CN116090157A - Polynomial approximation-based optimal power flow solving method of probability type transient stability constraint - Google Patents

Polynomial approximation-based optimal power flow solving method of probability type transient stability constraint Download PDF

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CN116090157A
CN116090157A CN202211267929.1A CN202211267929A CN116090157A CN 116090157 A CN116090157 A CN 116090157A CN 202211267929 A CN202211267929 A CN 202211267929A CN 116090157 A CN116090157 A CN 116090157A
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夏冰清
陈雨薇
戚海峰
杨文斌
陈晴
王霄鹤
杨林刚
施朝晖
郦洪柯
殷贵
徐晗
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PowerChina Huadong Engineering Corp Ltd
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Abstract

The invention provides an optimal power flow solving method based on a polynomial approximation probability type transient stability constraint, which comprises the following steps: s1, establishing a mathematical model of a probabilistic transient stability constrained optimal power flow (TSCOPE); s11, a general mathematical model of the probabilistic TSCOPF; s12, constructing probability type transient stability constraint based on a single machine equivalence method; s2, solving the optimal power flow of the probability type transient stability constraint based on the mathematical model established in the step S1 and a polynomial approximation method; s21, solving a polynomial approximation formula of a transient variable of the system; s22, converting the probability type inequality constraint into a deterministic inequality constraint to obtain an optimized mathematical model convenient to solve; s23, solving a solution of the optimal power flow of the probability type transient stability constraint based on an original dual interior point method, and solving the technical problem of complex solution of the probability type TSCOPF existing in the prior art.

Description

Polynomial approximation-based optimal power flow solving method of probability type transient stability constraint
Technical Field
The invention relates to the technical field of power systems, in particular to an optimal power flow solving method based on polynomial approximation and probability type transient stability constraint.
Background
The global exhaustion of fossil energy resources and their increasing concern over environmental impact has driven the rapid development of green and renewable energy sources in power systems, while the inherent volatility and intermittence of renewable energy sources lead to high uncertainties and also present new challenges to power systems. The impact of these uncertainties on system operation and control requires thorough analysis and investigation through probabilistic studies of static and transient safety. The optimal power flow (Transient Stability Constrained Optimal Power Flow, TSCOPF) of transient stability constraints is an effective tool to balance economic problems and stability requirements in power system operation. Given an incident that causes a transient instability in the system, TSCOPF aims to optimize the system operating state while preserving the ability to withstand the incident when it actually occurs. However, grid-connection of large-scale renewable energy sources exacerbates the uncertainty of power system operation, requiring the effects of random parameters to be considered in the solution of TSCOPF. According to the difference of the random parameter processing modes in the TSCOPF model, two main methods exist at present to solve the problem: 1) Robust TSCOPF is widely used because it does not require knowledge of probability distribution information of uncertain parameters. Robust optimization limits the values of all uncertain parameters to their range of variation and converts the problem to a deterministic optimization problem to meet the needs of all extreme scenarios. Robust optimization has been widely used to handle uncertain parameters in economic dispatch, set configuration for security constraints, and coordinated energy and reserve dispatch for multiple micro-grids. However, while robust optimization can effectively handle uncertain parameters, the optimization results tend to be very conservative due to the omission of distribution information of uncertain parameters. 2) The probabilistic TSCOPF assumes that the uncertainty parameter obeys a certain deterministic probability distribution. The method takes the power generation cost and the like as the expected value of an objective function to be minimized as an optimization target, and establishes a random probability model meeting certain probability level constraint. The research shows that the strategy derived from the probability planning can optimize the expectation of the objective function under the uncertainty, and the solution of the obtained probability planning can obtain the optimal control scheme taking the economy of the control scheme and the transient stability of the system into consideration on the premise of knowing the probability distribution of renewable energy sources. The invention provides an optimal power flow solving method based on a probability type transient stability constraint of polynomial approximation on the basis of a probability type TSCOPF.
Disclosure of Invention
The invention aims to provide an optimal power flow solving method based on a polynomial approximation probability type transient stability constraint, which aims to solve the technical problem of complex solving of probability type TSCOPF existing in the prior art.
The invention provides an optimal power flow solving method based on a polynomial approximation probability type transient stability constraint, which comprises the following steps: s1, establishing a mathematical model of an optimal power flow constrained by probability type transient stability; s11, a general mathematical model of the optimal power flow constrained by the probability type transient stability; s12, constructing a probability type transient stability constraint based on a single machine equivalence method; s2, solving the optimal power flow of the probability type transient stability constraint based on the mathematical model established in the step S1 and a polynomial approximation method; s21, solving a polynomial approximation formula of a transient variable of the system; s22, converting the probability type inequality constraint into a deterministic inequality constraint to obtain an optimized mathematical model convenient to solve; s23, solving a solution of the optimal power flow of the probability type transient stability constraint based on a primary dual interior point method.
Further, the general mathematical model in S11 includes:
min E{c(x 0 ,y 0 ,u,p)} (1.1)
probability type static stability constraint:
0＝g(x 0 ,y 0 ,u,p) (1.2)
0＝f(x 0 ,y 0 ,u,p) (1.3)
P{H 0 (x 0 ,y 0 ,u,p)≤H 0.max }≥β 0 (1.4)
transient constraint:
0＝g(x(t),y(t),u,p),t∈(t 0 ,t end ] (1.5)
Figure BDA0003893821230000021
probability type transient stability constraint:
P{H(x(t),y(t))≤H max }≥β,t∈(t 0 ,t end ] (1.7)
wherein x is a state variable related to transient state dynamics of the system, y is an algebraic variable comprising node voltage and injection current, u is a control variable with upper and lower limits, p is a random parameter, x (t), y (t) is the state variable of the system and the algebraic variable are in transient simulation time domain tE (t) 0 ,t end ]Values of x 0 ,y 0 For them at t=t 0 The initial value of the moment, E {.cndot. }, is the desired operator, c (x 0 ,y 0 U, p) is the objective function, g (x) 0 ,y 0 U, p) is a function vector representing a mathematical model of the system steady state, f (x) 0 ,y 0 ,u,P) is a function vector representing a transient mathematical model of the system, P {. Cndot. } is a probability operator, H 0 (x 0 ,y 0 U, p) and H (x (t), y (t)) are function vectors representing the static stability constraint and the transient stability constraint of the system, respectively, H 0.max And H max For its upper limit value, beta 0 And β is the lower limit that the corresponding probability stabilization constraint needs to satisfy.
Further, the construction of the probabilistic transient stability constraint (1.7) based on the stand-alone equivalence method in S12 includes:
the transient stability margin is:
η＝A dec -A acc ＝-M(ω(T u )) 2 /2 (1.12)
wherein eta is transient stability margin, A dec For decelerating area A acc For accelerating area, M is mechanical inertia, ω is relative rotor angle of the generator, T u The time when the electromagnetic power of the equivalent unit is just equal to the mechanical power of the equivalent unit in the transient process;
based on the above, the probabilistic transient stability constraint is structured as:
P{η＞0}≥β (1.13)。
further, the polynomial approximation of the system transient variable in S21 is:
Figure BDA0003893821230000031
wherein p' = [ u ] T ,p T ] T Is a compact form of control parameters and random parameters,
Figure BDA0003893821230000032
approximation, N, of system state variable x and algebraic variable y, respectively b Is the number of polynomial basis functions, +.>
Figure BDA0003893821230000033
And->
Figure BDA0003893821230000034
Is the time-varying approximation coefficient corresponding to the kth polynomial basis function, k is the counting variable, and phi (p) is the polynomial basis function.
Further, the converting the probabilistic inequality constraint into the deterministic inequality constraint in S22 includes:
based on Gram-Charlier expansion, uncertainty of random parameters in inequality constraint is quantified, and a series of configuration points u of control parameters are obtained according to a sparse grid method (m) M=1,..m, for each configuration point u (m) Calculating H (x, y) is less than or equal to H based on Gram-Charlier expansion max Probability of (2):
Figure BDA0003893821230000035
in the method, in the process of the invention,
Figure BDA0003893821230000041
mu and sigma are the mean and standard deviation of the functions H (x, y), N (x) is the normal distribution function, K i A cumulative amount derived from each moment of H (x, y);
obtaining a quantitative function relation of probability of line power flow meeting safety constraint along with control parameter change:
Figure BDA0003893821230000042
in the method, in the process of the invention,
Figure BDA0003893821230000043
is approximate probability, coefficient->
Figure BDA0003893821230000044
The solution can be found by:
Figure BDA0003893821230000045
the probabilistic inequality constraint translates into a deterministic inequality constraint:
Figure BDA0003893821230000046
further, the optimizing mathematical model in S22 includes:
min E{c(x 0 ,y 0 ,u,p)} (2.7)
static stability constraints:
0＝g(x 0 ,y 0 ,u,p) (2.8)
0＝f(x 0 ,y 0 ,u,p) (2.9)
Figure BDA0003893821230000047
transient constraint:
Figure BDA0003893821230000048
Figure BDA0003893821230000049
transient stability constraints:
Figure BDA0003893821230000051
in order to achieve the above object, according to a second aspect of the present invention, the present invention adopts the following technical scheme:
an electronic device comprises a memory, a processor and a computer program stored on the memory and capable of running on the processor, wherein the processor realizes the steps of the optimal power flow solving method based on the probability type transient stability constraint of polynomial approximation when executing the computer program.
In order to achieve the above object, according to a third aspect of the present invention, the following technical solutions are adopted:
a non-transitory computer readable storage medium having stored thereon a computer program which, when executed by a processor, implements the above-described polynomial approximation based probabilistic transient stability constraint optimal power flow solving method steps.
According to the optimal power flow solving method based on the probability type transient stability constraint of polynomial approximation, transient equation constraint in a differential algebra equation form which is difficult to process in an optimization model is converted into equation constraint in a group of simple polynomial form, uncertainty of random parameters in a probability inequality constraint formula is quantized, the uncertainty is converted into inequality constraint of certainty, solving of probability type TSCOPF is simplified, and the obtained solution of probability type TSCOPF can obtain an optimal control scheme which simultaneously considers economy of a control scheme and transient stability of a system on the premise of knowing probability distribution of renewable energy sources, solves the technical problem of complex solution of the probability type TSCOPF existing in the prior art, has higher calculation efficiency, and can provide a scientific and reasonable analysis scheme for safe transient stability analysis and control of a power system.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings that are needed in the description of the embodiments or the prior art will be briefly described, and it is obvious that the drawings in the description below are some embodiments of the present invention, and other drawings can be obtained according to the drawings without inventive effort for a person skilled in the art.
Fig. 1 is a topology diagram of an IEEE145 node system provided in this embodiment, where numbers in rectangular boxes in the diagram represent numbers of corresponding nodes, and numbers on lines represent the number of connection lines;
FIG. 2 is a plot of expected and standard deviation of generator rotor angle after IEEE145 node system failure provided in this embodiment;
fig. 3 is a graph of the maximum value of the generator rotor angle at node 104 as a function of control scheme cost provided by the present embodiment.
Detailed Description
The technical solutions of the present invention will be clearly and completely described in connection with the embodiments, and it is apparent that the described embodiments are some embodiments of the present invention, but not all embodiments. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
The embodiment provides an optimal power flow solving method based on a polynomial approximation probability type transient stability constraint. The present embodiment takes an IEEE145 node system power system as an example. To verify the effectiveness of the proposed method, wind farms were added at nodes 101, 105 and 112, with a range of [0, 150] MW of wind farm power generation as a function of the environment. The topology and detailed data of the system can be obtained in fig. 1. The controllable parameters in this embodiment are the output of the synchronous generators at nodes 89, 98, 100 and 104.
The optimal power flow solving method based on the probability type transient stability constraint of polynomial approximation provided by the embodiment comprises the following steps:
s1, establishing a mathematical model of a probabilistic transient stability constrained optimal power flow (TSCOPE).
S11, general mathematical model of probabilistic TSCOPF.
In order to adjust the control variables to improve the transient stability of the power system and to have a satisfactory safety level, the probabilistic TSCOPF can be expressed as a nonlinear programming problem in the form of differential algebra:
min E{c(x 0 ,y 0 u, p) } (1.1) probabilistic static stability constraint:
0＝g(x 0 ,y 0 ,u,p) (1.2)
0＝f(x 0 ,y 0 ,u,p) (1.3)
P{H 0 (x 0 ,y 0 ,u,p)≤H 0.max }≥β 0 (1.4)
transient constraint:
0＝g(x(t),y(t),u,p),t∈(t 0 ,t end ] (1.5)
Figure BDA0003893821230000071
probability type transient stability constraint:
P{H(x(t),y(t))≤H max }≥β,t∈(t 0 ,t end ] (1.7)
wherein x is a state variable related to transient dynamics of the system, such as generator rotor angle, adjustment of exciter and prime mover, etc.; y is algebraic variable including node voltage and injection current; u is a control variable with upper and lower limits, such as the active output of the generator and the terminal voltage; p is a random parameter; x (t), y (t) is the system state variable and algebraic variable in transient simulation time domain t E (t) 0 ,t end ]Values of x 0 ,y 0 For them at t=t 0 Initial value of time. E {.cndot. } is the desired operator, c (x 0 ,y 0 U, p) is the objective function, g (x) 0 ,y 0 U, p) is a function vector representing a mathematical model of the system steady state, f (x) 0 ,y 0 U, p) is a function vector representing a transient mathematical model of the system; p {.cndot. } is a probabilistic operator, H 0 (x 0 ,y 0 U, p) and H (x (t), y (t)) are function vectors representing the static stability constraint and the transient stability constraint of the system, respectively, H 0.max And H max Is the upper limit value thereof; beta 0 And β is the lower limit that the corresponding probability stabilization constraint needs to satisfy.
In this embodiment, the probabilistic TSCOPF can be expressed as a nonlinear programming problem in the form of differential algebra:
Figure BDA0003893821230000072
static stability constraints:
Figure BDA0003893821230000073
Figure BDA0003893821230000074
Figure BDA0003893821230000075
P{V i min ≤V≤V i max }≥β,i∈S B
Figure BDA0003893821230000076
transient stability constraints:
0＝g(x(t),y(t),u,p),t∈(t 0 ,t end ]
Figure BDA0003893821230000081
P{H(x(t),y(t))≤H max }≥β,t∈(t 0 ,t end ]
wherein E {.cndot. } is the desired operator, N G Is a set of adjustable generators; c 2,i ，c 1,i ，c 0,i The correlation coefficient of the power generation cost of the node i, P Gi Active power of the generator of the node i; s is S B For node collection, P Gi Active power, Q, generated by a conventional power supply for node i Ri Reactive power sent by various reactive power sources of the node i; p (P) Li And Q Li Active power and reactive power of the node i load respectively; v (V) i The voltage amplitude of the node i; v (V) j The voltage amplitude at node j; y is Y ij Is a node admittance matrix element. P {.cndot }' is the probabilistic operator,
Figure BDA0003893821230000082
for unit i minimum and maximumActive force, < >>
Figure BDA0003893821230000083
For minimum and maximum reactive output of the machine set, V i min ，V i max For node i minimum and maximum voltage amplitude, +.>
Figure BDA0003893821230000084
For minimum and maximum power transmissible by branch i, S l Is the set of all branches. x is a state variable related to transient dynamics of the system, such as generator rotor angle, adjustment of an exciter and a prime motor, and the like; y is algebraic variable including node voltage and injection current; u is a control variable with upper and lower limits, such as the active output of the generator and the terminal voltage; p is a random parameter; x (t), y (t) is the system state variable and algebraic variable in transient simulation time domain t E (t) 0 ,t end ]Values of x 0 ,y 0 For them at t=t 0 Initial value of time. g (x) 0 ,y 0 U, p) is a function vector representing a mathematical model of the system steady state, f (x) 0 ,y 0 U, p) is a function vector representing a transient mathematical model of the system; h (x (t), y (t)) is a function vector representing the transient stability constraint of the system, H max Is the upper limit value thereof; beta is the lower limit that the corresponding probability stability constraint needs to satisfy.
S12, constructing a probability type transient stability constraint based on a single machine equivalence method.
The single machine equivalence method is an effective algorithm for transient stability analysis of a multi-machine system. Based on the time domain simulation result, the system stability in the single-machine equivalent frame is observed by checking the states of the equivalent critical unit and the non-critical unit, and then the stability margin is determined according to the equal area criterion. The single-machine equivalent method is widely used for transient stability evaluation and TSCOPF (transient stability coefficient of performance) problems in an actual system, so that the probability-type transient stability constraint is constructed by adopting the single-machine equivalent method. According to a single-machine equivalent method, the electromagnetic power P of an equivalent single-machine infinite system e And mechanical power P m The method comprises the following steps:
Figure BDA0003893821230000091
Figure BDA0003893821230000092
wherein, the liquid crystal display device comprises a liquid crystal display device,
Figure BDA0003893821230000093
M CM is equivalent mechanical inertia of critical unit, M NM Is equivalent mechanical inertia of a non-critical unit, M is mechanical inertia of the equivalent unit, and P ei Electromagnetic power of the ith generator, P mi Mechanical power of the ith generator; subscripts CM and NM correspond to critical units and non-critical units respectively, M i Is the mechanical inertia of the ith generator. Therefore, the angular velocity and the rotor angle of the equivalent single-machine infinity system are respectively:
Figure BDA0003893821230000094
wherein ω and δ are the angular velocity and rotor angle, ω, respectively, of an equivalent stand-alone infinity system i For angular velocity, delta, of generator i i Is the rotor angle of the generator i.
According to the equal area rule, at T u At the moment, if the electromagnetic power P e And mechanical power P m The original power system is transient unstable if the following conditions are satisfied:
Figure BDA0003893821230000095
wherein P is a Is the acceleration power of an equivalent single machine infinite system, T u The electromagnetic power of the equivalent unit is just equal to the moment of the mechanical power of the equivalent unit in the transient process.
The transient stability margin may be defined as the difference between the deceleration area and the acceleration area as follows:
Figure BDA0003893821230000096
wherein eta is transient stability margin, A dec For decelerating area A acc For acceleration area, M is the mechanical inertia and ω is the relative rotor angle of the generator. Based on this, the probabilistic transient stability constraint may be constructed as follows:
P{η＞0}≥β (1.13)
the expected rotor angle of a part of the synchronous generator and its standard deviation are plotted over time during a post-fault transient when no control measures are applied as shown in fig. 2. As can be seen from fig. 2, the maximum value of the desired trajectory of the rotor angle of the generator at node 104 is 136.152 ° and the standard deviation is 23.653 °. From chebyshev, it is known that the maximum value of the rotor angle of the generator at node 104 is 254.417 deg., much greater than the critical value 180 deg., under the influence of random parameters. Therefore, it is necessary to solve the probabilistic TSCOPF to obtain an economic scheduling scheme that can improve the transient stability of the system.
S2, solving the optimal power flow of the probability type transient stability constraint based on the mathematical model established in the step S1 and the polynomial approximation method.
S21, solving a polynomial approximation formula of the transient state variable of the system.
The present embodiment will construct a polynomial approximation describing the quantitative relationship of system state variables with random parameters and control parameters based on the point-matching method. Firstly, based on the power system transient constraint formulas (1.5) - (1.6), the relation between the state variable and algebraic variable in the transient process of the system, the control parameter and the random parameter can be solved. The system variables can be expressed as a linear combination of polynomial basis functions { Φ (p) } with respect to parameters as follows:
Figure BDA0003893821230000101
wherein p' = [ u ] T ,p T ] T Is a compact form of control parameters and random parameters,
Figure BDA0003893821230000102
approximation, N, of system state variable x and algebraic variable y, respectively b Is the number of polynomial basis functions, +.>
Figure BDA0003893821230000103
And->
Figure BDA0003893821230000104
Is the time-varying approximation coefficient corresponding to the kth polynomial basis function, k is the counting variable, and phi (p) is the polynomial basis function.
Then, in order to solve the polynomial approximation coefficients, a point matching method can be adopted, and the coefficients are to be determined
Figure BDA0003893821230000105
The solution can be found by:
Figure BDA0003893821230000111
wherein χ is k ＝∫ D Φ k (p)Φ k (p) ω (p) dp is the modulus of the polynomial basis function, D is the domain of p, ω (p) is the probability density function of p; p is p (m) M=1,..m is a configuration point (also called integration point), y (p (m) ) Is the configuration point p (m) Sampled values of system state variables, alpha m Is p (m) Integral coefficient of (a), configuration point and corresponding integral coefficient alpha m Is determined by a sparse grid method. It should be noted that when the polynomial basis function is selected, χ k ，α m ，p (m) Has been designed in advance in the dot placement method, so
Figure BDA0003893821230000114
Will be subject to the sampling result x (t, p) (m) )，y(t,p (m) ) Is a function of (a) and (b). The undetermined coefficient +.in the formula (2.1) can be solved by the formula (2.2)>
Figure BDA0003893821230000115
And further obtaining a polynomial approximation formula describing quantitative relation between the line power flow and the random parameter and the control parameter.
S22, converting the probability inequality constraint into a deterministic inequality constraint to obtain an optimized mathematical model convenient to solve.
The embodiment provides a safe domain solving method of line power flow considering random parameter influence based on a point allocation method and Gram-Charlier expansion, which comprises the following specific processes:
first, based on Gram-Charlier expansion, the uncertainty of the random parameters in the inequality constraint is quantified. Obtaining a series of configuration points u of control parameters according to a sparse grid method (m) M=1, M; for each configuration point u (m) Calculating H (x, y) is less than or equal to H based on Gram-Charlier expansion max Probability of (2):
Figure BDA0003893821230000112
wherein, the liquid crystal display device comprises a liquid crystal display device,
Figure BDA0003893821230000113
mu and sigma are the mean and standard deviation of the functions H (x, y), N (x) is the normal distribution function, K i The cumulative amount is derived from the moments of each order of H (x, y).
Then, based on a polynomial approximation method, the quantitative relation of probability of meeting safety constraint of line power flow along with control parameters can be constructed as the following polynomial approximation:
Figure BDA0003893821230000121
wherein, the liquid crystal display device comprises a liquid crystal display device,
Figure BDA0003893821230000122
is approximate probability, coefficient->
Figure BDA0003893821230000123
The solution can be found by:
Figure BDA0003893821230000124
the probabilistic inequality constraint that considers the influence of the random parameters can be converted into the deterministic inequality constraint as follows:
Figure BDA0003893821230000125
the influence of random parameters in the original probabilistic TSCOPF mathematical model is quantified by using polynomial approximation and Gram-Charlier expansion, and the original random TSCOPF mathematical model is converted into an optimized mathematical model which is convenient to solve as follows:
min E{c(x 0 ,y 0 ,u,p)} (2.7)
static stability constraints:
0＝g(x 0 ,y 0 ,u,p) (2.8)
0＝f(x 0 ,y 0 ,u,p) (2.9)
Figure BDA0003893821230000126
transient constraint:
Figure BDA0003893821230000127
Figure BDA0003893821230000128
transient stability constraints:
Figure BDA0003893821230000129
s23, solving a solution of the optimal power flow of the probability type transient stability constraint based on a primary dual interior point method.
Deterministic optimization model types (2.7) - (2.13) are a group of nonlinear programming models, and can be solved by adopting a primary dual interior point method, and the obtained optimal solution is the approximate optimal solution of the primary probability type TSCOPF. The reconstructed probabilistic TSCOPF optimization mathematical model is solved by using the original dual interior point method, and the obtained optimization result is shown in figure 3. In fig. 3, the abscissa represents the objective function value of the optimized random wind power output, and the ordinate represents the maximum value of the transient rotor angle corresponding to the objective function value. Since the magnitude of the objective function characterizes the magnitude of the control cost, the condition of the change relation between the objective function value and the maximum value of the transient rotor angle shows that the higher the control cost is, the larger the maximum value of the transient rotor angle of the system is, and the more stable the transient state of the system is. From the results of the transient rotor angle maxima, it can be seen that, when the optimal control scheme is implemented on the power system, using the probabilistic TSCOPF, the maximum rotor angle delta that may occur during the post-fault transient of the generator at node 104 104.max 144 deg., below the threshold of 180 deg.. On one hand, the obtained result verifies the relation of the transient stability of the system along with the improvement of the control cost; on the other hand, after the proposed method is used, no matter how random parameters change in the definition domain, all the optimized control schemes can always keep the transient stability of the system, and the effectiveness of the proposed method is verified.
The invention also provides a non-transitory computer readable storage medium having stored thereon a computer program which, when executed by a processor, implements the method steps of solving an optimal power flow based on a probabilistic transient stability constraint of polynomial approximation as described hereinbefore.
The invention also provides an electronic device comprising a memory, a processor and a computer program stored on the memory and executable on the processor, the processor implementing the steps of the method for solving an optimal power flow based on a probabilistic transient stability constraint of polynomial approximation as described above when executing the computer program.
The computer readable storage medium may be any available medium or data storage device that can be accessed by a processor in an electronic device, including but not limited to magnetic memories such as floppy disks, hard disks, magnetic tapes, magneto-optical disks (MO), etc., optical memories such as CD, DVD, BD, HVD, etc., and semiconductor memories such as ROM, EPROM, EEPROM, nonvolatile memories (NAND FLASH), solid State Disks (SSD), etc.
The present invention is described with reference to flowchart illustrations and/or block diagrams of methods, apparatus (systems) and computer program products according to embodiments of the invention. It will be understood that each flow and/or block of the flowchart illustrations and/or block diagrams, and combinations of flows and/or blocks in the flowchart illustrations and/or block diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to a processor of a general purpose computer, special purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be stored in a computer-readable memory that can direct a computer or other programmable data processing apparatus to function in a particular manner, such that the instructions stored in the computer-readable memory produce an article of manufacture including instruction means which implement the function specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide steps for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.
While preferred embodiments of the present invention have been described, additional variations and modifications in those embodiments may occur to those skilled in the art once they learn of the basic inventive concepts. It is therefore intended that the following claims be interpreted as including the preferred embodiments and all such alterations and modifications as fall within the scope of the invention.
The above detailed description is intended to illustrate the present invention by way of example only and not to limit the invention to the particular embodiments disclosed, but to limit the invention to the precise embodiments disclosed, and any modifications, equivalents, improvements, etc. that fall within the spirit and scope of the invention as defined by the appended claims.

Claims (8)

1. The optimal power flow solving method based on the probability type transient stability constraint of polynomial approximation is characterized by comprising the following steps of:
s1, establishing a mathematical model of an optimal power flow constrained by probability type transient stability;
s11, a general mathematical model of the optimal power flow constrained by the probability type transient stability;
s12, constructing a probability type transient stability constraint based on a single machine equivalence method;
s2, solving the optimal power flow of the probability type transient stability constraint based on the mathematical model established in the step S1 and a polynomial approximation method;
s21, solving a polynomial approximation formula of a transient variable of the system;
s22, converting the probability type inequality constraint into a deterministic inequality constraint to obtain an optimized mathematical model convenient to solve;
s23, solving a solution of the optimal power flow of the probability type transient stability constraint based on a primary dual interior point method.
2. The method for solving the optimal power flow based on the probability type transient stability constraint of polynomial approximation according to claim 1, wherein the general mathematical model in S11 comprises:
min E{c(x 0 ,y 0 ,u,p)} (1.1)
probability type static stability constraint:
0＝g(x 0 ,y 0 ,u,p) (1.2)
0＝f(x 0 ,y 0 ,u,p) (1.3)
P{H 0 (x 0 ,y 0 ,u,p)≤H 0.max }≥β 0 (1.4)
transient constraint:
0＝g(x(t),y(t),u,p),t∈(t 0 ,t end ] (1.5)
Figure FDA0003893821220000011
probability type transient stability constraint:
P{H(x(t),y(t))≤H max }≥β,t∈(t 0 ,t end ] (1.7)
wherein x is a state variable related to transient state dynamics of the system, y is an algebraic variable comprising node voltage and injection current, u is a control variable with upper and lower limits, p is a random parameter, x (t), y (t) is the state variable of the system and the algebraic variable are in transient simulation time domain tE (t) 0 ,t end ]Values of x 0 ,y 0 For them at t=t 0 The initial value of the moment, E {.cndot. }, is the desired operator, c (x 0 ,y 0 U, p) is the objective function, g (x) 0 ,y 0 U, p) is a function vector representing a mathematical model of the system steady state, f (x) 0 ,y 0 U, P) is a function vector representing a transient mathematical model of the system, P {.cndot. } is a probability operator, H 0 (x 0 ,y 0 U, p) and H (x (t), y (t)) are function vectors representing the static stability constraint and the transient stability constraint of the system, respectively, H 0.max And H max For its upper limit value, beta 0 And β is the lower limit that the corresponding probability stabilization constraint needs to satisfy.
3. The method for solving the optimal power flow based on the probability-based transient stability constraint of polynomial approximation according to claim 1, wherein the construction of the probability-based transient stability constraint (1.7) based on the stand-alone equivalence method in S12 comprises:
the transient stability margin is:
η＝A dec -A acc ＝-M(ω(T u )) 2 /2 (1.12)
wherein eta is transient stability margin, A dec For decelerating area A acc For accelerating area, M is mechanical inertia, ω is relative rotor angle of the generator, T u The time when the electromagnetic power of the equivalent unit is just equal to the mechanical power of the equivalent unit in the transient process;
based on the above, the probabilistic transient stability constraint is structured as:
P{η＞0}≥β (1.13)。
4. the method for solving the optimal power flow based on the probability type transient stability constraint of polynomial approximation according to claim 1, wherein the polynomial approximation of the system transient variable in S21 is:
Figure FDA0003893821220000021
wherein p' = [ u ] T ,p T ] T Is a compact form of control parameters and random parameters,
Figure FDA0003893821220000022
approximation, N, of system state variable x and algebraic variable y, respectively b Is the number of polynomial basis functions, +.>
Figure FDA0003893821220000023
And->
Figure FDA0003893821220000024
Is the time-varying approximation coefficient corresponding to the kth polynomial basis function, k is the counting variable, and phi (p) is the polynomial basis function.
5. The method according to claim 1, wherein the converting the probabilistic inequality constraint into the deterministic inequality constraint in S22 comprises:
based on Gram-Charlier expansion, uncertainty of random parameters in inequality constraint is quantified, and a series of configuration points u of control parameters are obtained according to a sparse grid method (m) M=1,..m, for each configuration point u (m) Calculating H (x, y) is less than or equal to H based on Gram-Charlier expansion max Probability of (2):
Figure FDA0003893821220000031
in the method, in the process of the invention,
Figure FDA0003893821220000032
mu and sigma are the mean and standard deviation of the functions H (x, y), N (x) is the normal distribution function, K i A cumulative amount derived from each moment of H (x, y);
obtaining a quantitative function relation of probability of line power flow meeting safety constraint along with control parameter change:
Figure FDA0003893821220000033
in the method, in the process of the invention,
Figure FDA0003893821220000034
is approximate probability, coefficient->
Figure FDA0003893821220000035
The solution can be found by:
Figure FDA0003893821220000036
the probabilistic inequality constraint translates into a deterministic inequality constraint:
Figure FDA0003893821220000037
6. the method of claim 5, wherein the optimizing mathematical model in S22 comprises:
min E{c(x 0 ,y 0 ,u,p)} (2.7)
static stability constraints:
0＝g(x 0 ,y 0 ,u,p) (2.8)
0＝f(x 0 ,y 0 ,u,p) (2.9)
Figure FDA0003893821220000038
transient constraint:
Figure FDA0003893821220000041
Figure FDA0003893821220000042
transient stability constraints:
Figure FDA0003893821220000043
7. an electronic device comprising a memory, a processor and a computer program stored on the memory and executable on the processor, characterized in that the processor implements the polynomial approximation based probabilistic transient stability constraint optimal power flow solving method steps of any one of claims 1-6 when the computer program is executed.
8. A non-transitory computer readable storage medium, having stored thereon a computer program, which when executed by a processor, implements the polynomial approximation based probabilistic transient stability constraint optimal power flow solving method steps of any of claims 1-6.
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