents the current flowing from the energy bus to the jth equalizer;
when the battery is discharged, the current is directly discharged from the battery, and the following formula can be obtained:
wherein the content of the first and second substances,
representing the current between the jth battery and the equalizer;
u j,2 represents the current flowing from the jth battery to the jth equalizer;
substituting equations (9) (10) into equation (4) yields the following equation:
wherein the content of the first and second substances,
indicates the remaining capacity x of the battery j Differentiation of (1);
τ is used to describe the self-discharge rate of the cell;
x j representing the residual capacity of the j-th battery;
h(u j,1 ) Indicating the efficiency between the energy bus and the jth equalizer;
v bus is the bus voltage;
u j,1 represents the current flowing from the energy bus to the jth equalizer;
v j is the voltage of the jth cell;
u j,2 represents the current flowing from the jth battery to the jth equalizer;
s2-2, according to the battery equalization model with the nonlinear efficiency, the energy consumed by the jth battery in the process of an equalization period can be expressed as the following formula:
wherein, E represents the energy lost by the j-th battery in the process of a balancing period;
t eq represents an equalization period;
h(u j,1 ) Represents the efficiency between the energy bus and the jth equalizer;
u j,1 represents the current flowing from the energy bus to the jth equalizer;
v bus is the bus voltage;
f(u j,2 ) Is the efficiency between the jth cell and the jth equalizer;
u j,2 represents the current flowing from the jth battery to the jth equalizer;
v j is the voltage of the jth cell;
thus, a balance loss objective function of the bus type balance system in the balance process is obtained;
and S3, combining the residual electricity quantity change equations of the n batteries together according to the structure of the bus type equalizing system to obtain a state space model of the bus type equalizing system, then controlling the state space model of the bus type equalizing system by utilizing model prediction control, and establishing an equalizing model of nonlinear efficiency based on a model prediction control strategy.
3. The battery equalization modeling method based on the nonlinear efficiency model according to claim 2, wherein the solving of the nonlinear efficiency model, which is the equalization model based on the nonlinear efficiency of the model predictive control strategy in S3, belongs to nonlinear optimization, and comprises:
the transmission efficiency of the equalizer is set as a constant, a linear model, namely a linear efficiency model, is established, an optimal solution is solved through multi-objective linear programming, and the optimal solution is assigned to a trust domain reflection algorithm to be used as an initial value, so that the optimal solution of the nonlinear efficiency model is solved.
4. The battery equalization modeling method based on the nonlinear efficiency model according to claim 2, wherein the S1 comprises:
s1-1, establishing a relation between the SOC (state of charge) of the jth battery and the balance current according to the relation between the SOC of the jth battery and the balance current of the jth battery in a bus type balance system, so as to obtain a relation between the residual electric quantity of the battery and the balance current;
s1-2, adding the self-loss of the battery in the equation of the relation between the residual capacity of the battery and the balance current, and setting the residual capacity of the j-th battery as x j ＝Q j *SOC j Obtaining a complete equation of the relation between the residual battery capacity and the balance current;
according to the relation between the residual electric quantity of the lithium ion battery and the balance current, writing out an SOC change equation of the j-th battery as follows:
wherein Q is j The capacity of the jth battery;
SOC j (t) representing the SOC change equation of the jth battery at the moment of time t;
SOC j (t 0 ) Is represented at time t 0 At the moment, the SOC change equation of the j-th battery;
by differentiating equation (1), the following equation can be obtained:
wherein Q is j The capacity of the jth battery;
for the completeness of the equation, adding the self-depletion of the battery in equation (2) yields the following equation:
wherein Q is j The capacity of the jth battery;
τ is used to describe the self-discharge rate of the cell;
Q j the capacity of the jth battery;
SOC j represents the percentage of the remaining capacity of the jth battery;
let the residual capacity x of the jth battery j ＝Q j *SOC j And obtaining the relation between the residual battery capacity and the balance current as follows:
wherein the content of the first and second substances,
is the remaining capacity x of the battery j Differentiation of (1); />
τ is used to describe the self-discharge rate of the cell;
x j the residual capacity of the jth battery is;
a relationship between the remaining capacity of the battery and the equalizing current has been established so far.
5. The battery equalization modeling method based on the nonlinear efficiency model according to claim 2, wherein the S3 comprises:
s3-1, combining the residual electricity quantity change equations of n batteries together to obtain a state space model of the bus type equalization system according to the structure of the bus type equalization system, discretizing the state space model to obtain a discrete state space model, and obtaining an equalization loss target function and constraint of the target function in the discretization model;
and S3-2, controlling the state space model of the bus type equalization system by using a model prediction control strategy, so that a target function and a constraint condition of the target function which minimize equalization loss in model prediction control can be obtained, and according to the rule that the battery charge state changes along with time in the equalization process, the equalization speed is related to the area enclosed by a charge state change curve and a longitudinal axis in the equalization process, so that the time efficiency is combined into the target function with the minimum equalization loss.
6. The battery equalization modeling method based on the nonlinear efficiency model according to claim 5, wherein the S3-1 comprises:
combining the residual electricity quantity change equations of n batteries together to obtain a state space model of the bus type equalization system as shown in the following formula:
wherein, the first and the second end of the pipe are connected with each other,
representing a state space model formed by combining equations of n batteries together;
A 0 ＝-τI n×n is a matrix of n × n; τ is the self-discharge rate of the cell, I n×n Representing an n × n identity matrix, n being the total number of cells;
x＝[x 1 ,x 2 ,…x n ] Τ is a state vector of n x 1 dimension, representing the residual capacity of all batteries in the equalizing system, x 1 Indicates the remaining capacity, x, of the 1 st cell 2 Indicates the remaining capacity, x, of the 2 nd battery n Representing the residual capacity of the nth battery; a Τ Representing a transpose;
u＝[u 1,1 ,u 1,2 ,u 2,1 ,u 2,2 ,…u n,1 ,u n,2 ,] Τ is a control input of 2n × 1 dimension, representing the control action of the equalizer on the battery, u 1,1 Representing the current of the energy bus to the 1 st equalizer, u 1,2 Representing the current, u, flowing from the 1 st cell to the 1 st equalizer 2,1 Representing the current, u, of the energy bus flowing to the 2 nd equalizer 2,2 Representing the current of the 2 nd battery to the 2 nd equalizer, u n,1 Representing the current of the energy bus flowing to the nth equalizer, u n,2 Representing the current of the nth battery flowing to the nth equalizer;
B 0 the matrix for n × 2n is as follows:
wherein, h (u) 1,1 ) Efficiency between the energy bus and the 1 st equalizer;
v bus is the bus voltage;
v 1 voltage of the 1 st battery;
h(u 2,1 ) Represents the efficiency between the energy bus and the 2 nd equalizer;
v 2 the voltage of the 2 nd battery;
h(u n,1 ) Representing the efficiency between the energy bus and the nth equalizer;
v n is the voltage of the nth cell;
discretizing the formula (13) with kt as the sampling time 0 -t 0 Substituting the sampling instant into equation (13) may result in the following equation:
wherein the content of the first and second substances,
representing a time of kt 0 -t 0 The state space model of the bus type equalization system;
x(kt 0 -t 0 ) Representing the battery at time kt 0 -t 0 The remaining amount of power of;
u(kt 0 -t 0 ) Representing the battery at time kt 0 -t 0 The magnitude of the balancing current;
t 0 the sampling time of the discretization of the continuous state space model, and the control signal is unchanged in a sampling period;
when t is 0 Sufficiently small, then equation (15) may become the following equation:
wherein x (k) represents the remaining capacity of the battery at the kth sampling moment;
x (k-1) represents the remaining capacity of the battery at the k-1 sampling point;
t 0 the sampling time of the discretization of the continuous state space model, and the control signal is unchanged in a sampling period;
x (k-1) represents the remaining capacity of the battery at the k-1 th sampling point;
u (k-1) represents a control signal at the k-1 th sampling point;
the equation (16) can be simplified as shown in the following equation:
x(k)＝(I n×n +A 0 t 0 )x(k-1)+B 0 t 0 u(k-1) (17)
wherein x (k) represents the remaining capacity of the battery at the kth sampling moment;
I n×n an identity matrix of n × n;
t 0 the sampling time of the discretization of the continuous state space model, and the control signal is unchanged in a sampling period;
x (k-1) represents the remaining capacity of the battery at the k-1 th sampling point;
u (k-1) represents a control signal at the k-1 th sampling point;
so far, a discretization model of the formula (13) such as the formula (17) can be obtained; a = I n×n +A 0 t 0 、B＝B 0 t 0 Wherein A is 0 ＝-τI n×n Is a matrix of n × n, τ is the self-discharge rate of the cell, I n×n Representing an n × n identity matrix, B 0 Is a matrix of n × 2 n; for the discretized state space model, the time when equalization is completed is K, and the equalization loss objective function in the discretized state space model can be obtained as shown in the following formula:
wherein E represents the energy lost by the jth battery in the process of one balancing cycle;
k represents the time when equalization is completed;
t 0 a sampling time representing a discretization of the continuous state space model;
F 0 (k) The F matrix at the 0 th step of the kth sampling moment is obtained;
u (k) represents the control action of the equalizer on the battery at the k sampling moment;
v bus is the bus voltage;
h[u 1,1 (k)]the efficiency between the energy bus and the 1 st equalizer at the k sampling moment is shown;
h[u n,1 (k)]representing the efficiency between the energy bus and the nth equalizer at the kth sampling instant;
v 1 voltage of the 1 st battery;
f[u 1,2 (k)]the efficiency between the 1 st battery and the 1 st equalizer at the kth sampling moment is shown;
v n is the voltage of the nth cell;
f[u 1,2 (k)]the efficiency between the 1 st battery and the 1 st equalizer at the kth sampling moment is shown;
f[u n,2 (k)]the efficiency between the nth battery and the nth equalizer at the kth sampling moment is shown;
the constraints on the state variables are soft constraints, the constraints on the control inputs belong to hard constraints, and for equation (18) the following constraints exist: formula (19) is a soft constraint that the remaining capacity of each battery is equal at the time K of completing equalization, formula (20) is a hard constraint that the total current flowing into the bus and the total current flowing out of the bus are equal to ensure the bus voltage is stable during equalization, and formula (21) is a hard constraint on the control signal;
x 1 (K)＝x 2 (K)＝…＝x n (K) (19)
wherein x is 1 (K) Indicating the residual capacity of the 1 st battery at the moment K of finishing the balance;
x 2 (K) Indicating the residual capacity of the 2 nd battery at the moment K of finishing the balance;
x n (K) Indicating equalisation is being performedAt the moment K, the residual capacity of the nth battery;
wherein u is j,1 (k) Is the current from the bus to the equalizer at the kth sampling instant;
u j,2 (k) Efficiency between the equalizer and the battery at the kth sampling instant;
u j,2 (k) Is the current from the battery to the equalizer at the kth sampling instant;
v j is the voltage of the jth cell;
v bus is the bus voltage;
u j,1 (k)×u j,2 (k)＝0 k＝0,1,...,K-1,j＝1…,n (21)
wherein u is j,1 (k) Is the current from the bus to the equalizer at the kth sampling instant;
u j,2 (k) Is the current from the battery to the equalizer at the kth sampling instant;
k is the time for completing the equalization;
n means n batteries;
and establishing the discrete state space model and corresponding constraint conditions of the discrete state space model.
7. The battery equalization modeling method based on the nonlinear efficiency model according to claim 5, wherein the S3-2 comprises:
controlling a state space model of the bus type balance system by using a model prediction control strategy; and (3) controlling the balance system by adopting model prediction control, and obtaining a prediction state in a prediction step length according to a formula (17) as follows:
x(k+1|k)＝Ax(k)+Bu(k) (22)
the state for the next time is as follows:
x(k+2|k)＝Ax(k+1)+Bu(k+1) (23)
substituting equation (22) into equation (23) yields the following equation:
x(k+2|k)＝A 2 x(k)+ABu(k)+Bu(k+1) (24)
wherein u (k) means the control signal at the kth sampling instant, u (k + 1) means the control signal at the kth +1 sampling instant, a = I n×n +A 0 t 0 ，B＝B 0 t 0 Wherein A is 0 ＝-τI n×n Is a matrix of n × n, τ is the self-discharge rate of the cell, I n×n Representing an n × n identity matrix, B 0 Is a matrix of n × 2n, t 0 Is the sampling time of the discretization of the continuous state space model; the step size of the model predictive control is actually the time interval t between adjacent moments 0 (ii) a x (k) and x (k + 1) respectively represent the residual capacity of the battery at the kth sampling moment and the residual capacity of the battery at the kth +1 sampling moment; x (k +1 k) represents the residual capacity of the battery predicted in the step 1 at the kth sampling time, and x (k +2 k) represents the residual capacity of the battery predicted in the step 2 at the kth sampling time; similar iterations can result in the following prediction matrices:
wherein x (k + 1) is the predicted state in step 1 at the kth sampling time; x (k +2 purple k) is the predicted state of the 2 nd step at the kth sampling moment; x (k + N | k) is the prediction state of the Nth step at the k sampling moment, namely the prediction step length is N; (.) 2 Represents the square, (.) N Representing the power of N, x (k + i | k) being at time kt 0 The prediction state of the step i at the moment; u (k + i | k) is at time kt 0 Predicting and inputting at the ith moment;
b (k | k), u (k | k), B (k + N-1 luminance k) are equations derived from the above equations:
from equation (13), u = [ u = [ u ] 1,1 ,u 1,2 ,u 2,1 ,u 2,2 ,…u n,1 ,u n,2 ,] Τ U (k | k) is the control input at the kth sampling instant, and u (k +1 luminance k) isThe prediction input of the step 1 at the kth sampling moment; u (k + N-1) is predicted input of the step N-1 at the k sampling moment;
b = B as shown in formula (17) 0 t 0 (ii) a From equation (14), the matrices B and h (u) 1,1 )、h(u 2,1 )…h(u n,1 ) (ii) related; so matrix B also sums u = [ u ] 1,1 ,u 1,2 ,u 2,1 ,u 2,2 ,…u n,1 ,u n,2 ,] Τ Therefore, B (k | k) is a matrix obtained by substituting u (k | k) into the B matrix at the k-th sampling time; b (k + i | k) is a matrix obtained by substituting u (k + i | k) into the B matrix when predicting i steps at the kth sampling moment; b (k + N-1) is a matrix obtained by substituting u (k + N-1) into the B matrix when predicting the N-1 step at the kth sampling moment;
wherein k represents the kth sampling instant; n is the prediction step size, expressed for simplicity as:
wherein the content of the first and second substances,
for convenience of the following description, the following redefinitions are made; wherein the input variable matrix U (k) is a matrix of dimension (2 × N, 1), the state variable X (k + 1) is a matrix of dimension (N × N, 1), and F (k) is a row vector of dimension (1, 2 × N);
U(k)＝[u(k|k),u(k+1|k),…,u(k+N-1|k)] Τ (27)
X(k+1)＝[x(k+1|k),x(k+2|k),…,x(k+N|k)] Τ (28)
F(k)＝[F 0 (k|k),F 0 (k+1|k),…,F 0 (k+N-1|k)] (29)
wherein, F 0 (k | k) denotes the matrix F, F at the prediction of step 0 at the k-th sampling instant 0 (k +1 purple k) denotes the 1 st step at the kth sampling timeF matrix in time measurement, F 0 (k + N-1 caldus) represents the F matrix during the N-1 st prediction at the kth sampling time · Τ Representing a transpose; in order to minimize the loss during equalization, i.e., minimize equation (18); in the model predictive control strategy, it is equivalent to minimizing the loss within the whole prediction step; the objective function for minimizing the equalization loss in model predictive control can be obtained as follows:
min t 0 F(k)U(k) (30)
t 0 is the time interval between adjacent sampling instants of the discretization of the continuous-state space model, F (k) is a row vector of dimension (1, 2 × N), and the variable matrix U (k) is a matrix of dimension (2 × N, 1);
meanwhile, obtaining the constraint of the discrete state space model under a model prediction control strategy; as shown in equations (31) (32) (33) (34) (35); wherein equation (31) is the range limit for the input variable; equation (32) is the range limit for the state variables; formula (33) is a constraint that the current flowing into the bus is equal to the current flowing out of the bus in an equalization period; equation (34) is the limit for the battery charge to be equal after the equalization process is over; formula (35) is a constraint that the control signal has only one direction at a certain moment in the predicted step length;
0≤U(k)≤i m (31)
x l ≤X(k+1)≤x u (32)
x 1 (k+N|k)＝x 2 (k+N|k)＝…＝x n (k+N|k) (34)
u j,1 (k+h|k)u j,2 (h|k)＝0 h＝0,1,…N-1,j＝1,…,n (35)
wherein i m Is the maximum value of the input variable; x is a radical of a fluorine atom l 、x u Respectively the minimum residual capacity and the maximum residual capacity which can be allowed by the battery;
f[u j,2 (k+h|k)]the prediction efficiency of the h step between the jth battery and the jth equalizer at the kth sampling moment is shown; u. of j,1 (k + h | k) represents the prediction of the h step of the bus to equalizer current at the kth sample time; u. of j,2 (k + h | k) represents the current prediction of the h step between the jth battery and the jth equalizer at the kth sampling instant;
k represents sampling time, j represents a j-th battery, N represents total number of batteries, N represents prediction step length, h represents prediction step number at the k-th sampling time, v bus Is the bus voltage, v j Is the voltage of the jth cell;
x 1 (k + N | k) represents the remaining capacity of the 1 st battery at the k sampling moment when predicting N steps, x 2 (k + N | k) represents the remaining capacity of the 2 nd battery at the k sampling moment when predicting N steps, x n (k + N | k) represents the remaining capacity of the nth battery at the kth sampling moment when the nth battery predicts the N steps;
according to the prediction step size of N, the other coefficient matrix M = [ M = 0 ,…,M 0 ] (1,N×n) The objective function related to time efficiency can be obtained as shown in the following equation:
min β MX (k + 1) (38) where N is the total number of cells, β is the weight coefficient, and X (k + 1) is a matrix with dimension (N × N, 1);
adding equation (30) to equation (38) results in a final objective function of time efficiency and loss efficiency, as shown in the following equation:
min t 0 F(k)U(k)+βMX(k+1) (39)
the objective function can be reduced to min (F (k) t) by the equations (26) and (39) 0 + β M (k) S) U (k) + β MRx (k), where t 0 Is the interval duration between adjacent sampling points of the discretization of the continuous state space model, F (k) is a row vector with dimension (1, 2 × N), the variable matrix U (k) is a matrix with dimension (2 × N, 1), β is a weight coefficient, the state variable X (k + 1) is a matrix with dimension (N × N, 1), and since β MRx (k) belongs to a constant independent of the input U (k), the objective function can be simplified as shown in the following equation:
min(F(k)t 0 +βMS)U(k) (40)
thus, an objective function of the equilibrium model based on the nonlinear efficiency of the model predictive control strategy and a constraint condition thereof are obtained.
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