State-space equation discretization method based on capacitance and inductance element characteristics

Abstract

The application relates to the field of circuit state space equation solving technology in the real-time simulation field, and discloses a state space equation discretization method for capacitor and inductor element characteristics, which comprises the following steps: A) establishing a state space differential equation of a circuit; B) establishing a characteristic equation of a capacitor and an inductor element, determining a numerical integration method used by each capacitor and inductor element in the circuit, and obtaining a discrete time domain characteristic equation by using the numerical integration method; C) combining the discrete time domain characteristic equation of step B) and the state space differential equation of step A) to obtain a state space discrete equation of the circuit; and D) cyclic simulation. The state space equation discretization method for capacitor and inductor element characteristics improves the speed and precision of real-time simulation of a circuit system by independently discretizing the characteristic equations of each capacitor and inductor element and optimizing the discretization process of the state space equation.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of circuit state space equation solving in real-time simulation, and particularly relates to a state space equation discretization method based on characteristics of capacitive and inductive elements. BACKGROUND

[0002] With the continuous expansion of the scale of power systems and the adoption of new power electronic devices, the dynamic characteristics of the system become gradually complex. Unlike traditional power equipment such as generators and motors, power electronic equipment has the characteristics of high frequency and rapid response. The system with the introduction of power electronic equipment also presents the characteristics of containing numerous elements and complex dynamic characteristics, which brings new challenges to the research and testing of power systems.

[0003] Real-time simulation can study the electromagnetic transient behavior characteristics of the system, facilitate the design of system control methods, and replace physical experiments through hardware-in-the-loop testing to reduce research costs, development cycles and experimental risks. Therefore, real-time simulation is an important tool for studying power systems with power electronic devices. In this context, power electronic real-time simulation has attracted widespread attention from scholars.

[0004] Based on the current situation of the gradually complex system, considering that the time step of real-time simulation is a fixed value, the optimization goals of power electronic real-time simulation focus on simulation speed and calculation accuracy, mainly including: reducing the amount of calculation, improving the speed of real-time simulation; improving the accuracy of real-time simulation under the premise of ensuring the real-time of simulation.

[0005] To achieve the above goals, one of the key means to solve is to optimize the discretization process of system equations, reduce the amount of calculation, and improve the calculation accuracy.

[0006] Taking the trapezoidal method as an example, the traditional state space equation discretization method focuses on directly discretizing the state space equation as a whole by using numerical integration method, ignoring the characteristics of different capacitive and inductive elements in the circuit, and has the disadvantages of large amount of calculation and inflexible use of numerical integration method. SUMMARY

[0007] The purpose of the present application is to overcome the above technical problems, and provide a state space equation discretization method based on characteristics of capacitive and inductive elements, which improves the speed and accuracy of real-time simulation of circuit systems by independently discretizing the characteristic equations of each capacitive and inductive element and optimizing the discretization process of state space equations.

[0008] To achieve the above purpose, the state space equation discretization method based on characteristics of capacitive and inductive elements designed by the present application comprises the following steps:

[0009] A) establishing the state space differential equation of the circuit;

[0010] B) establishing characteristic equations of the capacitors and inductors, determining the numerical integration method used by each capacitor and inductor in the circuit, and using the numerical integration method to obtain the characteristic equations of the discrete time domain;

[0011] C) combining the discrete time domain characteristic equations of step B) and the state space differential equations of step A) to obtain the state space discrete equations of the circuit;

[0012] D) loop simulation.

[0013] Preferably, in step C), the relationship equations of the non-state quantities, state quantities and inputs are established according to the characteristic equations of the capacitors and inductors, the relationship equations of the non-state quantities of the capacitors, the non-state quantities of the inductors, the state quantities of the capacitors, the state quantities of the inductors and the history sources are established according to the solutions of step B), and then the relationship equations of the state quantities and the history sources, inputs are established, and finally the state space discrete equations based on the characteristics of the capacitors and inductors are established.

[0014] Preferably, in step A), the state space differential equations of the circuit are:

[0015]

[0016] In the formula, A and B are state space matrices, t is the time, X is the state quantity, which includes the current i flowing through the inductor and the voltage u across the capacitor, L C , U is the system input.

[0017] Preferably, in step B), the characteristic equations of the capacitors and inductors are respectively:

[0018]

[0019] In the formula, i C represents the current flowing through the capacitor, u L represents the voltage across the inductor, the characteristic equation of the capacitor is solved by using the trapezoidal method, and the following is obtained:

[0020]

[0021] In the formula, △t is the simulation step, the characteristic equation of the inductor is solved by using the 3rd order Adams-Moulton method, and the following is obtained:

[0022]

[0023] After calculation, the following is obtained:

[0024] In the formula, ​​

[0025] Preferably, in the step C), define i C , u L , the non-state variable X', the relationship equation of the non-state variable X', the state variable X and the input U is established:

[0026] X'(t)=EAX(t)+EBU(t) (3)

[0027] In the formula, E is a diagonal matrix containing the capacitance value C and the inductance value L;

[0028] Further, the relationship equation of the non-state variable X C ' of the capacitor, the non-state variable X L ' of the inductor, the state variable X C of the capacitor, the state variable X L of the inductor and the history source h is established:

[0029]

[0030] The history source h is a column vector composed of i history , u history , and the relationship equation of the state variable X, the history source h and the input U is established by combining formula (7) and formula (3):

[0031] FX(t)-h(t)=EAX(t)+EBU(t) (8)

[0032] In the formula, F is a diagonal matrix composed of Y C , Z L ;

[0033] Finally, the state space discrete equation based on the characteristics of the capacitor and the inductor element is established:

[0034] X(t)=Ph(t)+QU(t) (9)

[0035] In the formula, P=(F-EA) -1 , Q=(F-EA) -1 EB.

[0036] Preferably, in the step D), when the loop simulation is performed, the state variable X(t) at the initial time t and the history source h at the current time are calculated, and then t=t+1 until the simulation is completed.

[0037] Compared with the prior art, the present application has the following advantages:

[0038] 1. The traditional way of directly discretizing the whole state space equation is abandoned, so that the numerical integration method can be flexibly selected for each state variable in the system;

[0039] 2. When all state variables are discretized using the same third-order or higher implicit numerical integration method, the computational complexity of the method provided by this invention is less than that of traditional state space discretization methods.

[0040] 3. When numerical integration is selected for partitioning, that is, when most capacitors and inductors in the system use low-order numerical integration methods while the remaining capacitors and inductors use high-order numerical integration methods, the accuracy can be significantly improved at the cost of a small increase in computation. Attached Figure Description

[0041] Figure 1 This is a simulation flowchart of the state-space equation discretization method based on the characteristics of capacitors and inductors in this invention.

[0042] Figure 2 This is a schematic diagram of an RLC circuit system.

[0043] Figure 3 This is a comparison chart of the target center distances of the four schemes calculated using the gray target method in Experiment 2. Detailed Implementation

[0044] The present invention will be further described in detail below with reference to specific embodiments.

[0045] like Figure 1 As shown, a method for discretizing state-space equations based on the characteristics of capacitors and inductors includes the following steps:

[0046] A) Establish the state-space differential equations of the circuit;

[0047] B) Establish the characteristic equations of capacitors and inductors, determine the numerical integration method used for each capacitor and inductor in the circuit, and use the numerical integration method to obtain the characteristic equations in the discrete time domain.

[0048] C) Combining the discrete-time characteristic equation from step B) and the state-space differential equation from step A), the state-space discrete equation of the circuit is obtained.

[0049] D) Cyclic simulation.

[0050] In step C), the relationship equations between non-state quantities, state quantities, and inputs are established based on the characteristic equations of capacitors and inductors. Based on the solution in step B), the relationship equations between the non-state quantities of capacitors, non-state quantities of inductors, state quantities of capacitors, state quantities of inductors, and historical sources are established. Then, the relationship equations between state quantities, historical sources, and inputs are established. Finally, the state-space discrete equations based on the characteristics of capacitors and inductors are established.

[0051] by Figure 2 Taking a certain RLC circuit system as an example, in step A), the state-space differential equation of the circuit is:

[0052]

[0053] Where A, B are state space matrices, t is time, X is state variable, including current i flowing through inductor L and voltage u across capacitor C , including U is system input.

[0054] In step B), the characteristic equation of capacitor and inductor are respectively:

[0055]

[0056] Where i C represents current flowing through capacitor, u L represents voltage across inductor, the characteristic equation of capacitor is solved by trapezoidal method, and the following is obtained:

[0057]

[0058] Where △t is simulation step, the characteristic equation of inductor is solved by 3rd order Adams-Moulton method, and the following is obtained:

[0059]

[0060] After calculation, the following is obtained:

[0061] Where,

[0062] In step C), define i C , u L as non-state variable X', establish the relationship equation of non-state variable X', state variable X and input U:

[0063] X'(t) = EAX(t) + EBU(t) (3)

[0064] Where E is diagonal matrix including capacitor value C and inductor value L;

[0065] Further, establish the relationship equation of capacitor's non-state variable X C ', inductor's non-state variable X L ', capacitor's state variable X C , inductor's state variable X L and history source h:

[0066]

[0067] History source h is i history , uhistory The column vector consisting of Y, Z, and h(t) is combined with formula (7) and formula (3) to establish a relationship formula of state quantity X, history source h, and input U as

[0068] FX(t)-h(t)=EAX(t)+EBU(t) (8)

[0069] In the formula, F is a diagonal matrix consisting of Y C , Z L ;

[0070] Finally, the state space discrete equation based on the characteristics of capacitive and inductive elements is established:

[0071] X(t)=Ph(t)+QU(t) (9)

[0072] In the formula, P=(F-EA) -1 , Q=(F-EA) -1 EB.

[0073] In step D), when the loop simulation is performed, the state quantity X(t) at the initial time t and the history source h at the current time are calculated, and then t=t+1 until the simulation is completed.

[0074] In order to verify the correctness of the present application, the following comparative experiments are carried out:

[0075] Experiment one

[0076] Objective: To verify that when all state quantities are selected to be discretized by the same third-order and above implicit numerical integration method, the calculation amount of the method provided by the present application is less than that of the traditional state space discrete method

[0077] Scheme 1: The state space equation discrete method based on the characteristics of capacitive and inductive elements of the present application is used, and the 3rd-order Adams-Moulton numerical integration method is selected for all capacitive and inductive elements.

[0078] Scheme 2: The traditional state space equation discrete method is used, and the 3rd-order Adams-Moulton numerical integration method is selected.

[0079] The experimental results are as follows:

[0080] Multiply Add Scheme 1 50 40 Scheme 2 58 52

[0081] Therefore, the calculation amount of the state space equation discrete method based on the characteristics of capacitive and inductive elements of the present application is less than that of the traditional discrete method.

[0082] Experiment two

[0083] Objective: To verify that when the partition selects numerical integration, that is, most of the capacitors and inductors in the system select low-order numerical integration methods, and the rest of the capacitors and inductors select high-order numerical integration methods, the precision can be significantly improved at the cost of increasing less computation.

[0084] Scheme 1: All state variables use 3-order Adams-Multon numerical integration method;

[0085] Scheme 2: L1 uses 4-order Adams-Multon numerical integration method, and the rest uses 3-order Adams-Multon numerical integration method.

[0086] Scheme 3: C2 uses 4-order Adams-Multon, and the rest uses 3-order Adams-Multon

[0087] Scheme 4: C1, C2 use 4-order Adams-Multon, and the rest use 3-order Adams-Multon

[0088] The experimental results are as follows:

[0089] Multiply Add Simulation result error Scheme 1 50 40 2.2338*10 -7 ]]> Scheme 2 51 41 1.3729*10 -7 ]]> Scheme 3 51 41 1.4258*10 -7 ]]> Scheme 4 52 42 1.0380*10 -7 ]]

[0090] Taking the number of multiplication and the error of simulation results as the target, as shown in Figure 3 , the target center distance of the four schemes is calculated using the gray target method. The smaller the target center distance, the better the scheme.

[0091] From the above figure, schemes 2, 3, and 4 are better than scheme 1. Therefore, when using the state space equation discretization method based on the characteristics of capacitors and inductors to select numerical integration, it is determined that most of the capacitors and inductors in the system select low-order numerical integration methods, and the rest of the capacitors and inductors select high-order numerical integration methods. The goal of significantly improving the precision at the cost of increasing less computation can be achieved, which has superiority.

Claims

1. A state-space equation discretization method based on the characteristics of capacitive and inductive elements, characterized by: The method comprises the following steps: A) establishing state space differential equations of the circuit; B) establishing characteristic equations of the capacitive and inductive elements, determining numerical integration methods used by each capacitive and inductive element in the circuit, and obtaining the characteristic equations in discrete time domain by using the numerical integration methods, wherein the characteristic equations of the capacitive and inductive elements are respectively: where i C represents the current flowing through the capacitor, u L represents the voltage across the inductor, the characteristic equation of the capacitor is solved using the trapezoidal method, and the following is obtained: where △t is a simulation step, and the characteristic equation of the inductive element is solved by using a third-order Adams-Moulton method to obtain: The following were obtained after the calculation: In the formulae, C) combining the discrete time domain characteristic equations in the step B) and the state space differential equations in the step A) to obtain state space discrete equations of the circuit; D) performing cyclic simulation.

2. The method of claim 1, wherein: In the step C), relationship equations of non-state quantities, state quantities and inputs are established according to the characteristic equations of the capacitive and inductive elements, relationship equations of non-state quantities of the capacitive and inductive elements, state quantities of the capacitive and inductive elements and historical sources are established according to the solving in the step B), and then relationship equations of state quantities and historical sources and inputs are established, and finally state space discrete equations based on characteristics of the capacitive and inductive elements are established.

3. The method of claim 1, wherein: In the step A), the state space differential equations of the circuit are: In the formula, A, B are state space matrices, t is the time, X is the state quantity, including the current i flowing through the inductor L and the voltage u across the capacitor C , contains U is the system input.

4. The method of claim 1, wherein: In the step C), define i C , u L is the non-state variable X', the relationship equation of the non-state variable X', the state variable X and the input U is established: X'(t) = EAX(t) + EBU(t) (3) where E is a diagonal matrix containing a capacitive value C and an inductive value L; Further, the non-state quantity X of the capacitor is established C Further, the non-state quantity X of the inductor is established L Further, the state quantity X of the capacitor is established C Further, the state quantity X of the inductor is established L The relationship equation of the history source h X C '(t) = Y C X C (t) - i history (t) (7) X L (t) = Z L X L (t) - u history (t) history source h is i history , u history The relationship between the state quantity X, the history source h and the input U is established by combining equation (7) and equation (3) FX(t) - h(t) = EAX(t) + EBU(t) (8) where F is a diagonal matrix consisting of Y C , Z L ; Finally, the state space discrete equations based on characteristics of the capacitive and inductive elements are established: X(t) = Ph(t) + QU(t) (9) where P = (F - EA) -1 Q = (F - EA) -1 EB.

5. The method of claim 4, wherein: In the step D), when the cyclic simulation is performed, state quantities X(t) at an initial time t and a historical source h at a current time are calculated, then t = t + 1, and the simulation is completed until the simulation is completed.

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