Disclosure of Invention
Aiming at the defects in the background technology, the invention provides an active power distribution network energy sharing method based on communication reliability constraint, and solves the technical problem that the safe operation of a power system is influenced due to incorrect control commands possibly caused by errors of a communication end.
The technical scheme of the invention is realized as follows:
an active power distribution network energy sharing method based on communication reliability constraint comprises the following steps:
the method comprises the following steps: establishing a power distribution network system model, and aggregating N energy micro-grids by one power distribution system operator in a power distribution grid, wherein the energy micro-grids are matched with the communication base stations in a one-to-one correspondence manner, and all the communication base stations share the same channel;
step two: controlling energy sharing among the energy micro-grids through the communication base station, and constructing a communication reliability constraint model of an energy sharing mechanism and a power loss constraint model of the communication base station;
step three: constructing an optimal operation model of an energy sharing mechanism according to a power distribution network, controllable loads and a communication base station power loss constraint model shared by distributed energy in a given area;
step four: and carrying out linear solving on the communication reliability constraint model by adopting a least square method and a piecewise linear method to obtain model parameters of the reliability constraint model, and applying the model parameters to an optimal operation model of an energy sharing mechanism to obtain the minimum value of the overall cost of the power distribution network of the energy sharing mechanism.
The communication reliability constraint model of the energy sharing mechanism is as follows:
δn-i=(1-En-i)l
wherein, deltan-iIs a communication reliability index between the nth communication base station and the ith energy microgrid, En-iThe bit error rate between the nth communication base station and the ith energy microgrid is shown, l is the bit length of a data packet, i is 1,2, … …, and N is the number of all energy microgrids in the scheduling range;
the bit error rate En-iExpressed as:
where Q () is the integral tail function of the standard Gaussian distribution, BNIs the noise bandwidth of the wireless transmission transceiver, R is the data transmission speed, SNRn-iIs the signal-to-noise ratio of the ith energy piconet accessing the nth communication base station.
SNR of nth communication base station accessed by ith energy microgridn-iExpressed as:
wherein, Wn-iIs the power, sigma, received by the nth communication base station connected with the ith energy microgridm≠nWm-iThe sum of interference power brought by other communication base stations to the ith energy microgrid, N0Is thermal noise;
the thermal noise N0Expressed as:
N0=kTB
where B is the channel bandwidth of the wireless transmit transceiver and k is the Boltzmann constantAnd k is 1.3803 × 10-23J/K, T is the temperature in Kelvin;
the power W received by the nth communication base station connected with the ith energy microgridn-iExpressed as:
wherein,
is BS transmit power, PL is the transmission loss multiple, PL is the dB form of the transmission loss;
PL=h+g log10(dn-i)
where h and g are both coefficients of the path loss model, dn-iIs the distance between the nth communication base station and the ith energy microgrid.
The power loss constraint model of the communication base station is as follows:
δi′-i≥α
wherein,
is the transmission power of the ith' communication base station in the interval of time slot t under scene s,
is the power consumption of the ith' communication base station in the interval of time slot t under the scene s, and e is the total power consumption and communicationThe linear ratio coefficient between the transmission powers of the communication base stations, f is the fixed load of the communication base stations,
is the minimum value of the communication base station transmit power,
is the maximum value of the transmission power of the communication base station, delta
i′-iThe communication reliability index of the ith energy microgrid accessing the ith communication base station is shown, and alpha is the communication reliability requirement of the active power distribution network.
The optimal operation model of the energy sharing mechanism is as follows:
wherein, c
iRepresents the distribution system operator operating cost, gamma, for a given area
sIs the probability of the scene s,
is the hour controllable load of the ith energy microgrid within the time slot t interval under the scene s,
is the load consumption of the ith' communication base station in the interval of time slot t under the scene s,
is the electricity price of the residents,
is the electricity price of the communication base station, U
i(. is) the utility function of the ith energy microgrid, P
ijstAnd j is 1,2, … …, where N is the number of all energy micro-grids in the scheduling range.
The constraint conditions of the optimal operation model of the energy sharing mechanism are as follows:
Vmin≤Vist≤Vmax
θmin≤θist≤θmax
wherein,
is the charging power of the ith energy microgrid within the time slot t interval under the scene s,
is the discharge power of the ith energy microgrid within the time slot t interval under the scene s,
is the reactive power stored by the ith energy microgrid within the time slot t interval under the scene s,
is the controllable load of the ith energy microgrid in the time slot t interval under the scene s,
is the power, phi, of the photovoltaic system in the interval of time slot t under scene s
iIs a set of units, Q, connected to the ith energy microgrid
ijstIs the reactive power flow from the ith energy microgrid to the jth energy microgrid, g
ijIs the susceptance value of the line between the ith energy microgrid and the jth energy microgrid, b
ijIs the ith energy microgrid and the jth energy microgridConductance, v, of the lines between the networks
istIs the voltage amplitude, v, of the ith energy microgrid
jstIs the voltage amplitude, theta, of the jth energy microgrid
istIs the voltage phase angle theta of the ith energy microgrid
jstIs the voltage phase angle, V, of the jth energy microgrid
minIs the lower bound of the voltage amplitude, V
maxIs the upper bound of the voltage amplitude, θ
minIs the lower bound of the phase angle of the voltage, theta
maxIs the upper bound of the phase angle of the voltage,
is a photovoltaic power generation predicted value of the ith energy microgrid in the time slot t interval,
is the lower bound of the controllable load,
is the upper bound of the controllable load,
is the predicted daily load requirement for the load,
is the maximum charging power that can be charged,
is the maximum discharge power of the discharge lamp,
is the maximum reactive power of the energy storage system,
it is the efficiency of the energy storage system,
is the stored energy of the energy storage system,
is an energy storage systemThe upper limit of the capacity of (a),
is the lower limit of the capacity of the energy storage system,
is the initial energy level stored by the ith energy microgrid under the scene s,
is the final energy level, S, stored by the ith energy microgrid under the scene S
ijIndicating the apparent power from the ith to the jth energy microgrid.
The method for linearly solving the communication reliability constraint model by adopting the least square method and the piecewise linear method comprises the following steps:
s41, the signal-to-noise ratio of the ith energy microgrid accessing the nth communication base station is expressed by using a regression equation:
SNRn-i=WEC+ε
wherein the SNRn-iIs a k x 1 dimensional random vector, W, determined from observationsEIs a kx (n +1) matrix determined by the predictor variables, C is the (n +1) x 1 vector of unknown parameters, and epsilon is the k x 1-dimensional vector of random errors;
s42, solving the step S41 through a least square normal equation to obtain a regression coefficient:
wherein,
a vector, i.e., a regression coefficient, for least squares estimation;
s43, dividing the signal-to-noise ratio domain into equal interval intervals, and writing a communication reliability constraint model into the following steps in the q-th interval:
wherein, a
qAnd b
qAre all coefficients in the interval q,
representing the signal-to-noise ratio of the equal interval, and the SNR represents the signal-to-noise ratio;
s44, introducing new binary and continuous variables of q-1 and a new inequality of 4 (q-1) to rewrite the constraint conditions of the optimal operation model into linear constraints:
max{π1,π2}≥0,
introducing new variable t ═ max { pi-1,π2The following new constraints are released from the maximum operator:
π1≤t≤π1+vΩ
π2≤t≤π2+(1-v)
s45, applying the method of the step S44 to a minimum operator: t is min { pi ═ n1,π2}:
π1-vΩ≤t≤π1
π2-(1-v)Ω≤t≤π2
Wherein v is a binary variable and Ω is a positive scalar;
s46, converting the nonlinear constraint of the reliability constraint model into a specific inequality according to the steps S41-S45, and obtaining model parameters of the reliability constraint model through solving the specific inequality.
The beneficial effect that this technical scheme can produce: the invention provides a three-layer framework comprising a decision process and an information physical system: in the framework, an active power distribution network is connected with a microgrid and a communication base station which aggregate a large amount of energy of demand parties, interference and noise exist among the microgrids, communication reliability is a nonlinear function formed by transmitting power of different communication base stations, linearization is carried out on the microgrids by adopting a least square method and a piecewise linearity method, and the total cost of the power distribution network based on distributed energy sharing is the minimum by the proposed energy sharing mechanism on the premise that each microgrid meets the physical and communication reliability constraints.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be obtained by a person skilled in the art without inventive effort based on the embodiments of the present invention, are within the scope of the present invention.
The present invention contemplates a utility-level power distribution grid having N microgrid's. Through the connection of the communication network, each piconet operator manages several customers and distributed energy sources, including photovoltaic systems and battery storage. In addition, the microgrid shares distributed energy with other piconets through a communication network between different base stations. In order to evaluate the information transmission quality of a communication network, a communication reliability index is introduced. It should be noted that there are complex mechanisms for the interaction between the communication network and the distribution network: in order to cover all the customers in the microgrid, the local communication base stations should boost their power and therefore have a great impact on the power flow of the distribution grid. However, high transmission power may cause interference between adjacent communication base stations, possibly resulting in communication errors. These interactions complicate the power scheduling of multiple communication base stations in a power distribution network.
The system architecture of the energy sharing operation mechanism is composed of three layers, namely a physical layer, a communication layer and a decision layer, as shown in fig. 1.
The physical layer is composed of a plurality of nodes, and the nodes of the power distribution network are divided into two types: a connection node and a load node. The load nodes are responsible for bidirectional power transmission to the microgrid on the nodes, and the connecting nodes are hubs of the multiple sections of transmission lines and are not connected with the microgrid. Furthermore, the microgrid aggregates distributed energy sources and communication base stations, these components constituting a physical distribution network that enables energy sharing. Different constraints are defined in this layer to guarantee the quality and reliability of the power supply and to control the power flow. Constraints in the physical layer include voltage constraints, frequency constraints, active power constraints, and the like.
The communication layer consists of base station equipment, protocols, applications and information flows. Information flow refers to the sender, recipient and content of each message transmitted between communication devices. The technology of the internet of things enables a power distribution system operator to collect supply and demand information of a microgrid and realize optimal energy management. It should be noted that a communication base station belongs to both the physical layer and the information layer, since it requires power supply of the physical layer on the one hand and is responsible for information transmission of the information layer on the other hand. In the architecture of the present invention, it is the communication base station that reflects the coupling relationship between the physical system and the information system.
The decision layer decides how the power distribution system operator performs the optimal energy operation. Various models can be developed at this level to enable energy trading. The invention provides an energy trading scheme for energy sharing. Assuming that all microgrids are price makers, they purchase power from the distribution system operator and sell the remaining power. Based on the supply and demand information of different micro-grids, the power distribution system operator optimizes the power scheduling of the micro-grids while meeting the communication reliability requirement.
Each microgrid communicates with the power distribution system operator in the area and receives control signals through communication base stations. Reliable and stable communication networks are the prerequisite for energy sharing mechanisms. The invention introduces a communication reliability model and clarifies the relation between the transmission power of a communication base station and the communication reliability. Communication reliability constraints of the optimized operation model are set forth in combination with energy sharing requirements of communication reliability.
Based on the system architecture of the energy sharing operation mechanism, the embodiment of the invention provides an active power distribution network energy sharing method based on communication reliability constraint, which comprises the following steps:
the method comprises the following steps: establishing a power distribution network system model, and aggregating N energy micro-grids by one power distribution system operator in a power distribution grid, wherein the energy micro-grids are matched with the communication base stations in a one-to-one correspondence manner, and all the communication base stations share the same channel;
step two: controlling energy sharing among the energy micro-grids through the communication base station, and constructing a communication reliability constraint model of an energy sharing mechanism and a power loss constraint model of the communication base station;
the transmission of information between the microgrid and the operator of the power distribution system is efficient if the reliability of the communication between the Microgrid (MG) and the communication Base Station (BS) exceeds a certain value (α).
The calculation equation of the communication reliability is as follows:
δn-i=(1-En-i)l (1)
wherein, deltan-iIs a communication reliability index between the nth communication base station and the ith energy microgrid, En-iThe Bit Error Rate (BER) between the nth communication base station and the ith energy microgrid, l is the bit length of the data packet (usually 125bits), i is 1,2, … …, and N is the number of all energy microgrids within the scheduling range.
The bit error rate En-iExpressed as:
where Q () is the integral tail function of the standard Gaussian distribution, BNIs the noise bandwidth of the wireless transmission transceiver, R is a numberData transmission rate, SNRn-iIs the signal-to-noise ratio (SNR) of the ith energy piconet accessing the nth communications base station. According to the property of the integral tail function, the high signal-to-noise ratio represents the low bit error rate.
The signal-to-noise ratio (SNR) is the ratio of the received effective signal strength to the noise signal strength. The effective signal strength refers to the signal strength obtained by the user from the access communication base station. The noise signal of the invention comprises interference power and thermal noise interference of other communication base stations. The calculation equation of the signal-to-noise ratio is as follows:
wherein, Wn-iIs the power, sigma, received by the nth communication base station connected with the ith energy microgridm≠nWm-iThe sum of interference power brought by other communication base stations to the ith energy microgrid, N0Is thermal noise.
The thermal noise N0Expressed as:
N0=kTB (4)
where B is the channel bandwidth of the wireless transmit transceiver, k is Boltzmann (Boltzmann) constant, and k is 1.3803 × 10-23J/K, T is the kelvin temperature, and the temperature is typically T290K (17 ℃).
The path loss models under various propagation scenarios are different. Based on an analysis method and an empirical method, through the combination of the analysis method and the empirical method, a lognormal shadow path loss model is summarized in the LTE heterogeneous network physical layer specification (TR 36.814). The path loss model is generally expressed as:
PL=h+g log10(dn-i) (5)
where PL is the dB form of the transmission loss, h and g are the coefficients of the path loss model, dn-iIs the distance between the nth communication base station and the ith energy microgrid.
The power W received by the nth communication base station connected with the ith energy microgridn-iExpressed as:
wherein,
is the BS transmit power, PL is the transmission loss multiple converted from dB form (PL);
PL=h+g log10(dn-i) (7)
based on equations (3) to (7), the final calculation equation for the signal-to-noise ratio is shown in equation (8):
as is clear from equation (8), increasing the signal-to-noise ratio can reduce the bit error rate. Furthermore, based on the equation, reducing the bit error rate may improve the reliability of wireless communication. According to equation (8), increasing the transmission power of the communication base stations is a key approach to increase the signal-to-noise ratio, but increasing the transmission power of one communication base station increases the interference noise of other communication base stations, thereby decreasing the signal-to-noise ratio.
Although the communication base station can communicate through different channels by adopting the channel resource allocation method, network equipment using quasi-orthogonal channel resources is always in the same or adjacent areas, so that co-channel interference is difficult to avoid. Moreover, the power distribution network communication private network cannot necessarily distribute excessive channel resources, so that the problem of common channel interference is more prominent. The model of the invention therefore assumes that all communication stations are connected to the same channel. The analysis of the interaction between the communicating base stations is shown in figure 2. In fig. 2, all communication base stations share the same channel. When the transmit power of BS1 is low, its coverage area is also small (as shown by the inner circle in fig. 2), which does not affect the communication between other BSs. And all BSs communicate simultaneously, so that the overall network efficiency is high. When BS1 is operating improperly at high transmit power, although its SNR and CR are high, the wireless communication range is increased (as shown by the outer circles in fig. 2). The communications of BS2-9 are affected, reducing overall network efficiency.
The higher the transmit power, the more communication base stations are affected. Therefore, the BS transmit power needs to be properly planned to ensure its communication reliability without affecting neighboring base stations.
From the above conclusions, it is necessary to optimize the BS transmit power so that the communication of all MGs meets the reliability requirements.
In the optimized operation model, the BS transmission power must satisfy the power loss constraint model of the communication base station as follows:
δi′-i≥α (11)
wherein,
is the transmission power of the ith' communication base station in the interval of time slot t under scene s,
the power consumption of the ith' communication base station in the time slot t interval under the scene s, e is a linear ratio coefficient between the total power consumption and the transmission power of the communication base station, and the value of e is related to the loss of a radio frequency amplifier, the loss of a feeder line and the load consumption of related air conditioning equipment. The total power consumption of the BS and the transmitting power are in a linear increasing relationship, f is the fixed load of the communication base station, and the value of f is independent of the transmitting power of the BS. Mainly comprises a signal processing device, a storage battery and other load consumption of related parts of air conditioning equipment.
Is the minimum value of the communication base station transmit power,
is a communication baseMaximum value of station transmission power, delta
i′-iThe communication reliability index of the ith energy microgrid accessing the ith communication base station is alpha, which is a fixed value and is the communication reliability requirement of the active power distribution network. The constraint (10) represents a limit of the BS transmit power. The constraint (11) represents the CR requirement for the optimal operating model in the smart grid.
Step three: constructing an optimal operation model of an energy sharing mechanism according to a power distribution network, controllable loads and a communication base station power loss constraint model shared by distributed energy in a given area;
in the present invention, the Distribution System Operator (DSO) in a given area is responsible for optimal control of distributed energy, controllable loads and communication base station power consumption. And the DSO uniformly schedules all the MGs through an energy sharing scheme to optimize energy operation. Energy sharing requires that the MGs deviate from their respective optimal schedules to accommodate surplus or demand from neighboring piconets, but the overall cost may be minimized. Thus, the optimal operation model of the energy sharing mechanism is as follows:
wherein, c
iRepresents the distribution system operator operating cost, gamma, for a given area
sIs the probability of the scene s,
is the hour controllable load of the ith energy microgrid within the time slot t interval under the scene s,
is the load consumption of the ith' communication base station in the interval of time slot t under the scene s,
is the electricity price of the residents,
is the electricity price of the communication base station, U
i(. is) the utility function of the ith energy microgrid, P
ijstAnd j is 1,2, … …, where N is the number of all energy micro-grids in the scheduling range. In consideration of different pricing systems, the model takes the electricity prices of residents and BS (base station) as two different variables, can be reasonably set according to different regulations, does not lose generality, and applies a concave quadratic utility function.
The constraint conditions of the optimal operation model of the energy sharing mechanism are as follows:
Vmin≤Vist≤Vmax (18)
θmin≤θist≤θmax (19)
wherein,
is the charging power of the ith energy microgrid within the time slot t interval under the scene s,
is the discharge power of the ith energy microgrid within the time slot t interval under the scene s,
is the reactive power stored by the ith energy microgrid within the time slot t interval under the scene s,
is the ith energy micro-meter in the interval of time slot t under the scene sThe controllable load of the network is controlled by the load,
is the power, phi, of the photovoltaic system in the interval of time slot t under scene s
iIs a set of units, Q, connected to the ith energy microgrid
ijstIs the reactive power flow from the ith energy microgrid to the jth energy microgrid, g
ijIs the susceptance value of the line between the ith energy microgrid and the jth energy microgrid, b
ijIs the conductance value, v, of the line between the ith energy microgrid and the jth energy microgrid
istIs the voltage amplitude, v, of the ith energy microgrid
jstIs the voltage amplitude, theta, of the jth energy microgrid
istIs the voltage phase angle theta of the ith energy microgrid
jstIs the voltage phase angle, V, of the jth energy microgrid
minIs the lower bound of the voltage amplitude, V
maxIs the upper bound of the voltage amplitude, θ
minIs the lower bound of the phase angle of the voltage, theta
maxIs the upper bound of the phase angle of the voltage,
is a photovoltaic power generation predicted value of the ith energy microgrid in the time slot t interval,
is the lower bound of the controllable load,
is the upper bound of the controllable load,
is the predicted daily load requirement for the load,
is the maximum charging power that can be charged,
is the maximum discharge power of the discharge lamp,
is the maximum reactive power of the energy storage systemThe ratio of the total weight of the particles,
it is the efficiency of the energy storage system,
is the stored energy of the energy storage system,
is the upper limit of the capacity of the energy storage system,
is the lower limit of the capacity of the energy storage system,
is the initial energy level stored by the ith energy microgrid under the scene s,
is the final energy level, S, stored by the ith energy microgrid under the scene S
ijIndicating the apparent power from the ith to the jth energy microgrid.
Constraints (13) - (14) represent the power balance of the entire system. Constraints (15) - (17) and (18) - (19) account for line flow constraints and node phase angle and voltage constraints, respectively. The constraint (20) indicates that the photovoltaic power generation power of MGi is limited by the predicted value. At (21), the hour load of MGi is limited by a lower and an upper bound. Constraint (22) reveals the daily minimum load requirement of MGi. (26) And (27) limit the charge and discharge power, respectively. (26) And (27) using the constraints to give the energy balance equation and capacity limit of the ESS, respectively. The constraint (28) ensures that the energy stored in the end state is the same as in the initial state.
Step four: and carrying out linear solving on the communication reliability constraint model by adopting a least square method and a piecewise linear method to obtain model parameters of the reliability constraint model, and applying the model parameters to an optimal operation model of an energy sharing mechanism to obtain the minimum value of the overall cost of the power distribution network of the energy sharing mechanism.
For the optimal operating model, the variables are
And
the objective function is intended to determine the minimum of the sum of the load consumption costs of all BSs and all MGs. Under the condition of ensuring CR, the energy and the operation strategy of the BS can be optimized in the model. However, the bit error rate function in CR is an integral tail function of the standard gaussian distribution, which is a non-linear constraint. Since the solution of non-linear problems usually involves a high computational burden, the present invention seeks to reduce them to linear constraints that are easy to represent, i.e. to replace the non-linear constraints with its linearized equivalent constraint set.
To represent the constraints (11), a linear form needs to be found
The value ranges from 50 to 150, delta
n-iLess than or equal to 1. However, as an intermediate variable, SNR
n-iIs very high, from 0 to 10
7Are not equal. To obtain a better linear fit, it is split into two equations, δ
n-i(SNR
n-i) And
δ
n-i(SNR
n-i) Is a non-linear function that increases monotonically with respect to SNR. The equation is:
according to the equation (2),
is a non-linear function of n variables. Equations (2) and (29) are expressed as linear functions using the least squares method and the piecewise linear method (PWL), respectively.
For equation (2), there are n nonlinear equations to fit. Taking one as an example, the others can be obtained in a similar way.
S41, the signal-to-noise ratio of the ith energy microgrid accessing the nth communication base station is expressed by using a regression equation:
SNRn-i=WEC+ε (30)
wherein the SNRn-iIs a k x 1 dimensional random vector, W, determined from observationsEIs a kx (n +1) matrix determined by the predictor variables, C is the (n +1) x 1 vector of unknown parameters, and epsilon is the k x 1-dimensional vector of random errors;
in a matrix representation can be written:
in a first step, a vector of least squares estimates is determined
Giving linear combinations
The minimum error vector length is minimized. Basically, a vector is estimated
Characterization of
And SNR
n-iThe smallest possible value of the sum of variances. Variables of
Are linearly independent. Now, since the objective of multivariate regression is to minimize the sum of variances, the regression coefficients that satisfy this condition are determined by solving the least squares normal equation:
if variable
Is linearly independent, then (W)
E)
TW
EI.e., [ (W)
E)
TW
E]
-1Will be present.
S42, solving the step S41 by the least square normal equation, and multiplying the two sides of the equation (32) [ (W)E)TWE]-1Obtaining a regression coefficient:
wherein,
a vector, i.e., a regression coefficient, for least squares estimation;
to examine the linear fitting effect of the least square method, assuming that 27 base stations are uniformly distributed at random in an area with a diameter of 4 km, linear fitting of the signal-to-noise ratio is performed on the basis of B ═ 10MHz, and fig. 3 shows that
And approximation thereof
And
the gray curved surface is linear approximation, and the color curved surface is an original function.
S43, equation (29) is a one-dimensional nonlinear function. In order to apply the piecewise linearity method, a signal-to-noise ratio domain is divided into equal interval intervals, and in a q-th interval, a communication reliability constraint model is written as follows:
wherein, a
qAnd b
qAre all coefficients in the interval q,
representing the signal-to-noise ratio of the equal interval, and the SNR represents the signal-to-noise ratio;
applying piecewise linear method to equation (29), where BNand/R is 0.002898, and l is 125 bits. From the above results, the signal-to-noise ratio was in the range of 0 to 107In the meantime. The original function is shown in FIG. 4, with a constant length set to 106,aqAnd bqDetermined by break points at two ends of the interval. In fig. 4, the present invention obtains a set of linear functions as follows:
a.1 linearized Power flow model
In order to solve the mixed integer programming model, a linearized power distribution network power flow method is adopted.
Regardless of network losses, the power flow equation can be approximated in the following linear form:
wherein, bij、gijRespectively the susceptance value and the conductance value of the line between MGi and j. Pij、QijRepresenting the real and reactive power of the line between MGi and MGj, V being the square of the node voltage magnitude and θ being the voltage phase angle.
The line power constraint may be approximated as a piecewise linear form,
where N is the number of stages.
a.2 piecewise linear equation (32)
The set of linear functions is:
from all intervals and slopes of the equation, the constraint can be expressed as:
π
jas a linear function of the interval j. The function δ is again piecewise linearly represented using a method in section 5.2, where equations (35a) - (35d) avoid the use of the max-min operator. In addition, define t
1=max(π
2,π
3),t
2=max(π
1,t
1),t
3=min(π
6,t
2),t
4=min(π
5,t
3),
By this splitting method, t
5The values are the same in equation (33). Constraint conditions are as follows:
π2≤t1≤π2+v1Ω (A6)
π3≤t1≤π3+(1-v1)Ω (A7)
π1≤t2≤π1+v2Ω (A8)
t1≤t2≤t1+(1-v2)Ω (A9)
π6-v3Ω≤t3≤π6 (A10)
t2-(1-v3)Ω≤t3≤t2 (A11)
π5-v4Ω≤t4≤π5 (A12)
t3-(1-v4)Ω≤t4≤t3 (A13)
π4-v5Ω≤t5≤π4 (A14)
t4-(1-v5)Ω≤t5≤t4 (A15)
nonlinear constraint transformation t5≥α,t5δ. These 5 binary variables v1,v2,v3,v4v5There are 32 possible combinations. However, 16 of these 32 combinations result in infeasible constraints that will be captured by the MILP solver.
However, non-linear functions are constraints in the optimization problem. Therefore, the approximation of the non-linear function makes the problem unsmooth. Therefore, a method is introduced for range division before piecewise linear approximation.
S44, introducing new binary and continuous variables of q-1 and a new inequality of 4 (q-1) to rewrite the constraint conditions of the optimal operation model into linear constraints:
max{π1,π2}≥0, (35)
introducing new variable t ═ max { pi-1,π2The following new constraints are released from the maximum operator:
π1≤t≤π1+vΩ (36)
π2≤t≤π2+(1-v) (37)
s45, applying the method of the step S44 to a minimum operator: t is min { pi ═ n1,π2}:
π1-vΩ≤t≤π1 (38)
π2-(1-v)Ω≤t≤π2 (39)
Wherein v is a binary variable and Ω is a positive scalar;
s46, converting the nonlinear constraint of the reliability constraint model into a specific inequality according to the steps S41-S45, and obtaining model parameters of the reliability constraint model through solving the specific inequality.
The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents, improvements and the like that fall within the spirit and principle of the present invention are intended to be included therein.