Electric network analytical model order reduction method based on non-integer order singular perturbation

Abstract

The invention discloses an electric network analytical model order reduction method based on non-integer order singular perturbation, and relates to the field of electric networks, comprising the following steps: step 1, electric network modeling: performing split-phase modeling for an electric network to form a state space model containing three-phase line information; 2, solving fractional derivative analytic expressions of the fast state variables and the rapid state variables; and 3, simplifying the model. According to the method, 1.618-order derivatives and 1.309-order derivatives of two groups of variables which change rapidly and rapidly in the state space model are set to be 0, so that original differential equations corresponding to the 1.618-order derivatives and the 1.309-order derivatives are approximately replaced, the model of the system is fully simplified, and high precision is kept.

Description

Technical Field

[0001] The present invention relates to the field of electrical networks, and specifically to a method for reducing the order of an analytical model of an electrical network based on non-integer order singular perturbation. Background Art

[0002] The stability analysis of electrical networks plays a crucial role and can guide the topology and parameter design of the system to enhance its ability to resist external disturbances and maintain a stable state. The stability analysis of contemporary electrical networks is very difficult because a large number of power electronic devices are connected, and their non-linear characteristics make the system model exhibit high-order non-linear characteristics, exceeding the processing capabilities of current stability analysis methods. Therefore, it is necessary to effectively reduce the order of the electrical network model to simplify it to a range that can be processed by stability analysis methods and fully retain the accuracy of the model. Summary of the Invention

[0003] The purpose of the present invention is to provide a method for reducing the order of an analytical model of an electrical network based on non-integer order singular perturbation, taking into account both model accuracy and simplification degree, and providing an efficient and reliable basis for the quantitative analysis of the stability of electrical network systems, so as to solve the problems raised in the above background art.

[0004] To achieve the above purpose, the present invention provides the following technical solutions:

[0005] A method for reducing the order of an analytical model of an electrical network based on non-integer order singular perturbation, characterized by comprising the following steps:

[0006] Step 1: Electrical network modeling: Phase-by-phase modeling is performed on the electrical network to form a state space model containing three-phase line information, as shown in Equation (1):

[0007]

[0008] Where x = [x s1 ,..., x sm , x f1 ,..., x fn , x v1 ,..., x vk T , is a vector composed of state variables of the electrical network system;

[0009] x v1 ~x vk are the k rapid state variables of the electrical network system. When the electrical network system is disturbed and tends to be stable, the rapid state variables stabilize first and lead the electrical network system into a stable state;

[0010] x f1 ~x fn ​They are n fast state variables of the electrical network system, and they enter the stable state after the rapid variables are stable and before the electrical network system is stable;

[0011] x s1 ~x sm are other state variables excluding the rapid variables and the fast variables;

[0012] u = [u1, u2,..., u p T is the input vector of the electrical network system;

[0013] Step 2: Solve the analytical expressions of the fractional-order derivatives of the fast and rapid state variables

[0014] 2.1 Solve the analytical expression of the 1.382-order Riemann-Liouville derivative of the n fast state variables in Equation (1). Taking the i-th (1 ≤ i ≤ n) fast state variable as an example, its 1.382-order derivative is f fi (x, u) is differentiated with respect to time by 0.382 order:

[0015]

[0016] where

[0017] 2.2 Solve the analytical expression of the 1.618-order Riemann-Liouville derivative of the k rapid state variables in Equation (1). Taking the j-th (1 ≤ j ≤ k) rapid state variable as an example, its 1.618-order derivative is f vj (x, u) is differentiated with respect to time by 0.618 order:

[0018]

[0019] where

[0020] Step 3: Model simplification

[0021] Set the n 1.382-order derivatives and the k 1.618-order derivatives obtained in Steps 2.1 and 2.2 to zero to form the following algebraic equations:

[0022]

[0023] Solve the above equations to obtain the analytical expressions of the fast and rapid state variables;

[0024] Take the differential equations corresponding to the slow state variables in Equation (1):

[0025]

[0026] ​Substitute the analytical expressions of the fast and rapid state variables into it, and the original model of order m + n + k is reduced to an m-order model.

[0027] Compared with the prior art, the beneficial effects of the present invention are as follows: For two groups of variables with fast changes and rapid changes in the state space model of the present invention, 1.618 the 1.309 th derivative and the th derivative are set to 0 to approximately replace the original differential equations corresponding to them, so as to fully simplify the model of the system and retain a high accuracy.

[0028] The technical solutions in the embodiments of the present invention will be clearly and completely described below. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0029] In an embodiment of the present invention, a method for reducing the order of an analytical model of an electric network based on non-integer order singular perturbation is characterized by including the following steps:

[0030] Step 1: Electric network modeling: Perform phase-by-phase modeling on the electric network to form a state space model including three-phase line information, as shown in Equation (1):

[0031]

[0032] Where x = [x s1 ,..., x sm , x f1 ,..., x fn , x v1 ,..., x vk T , which is a vector composed of the state variables of the electric network system;

[0033] x v1 ~x vk are the k rapid state variables of the electric network system. When the electric network system tends to be stable after being disturbed, the rapid state variables stabilize first and lead the electric network system into a stable state;

[0034] x f1 ~x fn are the n fast state variables of the electric network system. They enter a stable state after the rapid variables are stable and before the electric network system is stable;

[0035] x s1 ~x sm are the other state variables excluding the rapid variables and the fast variables; ​

[0036] u = [u1, u2,..., u p T is the input vector of the electric network system;

[0037] Step 2: Solve the analytical expressions of the fractional derivatives of the fast and rapid state variables

[0038] 2.1 Solve the analytical expression of the 1.382-order Riemann-Liouville derivative of the n fast state variables in Equation (1). Taking the i-th (1 ≤ i ≤ n) fast state variable as an example, its 1.382-order derivative is f fi (x, u) is differentiated with respect to time by 0.382 order:

[0039]

[0040] where

[0041] 2.2 Solve the analytical expression of the 1.618-order Riemann-Liouville derivative of the k rapid state variables in Equation (1). Taking the j-th (1 ≤ j ≤ k) rapid state variable as an example, its 1.618-order derivative is f vj (x, u) is differentiated with respect to time by 0.618 order:

[0042]

[0043] where

[0044] Step 3: Model simplification

[0045] Set the n 1.382-order derivatives and k 1.618-order derivatives obtained in Steps 2.1 and 2.2 to zero, forming the following algebraic equations:

[0046]

[0047] Solve the above equations to obtain the analytical expressions of the fast and rapid state variables;

[0048] Take the differential equations corresponding to the slow state variables in Equation (1):

[0049]

[0050] Substitute the analytical expressions of the fast and rapid state variables into them, and the original m + n + k-order model is reduced to an m-order model.

[0051] ​For those skilled in the art, it is obvious that the present invention is not limited to the details of the above-described exemplary embodiments, and the present invention can be implemented in other specific forms without departing from the spirit or basic characteristics of the present invention. Therefore, from any point of view, the embodiments should be regarded as exemplary and non-limiting. The scope of the present invention is defined by the appended claims rather than the above description. Therefore, all changes falling within the meaning and scope of the equivalent elements of the claims are intended to be embraced within the present invention.

[0052] In addition, it should be understood that although this specification is described in terms of embodiments, not every embodiment only contains an independent technical solution. This narrative manner of the specification is only for clarity. Those skilled in the art should regard the specification as a whole, and the technical solutions in each embodiment can also be appropriately combined to form other embodiments that can be understood by those skilled in the art.

Claims

1. A method for reducing the order of an analytical model of an electric network based on non-integer order singular perturbation, characterized in that, It includes the following steps: Step 1: Electrical network modeling: Phase-separated modeling is performed on the electrical network to form a state-space model containing three-phase line information, as shown in Equation (1): where x = [x s1 ,..., x sm , x f1 ,..., x fn , x v1 ,..., x vk , ] T , is a vector composed of the state variables of the electrical network system; x v1 ~x vk are the k rapid state variables of the electrical network system. When the electrical network system tends to be stable after being disturbed, the rapid state variables stabilize first and lead the electrical network system into a stable state; x f1 ~x fn are the n fast state variables of the electrical network system, and they enter the stable state after the rapid variables are stable and before the electrical network system is stable; x s1 ~x sm Other state variables for removing rapid variables and fast variables; u = [u1, u2,..., u p T is the input vector of the electrical network system;​ Step 2: Solve the analytical expressions of the fractional-order derivatives of the fast and rapid state variables 2.1 Solve the analytical formula of the 1.382-order Riemann-Liouville derivative of the n fast state variables in formula (1). Taking the i-th (1 ≤ i ≤ n) fast state variable as an example, its 1.382-order derivative is f fi (x, u) is differentiated with respect to time by 0.382 order: Among them, 2.2 Solve the analytical formula of the 1.618-order Riemann-Liouville derivative of the k rapid state variables in formula (1). Taking the jth (1 ≤ j ≤ k) rapid state variable as an example, its 1.618-order derivative is f vj (x, u) is differentiated with respect to time by 0.618 order: Among them, Step 3: Model simplification Set the n 1.382-order derivatives and k 1.618-order derivatives obtained in Steps 2.1 and 2.2 to 0, forming the following algebraic equations: Solve the above equations to obtain the analytical expressions of the fast and rapid state variables; Take the differential equations corresponding to the slow state variables in Equation (1): Substitute the analytical expressions of the fast and rapid state variables into them, and the original m + n + k-order model is reduced to an m-order model.

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