JP2011081674A - Method for obtaining operating point of system with nonlinear element characteristics approximated into divided polygonal line by electronic computer, program for the same, storage medium storing the program, and simulation device - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a method and the like which can rapidly and surely converge an operating point of a nonlinear system with nonlinear element characteristics approximated into divided polygonal line. <P>SOLUTION: The method includes an operation section specifying process which specifies a nonlinear element operation section to which a solution x<SP>(n)</SP>belongs, a process which calculates a slope and an intercept of a straight line of the nonlinear element and obtains the solution x<SP>(n)</SP>based on a formula into which an equation expressing a nonlinear system through the slope and the intercept is linearized, and a process which decides that the solution x<SP>(n)</SP>is a solution to be obtained when the nonlinear element operation section to which the solution x<SP>(n)</SP>belongs and an operation section to which a solution x<SP>(n-1)</SP>belongs coincide with each other. In the operation section specifying process, a first operation section to which the solution x<SP>(n)</SP>belongs when an X axis is a reference and a second operation section to which the solution x<SP>(n)</SP>belongs when a Y axis is a reference are obtained respectively, and the first or the second operation section closer to the operation section to which the solution x<SP>(n-1)</SP>belongs is specified as the nonlinear element operation section. <P>COPYRIGHT: (C)2011,JPO&INPIT

Description

  The present invention relates to a method for obtaining an operating point of a system in which the characteristic of a nonlinear element is approximated by a piecewise broken line by an electronic computer, a program thereof, a recording medium recording the program, and a simulation apparatus.

  A power system (power system) will be described as an example of a system in which the characteristics of nonlinear elements are approximated by piecewise broken lines. In addition, this invention is applicable not only to an electric power system but to all systems including an electric / electronic circuit.

  In recent years, many loads such as a power load, a lighting load, and an information equipment load are connected to a power system using a power electronics (hereinafter abbreviated as PE) circuit. In addition, natural energy power generation facilities and power storage devices are almost always connected to a power system by a PE circuit, and introduction of a system stabilizing device using PE technology has begun.

  The PE circuit has an operation principle of switching on and off current at high speed using a switching device. Therefore, the analysis of the system including the PE circuit requires not only the analysis of the conventional effective value level but also the analysis of the waveform level, that is, the instantaneous value analysis (also called transient phenomenon analysis or transient analysis in other fields). It becomes. In addition to the analysis related to the PE circuit, the instantaneous value analysis is mainly used for the analysis related to the power quality whose importance has been recognized in recent years. In addition, with regard to analysis on various overvoltages, overcurrents, and abnormal resonances to which instantaneous value analysis has conventionally been applied, there are increasing opportunities to carry out these analyzes after a design review accompanying equipment replacement.

  As an example of instantaneous value analysis, a circuit equation is created for an electric circuit to be analyzed, and the linear equation obtained by linearizing the circuit equation using the Newton-Raphson method (hereinafter also abbreviated as NR method) is used. There is a method of obtaining the operating point of the electric circuit by solving (see, for example, Patent Document 1). Non-Patent Document 1 describes that the NR method is applied to an electronic circuit whose characteristics of nonlinear elements are approximated by piecewise broken lines.

  However, the NR method may not converge depending on the initial value. In particular, the NR method may be applied to a circuit equation related to a highly nonlinear circuit such as an electric circuit or the like including a nonlinear element such as an IGBT (Insulted Gate Bipolar Transistor). In this case, the problem of convergence becomes remarkable. On the other hand, there is a method called Katzenelson method (see Non-Patent Document 2, for example) that always converges when the characteristics of all the nonlinear elements are monotonically increasing and are approximated by piecewise broken lines. However, the number of iterations necessary for convergence is large. It is known to be impractical.

  Such problems exist not only when the NR method is applied to circuit equations relating to highly nonlinear electrical and electronic circuits, but also generally in nonlinear systems in which the characteristics of nonlinear elements are approximated by piecewise broken lines. To do.

JP 2002-197401 A

L.O. Chua, “Efficient computer algorithms for piecewise-linear analysis of resistive nonlinear networks,” IEEE Trans., Circuit Theory, vol.CT-18, no.1, pp 73-85, Jan. 1971. J. Katzenelson, “An Algorithm for solving nonlinear resistor networks,” Bell Syst. Tech. J., pp. 1605-1620, Oct. 1965.

  In view of the above prior art, the present invention provides a method capable of quickly and reliably converging the operating point of a nonlinear system in which the characteristics of nonlinear elements are approximated by piecewise broken lines, a program thereof, a recording medium recording the program, and a simulation apparatus The purpose is to provide.

  In the present invention, the operating point of the nonlinear system in which the characteristic of the nonlinear element is approximated by piecewise broken line is obtained by iterative calculation. In the following explanation, the solution at each iteration calculation step (solution processing) is simply referred to as “solution”. As a result of iterative calculation (convergence determination processing), a solution that is finally obtained is referred to as a “required solution”.

のＦ^{（ｎ−１）}の値及びｙ^{（ｎ−１）}の値を計算し、当該Ｆ^{（ｎ−１）}の値及びｙ^{（ｎ−１）}の値を用いて（ａ）式を解いて解ｘ^(ｎ)を求める求解処理と、解ｘ^（ｎ）が属する前記非線形要素の動作区間と、解ｘ^{（ｎ−１）}が属する動作区間とが一致した場合に解ｘ^（ｎ）を求める解とし、一致しない場合は前記動作区間特定処理及び前記求解処理を再実行させる収束判定処理とを備え、前記動作区間特定処理では、解ｘ^（ｎ）（ただし、ｎは１以上）の動作区間を特定する際には、Ｘ軸を基準としたときの解ｘ^（ｎ）が属する前記非線形要素の第１動作区間と、Ｙ軸を基準としたときの解ｘ^（ｎ）が属する前記非線形要素の第２動作区間とをそれぞれ求め、第１動作区間及び第２動作区間のうち解ｘ^{（ｎ−１）}が属する前記非線形要素の動作区間に近い方を前記非線形要素の動作区間として特定することを特徴とする非線形要素の特性が区分折れ線近似されたシステムの動作点を電子計算機により求める方法にある。 In order to achieve the above object, a first aspect of the present invention is a method for obtaining an operating point of a nonlinear system in which a characteristic of a nonlinear element is approximated by a piecewise broken line by an electronic computer, wherein the characteristic of the nonlinear element is determined by a plurality of operating sections. Initial setting processing for inputting characteristic information approximated by a straight line and an initial solution x ⁽⁰⁾ of an equation representing the nonlinear system, and motion interval specifying processing for specifying the motion interval of the nonlinear element to which the solution x ⁽ⁿ⁾ belongs And a linear slope and intercept of the nonlinear element in the motion section, and an equation representing the nonlinear system is linearized from the slope and intercept (a)

F ^(n-1) value and y ^(n-1) value are calculated, and equation (a) is solved using the F ^(n-1) value and y ^(n-1) value. obtaining a solving process to find the solution x ^(n), the operation section of the nonlinear element the solution x ⁽ⁿ⁾ belongs, the solution x ⁽ⁿ⁾ when the operation period the solution x ^(n-1) belongs matches A solution determination process that re-executes the action section specifying process and the solution-finding process if they do not match. In the action section specifying process, the action section of the solution x ⁽ⁿ⁾ (where n is 1 or more) , The first motion section of the nonlinear element to which the solution x ⁽ⁿ⁾ when the X axis is a reference belongs, and the nonlinear element to which the solution x ⁽ⁿ⁾ when the Y axis is a reference the second obtains the operation period, respectively, the nonlinear the solution x ^(n-1) belongs among the first operation period and the second operation section Certain operating point of the system characteristics of the nonlinear element is divided polygonal line approximation, characterized by identifying the closer to the operation section of the unit as an operation section of the nonlinear element to the method determined by the electronic computer.

In the first aspect, when the motion section to which the solution x ⁽ⁿ⁾ belongs is specified, motion sections are obtained not only for one axis but also for both the X axis and the Y axis, and among them, the solution x ^{(n− The one} closer to the operation section ¹⁾ is specified as the operation section. That is, according to the present invention, the number of operation sections that can be skipped in one iteration is smaller than that of the conventional NR method. For this reason, according to the present invention, it can be avoided that the solution moves to an operation section far from the operation section to which the solution to be found belongs depending on the characteristics of the nonlinear element, and the convergence can be improved.

  According to a second aspect of the present invention, in the method for obtaining an operating point of a system in which the characteristic of the nonlinear element described in the first aspect is approximated by a piecewise broken line by an electronic computer, the Newton-Raphson method is performed after the initial setting process is executed. A system in which characteristics of a nonlinear element are approximated by piecewise broken lines, wherein the convergence determination process is executed from the action section specifying process when a solution is not obtained within a predetermined number of iterations. The operating point is obtained by a computer.

  In the second aspect, as described in the first aspect, the calculation time is reduced overall by using the conventional NR method with a short calculation time while improving the convergence by obtaining the operation sections for both axes. It can be shortened.

  According to a third aspect of the present invention, in the method for obtaining an operating point of a system in which the characteristic of the nonlinear element described in the second aspect is approximated by a piecewise broken line by an electronic computer, the convergence determination process performs convergence within a predetermined number of times. If not, there is a method for obtaining the operating point of a system in which the characteristic of the nonlinear element is approximated by a piecewise broken line, which is characterized in that the solution is obtained by switching to the Katzenelson method.

  In the third aspect, it is possible to ensure that a solution can be reliably obtained by using the Katzenelson method that is guaranteed to converge, and to shorten the calculation time compared to the case of using only the Katzenelson method.

  According to a fourth aspect of the present invention, there is provided a program for causing an electronic computer to execute a method for obtaining an operating point of a system in which the characteristic of the nonlinear element described in any one of the first to third aspects is approximated by a piecewise broken line. It is in.

  A fifth aspect of the present invention resides in a recording medium readable by an electronic computer in which the program described in the fourth aspect is recorded.

  A sixth aspect of the present invention is configured to execute a method for obtaining an operating point of a system in which the characteristic of the nonlinear element described in any one of the first to third aspects is approximated by a piecewise broken line by an electronic computer. A simulation apparatus having a computer.

  In any of the fourth to sixth aspects, when these are used to determine the operating point of the nonlinear system, the calculation time can be shortened and the convergence can be improved.

  According to a seventh aspect of the present invention, in the simulation apparatus according to the sixth aspect, the nonlinear system is a virtual electric / electronic circuit including a nonlinear element, and the control for controlling the electric / electronic circuit is performed. A control signal from the apparatus is input as input data used for simulation of the electric / electronic circuit via an A / D converter, and a calculated value relating to a voltage or current of a predetermined portion of the electric / electronic circuit is output by the D / A converter. A simulation that is configured to calculate the operation of the electric / electronic circuit in synchronization with real time in accordance with the operation of the control device by converting the voltage or current to the control device and outputting it to the control device. In the device.

  In the seventh aspect, when calculating the operating point of the electric / electronic circuit, the solution can be obtained with a short calculation time and high convergence, so that the operation of the actual control device or the like can be verified in real time. can do.

  According to the present invention, there are provided a method capable of rapidly and reliably converging operating points of a nonlinear system whose characteristics of nonlinear elements are approximated by piecewise broken lines, a program thereof, a recording medium recording the program, and a simulation apparatus. .

It is a figure which shows the circuit diagram of the electronic circuit I, and the voltage-current characteristic of a nonlinear resistance. It is a block diagram which shows the structure of the simulation apparatus which concerns on Embodiment 1 of this invention. It is a block diagram which shows the hardware constitutions of the computer which concerns on the simulation apparatus which concerns on Embodiment 1 of this invention. It is a figure which shows the flow of the method of calculating | requiring the operating point of the system by which the characteristic of the nonlinear element which concerns on Embodiment 1 of this invention was piecewise broken line approximated with an electronic computer. FIG. 3 is a diagram illustrating voltage-current characteristics of an electronic circuit I. It is a figure which shows the operation area selection method in the conventional NR method. It is a figure which shows the convergence of the conventional NR method. It is a figure which shows the convergence of the method of calculating | requiring the operating point of the system by which the characteristic of the nonlinear element which concerns on Embodiment 1 of this invention was piecewise broken line approximated with an electronic computer. It is a figure which shows the convergence of Katzenelson method. It is a figure which shows the flow of the method of calculating | requiring the operating point of the system by which the characteristic of the nonlinear element which concerns on Embodiment 2 of this invention was piecewise broken line approximation with an electronic computer. It is a figure which shows the voltage-current characteristic of the nonlinear resistance which concerns on the electronic circuit II and the electronic circuit III. It is a figure which shows the locus | trajectory of iterative calculation in the electronic circuit II. It is a figure which shows the locus | trajectory of iterative calculation in the electronic circuit III. It is a power conditioner circuit diagram of photovoltaic power generation equipment. It is a figure which shows the model of a diode and IGBT. It is a figure which shows the nonlinear characteristic of each element which concerns on the power conditioner circuit of photovoltaic power generation equipment. It is the inverter output voltage and current waveform calculated | required by the method of calculating | requiring the operating point of the system by which the characteristic of the nonlinear element which concerns on Embodiment 2 of this invention was piecewise broken line approximated with an electronic computer. It is a figure which shows the change of the repetition frequency | count of 1 period. It is a figure which shows the relationship between the voltage waveform of IGBT, and the frequency | count of repetition. It is the figure which compared the inverter output voltage waveform each calculated by this invention and the conventional NR method. It is a block diagram which shows the structure of the simulation apparatus which concerns on Embodiment 3 of this invention.

<Embodiment 1>
A nonlinear system targeted by a “method for obtaining an operating point of a nonlinear system in which the characteristics of nonlinear elements are approximated by piecewise broken lines” (hereinafter also referred to as an operating point solution method) according to an embodiment of the present invention is a nonlinear system. The system can be expressed by the following simultaneous equations and can be generally derived by the following equation (1).

  x is a vector composed of N unknown variables to be obtained, and f (·) is a vector function giving N nonlinear equations. Since equation (1) is a nonlinear simultaneous equation, iterative calculation is required to solve it. If the equation (1) is linearized with respect to x, the equation (2) of the iterative calculation is obtained.

The number in parentheses on the right shoulder of the variable indicates the iteration step, and F is an N × N matrix obtained by partial differentiation of f (•) with respect to x. Starting from the initial value x ^{(0), an} updated solution x ⁽ⁿ⁾ is obtained by sequentially solving the simultaneous linear equations of equation (2). However, F ^(n-1) and y ^(n-1) can be calculated from the solution x ^(n-1) in the previous iteration step. When the solution x ⁽ⁿ⁾ converges, the solution is obtained and the iterative calculation is terminated.

  Actually, the formula (1) is not constructed, and it is usual to construct the formula (2) directly and perform the iterative calculation. For example, in the case of an electric / electronic circuit, there is a sputter blow method as one method for constructing the equation (2). Of course, as long as the expression (1) can be configured in the form of the expression (2), a technique other than the sputter blow method may be used.

  The operating point solving method according to the present invention is based on the premise that the characteristics of the nonlinear elements included in the nonlinear system are approximated by piecewise broken lines, and solves the equation representing the nonlinear system by the above iterative calculation. Note that the characteristic of the nonlinear element being approximated by a piecewise broken line means that the characteristic of the nonlinear element is composed of a plurality of straight lines.

  Hereinafter, an electronic circuit will be described as an example of a system in which characteristics of nonlinear elements are approximated by piecewise broken lines. FIG. 1 shows an electronic circuit I including a nonlinear resistance and characteristics (vi characteristics) of a nonlinear resistance which is an example of a nonlinear element.

As shown in FIG. 1A, an electronic circuit I that is an example of a non-linear system includes a DC power source, a resistor, and a non-linear resistance that is an example of a non-linear element. Further, in FIG. 1 (b), v-i characteristic of the nonlinear resistor _{R N} is shown. V-i characteristic of the nonlinear resistor R _N is divided polygonal line approximation. That is, the vi characteristic is composed of a plurality of straight lines. A range of each straight line in the voltage axis v direction and the current axis i direction is referred to as an operation section. Figure 1 In the example of (b), v-i characteristic of the nonlinear resistor _{R N,} the operation section 1 (v <-1, i < -1) and the straight line portion in the operating range 2 (-1 ≦ v <1, -1 ≦ i <1) and a linear portion in the operation section 3 (1 ≦ v, 1 ≦ i).

  A method for obtaining an operating point of a nonlinear system in which the characteristics of such nonlinear elements are approximated by piecewise broken lines by an electronic computer and a simulation apparatus configured to execute the method will be described in detail.

  As shown in FIG. 2, the simulation apparatus 1 is an apparatus having a computer 20 that includes units 11 to 14 that execute an initial setting process, an operation section specifying process, a solution finding process, and a convergence determination process, which will be described later. As shown in FIG. 3, the computer 20 includes storage means such as a CPU 21, a RAM 22, a ROM 23, and a hard disk 24, and an input device 30 such as a keyboard and a storage medium reading device and an output device 40 such as a display and a printer are connected to the computer 20. ing.

  As shown in FIG. 2, each means 11 to 14 that can be executed by the computer 20 is a means for executing each process in an operating point solution method to be described later. Specifically, the initial setting means 11 and the operation section specifying Means 12, solution finding means 13, and convergence determination means 14 are provided. Each of these means 11 to 14 reads and creates a program 10 for causing the computer 20 to execute the operating point solution method. Each of these means 11 to 14 may be implemented as an electronic circuit.

Here, the process of the operating point solution method will be described with reference to FIG. As shown in the figure, first, initial setting processing for inputting characteristic information obtained by approximating a characteristic of a nonlinear element with a straight line for each of a plurality of motion sections and an initial solution x ⁽⁰⁾ (iterator n = 0) of an equation representing a nonlinear system. (Step S1). The number of nonlinear elements is one or more, and characteristic information for each nonlinear element is input. In this initial setting process, characteristic information about other elements such as a power supply is also input according to the circuit configuration.

This process is executed by the initial setting means 11. Specifically, the characteristic information and the initial solution x ⁽⁰⁾ recorded in the hard disk 24, or the characteristic information and the initial solution x ⁽⁰⁾ are input via the input device 30. Recorded in the RAM 22. The characteristic information includes a range of voltage value v and current value i of each operation section, and a slope and an intercept representing a straight line of each operation section. If an operation section is specified, the slope and intercept of the straight line of the operation section are It is recorded in the RAM 22 so as to be obtained.

Next, the motion section specifying means 12 performs a motion section specifying process for specifying the motion section of each nonlinear element to which the solution x ⁽ⁿ⁾ (iterator n is 0 when the first motion section specifying process is executed) ( Step S2). With this process, for each non-linear element, an operation section used in the next solution calculation process is specified among the plurality of operation sections of the non-linear element. Details of the operation section specifying process will be described later.

Next, the solving means 13 increments the iterator n, and solves the equation (2) to obtain a solution x ⁽ⁿ⁾ (iterator 1 is 1 when the first solving process is executed) (step S3). ). More specifically, the slope and intercept of the straight line of each nonlinear element in the motion section identified in the motion section identification process are calculated. This calculation can be calculated from the characteristic information input in the initial setting process. And the value of F ^(n-1) and y ^(n-1) of (2) Formula is calculated from the inclination and intercept of each nonlinear element. The equation (2) is solved using F ⁽ⁿ⁻¹⁾ and y ⁽ⁿ⁻¹⁾ obtained in this way to obtain a solution x ⁽ⁿ⁾ , and the solution x ⁽ⁿ⁾ is recorded in the RAM 22.

Next, the convergence determination means 14 performs a convergence determination process for determining whether or not the solution x ⁽ⁿ⁾ has converged (step S4). Specifically, for each of a plurality of nonlinear elements, it is determined whether or not the motion section of the nonlinear element to which the solution x ⁽ⁿ⁾ belongs and the motion section to which the solution x ⁽ⁿ⁻¹⁾ belongs. For all nonlinear elements, if the motion sections match (step S5: Yes), the solution x ⁽ⁿ⁾ is determined as a solution. If there is even one nonlinear element whose motion sections do not match, the solution is not converged. (Step S5: No), the operation section specifying process (Step S2), and the solution finding process (Step S3) are re-executed. That is, until the time of convergence, the operation section specifying process, the solution finding process, and the convergence determining process are repeatedly executed.

  As described above, it is possible to obtain a solution of an equation representing a nonlinear system. However, in the operation section specifying process, the convergence and the calculation amount differ depending on the method for specifying the operation section of the nonlinear element. The operating point solution method according to the present invention has a higher convergence than the conventional NR method by devising this operating section specifying method. Hereinafter, in the case of solving the equation representing the electronic circuit I shown in FIG. 1, the description will focus on the method of specifying the operation section, and it will be shown that the convergence is better than the conventional NR method.

[Conventional NR method]
The electronic circuit I shown in FIG. 1A can be reduced to the following formulas (3) and (4) even when the sputter blow method or another formulation method is used.

  Here, e is the power supply voltage, R is the value of the series resistance, and v and i are the voltage and current of the nonlinear resistance. In addition, g (•) is the vi characteristic of the nonlinear resistance (see FIG. 1B). FIG. 5 shows a superposition of these equations (3) and (4). It can be seen that the intersection of equations (3) and (4) gives the solution of electronic circuit I.

First, the characteristic information of the nonlinear resistance and the initial solution x ⁽⁰⁾ (iterator n = 0) are input and recorded in the RAM 22 (initial setting process). Next, the operation section of the nonlinear resistance to which the solution x ⁽ⁿ⁾ (iterator n is 0 when the first operation section specifying process is executed) belongs is specified (operation section specifying process). A method for specifying this operation section will be described with reference to FIG. FIG. 6 shows the vi characteristic of the non-linear resistance. The vi characteristic is composed of a plurality of straight lines and is approximated by a piecewise broken line. Focusing on the voltage axis v of such v-i characteristic, the solution x ⁽ⁿ⁾ is to compute whether there in any of the range of each operation period, it identifies the operation period the solution x ⁽ⁿ⁾ belongs. That is, an operation interval including the voltage value v of the solution x ⁽ⁿ⁾ is obtained from a plurality of operation intervals. In this example, the voltage value v of the solution x ^(n), by comparing the respective operation sections recorded on the RAM 22, the solution x ⁽ⁿ⁾ is determined to belong to the operation section S _1.

However, it is not always necessary to focus on the voltage axis v, and the current axis i may be focused. That is, an operation interval including the current value i of the solution x ⁽ⁿ⁾ may be obtained from a plurality of operation intervals. In any case, in the conventional NR method, only one of the axes is used when specifying the operation section.

Next, the iterator n is incremented, and a solution x ⁽ⁿ⁾ (iterator n is 1 at the time of execution of the first solution processing) is calculated (solution processing). Of the vi characteristics of the non-linear resistance approximated by piecewise broken line, the slope and intercept of the straight line of the operation section specified in the operation section specifying process are obtained. And F ^(n-1) , y ^(n-1) of (2) Formula is calculated from this inclination and intercept, (2) using these F ^(n-1) , y ^(n-1 ). Solve the equation to find the solution x ⁽ⁿ⁾ . Then, if the motion section to which the solution x ⁽ⁿ⁾ belongs and the motion section to which the solution x ⁽ⁿ⁻¹⁾ belong are the same, the calculation is terminated. repeat.

The above-described motion section specifying process, solution process, and convergence determination process will be described visually with reference to FIG. FIG. 7 is a graph showing the expressions (3) and (4). The expression (4) is expressed as three operation sections (in the figure, “section 1”, “section 2”, and “section 3”). It is composed of straight lines. Since the initial solution x ⁽⁰⁾ is given in advance, the operation interval to which the solution x ⁽⁰⁾ belongs is determined. As shown in the figure, the operation section 3 is specified.

Next, among the vi characteristics of the nonlinear resistance, the slope and intercept of the straight line in the operation section 3 are obtained, and F ⁽ⁿ⁻¹⁾ and y ⁽ⁿ⁻¹⁾ are calculated. As a result, the equation (4) is linearized as av + b = i, and the simultaneous equation composed of the linearized equation and the equation (3) is solved to obtain a solution x ⁽¹⁾ . Since the motion section to which the solution x ⁽¹⁾ belongs is the motion section 1, since it is different from the motion section 3 of the solution x ⁽⁰⁾ , it is determined that it has not converged and the process proceeds to the next step.

When the motion section to which the solution x ⁽¹⁾ belongs is obtained, the motion section 1 is obtained. Among the vi characteristics of the non-linear resistance, the slope and intercept of the straight line in the operation section 1 are obtained, F ^(n-1) and y ^(n-1) are calculated, and av + b = i obtained by linearizing the equation (4) And (3) are solved to obtain a solution x ⁽²⁾ . Then, a convergence determination process is performed for the solution x ⁽²⁾ .

As described above, although the solution ^{x (n)} is going to be solving, further followed, if solving ^{x (3)} from the solution ^{x (2),} the solution ^{x (2)} is the initial solution ^{x ( Since} it belongs to the same operation section 3 as ⁰⁾ , the solution x ⁽³⁾ becomes the same as the solution x ⁽¹⁾ and does not converge forever. Thus, in the conventional NR method, the solution may not converge. In the case of this non-linear resistance, convergence is achieved by selecting the operation section by the current axis. However, if the characteristic of the non-linear resistance is not the characteristic that the current is saturated with respect to the voltage (see FIG. 1B), and the characteristic that the voltage is saturated with respect to the current, it is only necessary to follow the path of the solution cyclically. Can't reach the solution forever.

[Operating point solution method (biaxial NR method)]
In the above-described conventional NR method, when specifying the operation section to which the solution x ⁽ⁿ⁾ belongs, the operation section is specified considering only the X axis (voltage axis v). However, the point which considers an operation area about each of the X-axis (voltage axis v) and the Y-axis (current axis i) is different. Because of these characteristics, the operating point solution method according to the present invention is also referred to as a biaxial NR method.

With reference to FIG. 8, a description will be given of a method of specifying an operating point in the biaxial NR method. Up to the point where the solution x ⁽¹⁾ is obtained, it is the same as in the conventional NR method (see FIG. 7). In solving ^{x (2)} from the solution ^{x (1),} when determining the operation period of the solution ^{x (1),} the solution ^{x (n)} is non-linear elements belonging when a reference X-axis (voltage axis v) The first operation interval is obtained, and the second operation interval of the nonlinear element to which the solution x ⁽ⁿ⁾ when the Y axis (current axis i) is used as a reference is also obtained.

That is, among a plurality of operation period, the operation section including a voltage value v of the solution x ⁽¹⁾ and the first operation period, the operation section including the current value i of the solution x ⁽¹⁾ and the second operation period. In the example shown in FIG. 8, the first operation period the solution x ⁽¹⁾ belongs is the section 1, second operation period the solution x ⁽¹⁾ belongs is interval 2. On the other hand, since the motion section to which the solution x ⁽⁰⁾ belongs is the section 3, the second motion section that is closer to the section 3 is specified as the motion section.

Thereafter, the slope and intercept of the nonlinear resistance straight line in the operation section (section 2) specified in this way are obtained, and F ⁽ⁿ⁻¹⁾ and y ⁽ⁿ⁻¹⁾ are calculated. As a result, the equation (4) is linearized as av + b = i, and the simultaneous equation composed of the linearized equation and the equation (3) is solved to obtain a solution x ⁽²⁾ . Since the motion section to which the solution x ⁽²⁾ belongs is on the straight line of the section 2, and the motion section of the solution x ⁽¹⁾ is the section 2 as specified above, it is determined here that the solution has converged.

  As described above, the biaxial NR method can converge even on an electronic circuit including a nonlinear resistance that does not converge in the conventional NR method. In the biaxial NR method, when the operation interval is specified, the one closer to the operation interval of the previous step is selected, so that the number of operation intervals that can be skipped in one iteration is smaller than that of the conventional NR method. For this reason, the biaxial NR method can avoid that the solution moves to an operation section far from the operation section to which the solution to be found belongs depending on the characteristics of the nonlinear element. According to such a feature of the biaxial NR method, for example, a solution to be obtained without falling into a cyclic trajectory for a system including a nonlinear element having a characteristic that changes sharply as shown in FIG. It can be obtained and the convergence is better than the conventional NR method.

[Katzenelson method]
The Katzenelson method is an algorithm in which the characteristics of all nonlinear elements are expressed by piecewise line approximation, and in the case of monotonic increase, it always converges to a desired solution even if starting from any initial value (for details, see Non-Patent Document 2). reference). The outline of this method will be described with reference to FIG.

  The difference between the conventional NR method and the Katzenelson method lies in the part for correcting the solution in the next iteration step in addition to the part for selecting the operation section.

  In the conventional NR method, a new operation section is determined as it is from the voltage (or current) of the nonlinear resistance given by the solution, no matter how far the current operation section and the newly selected operation section are separated. On the other hand, the Katzenelson method moves in the direction of the operation section to which the voltage (current) given by the solution belongs, but stops at the boundary between the section and the section that crosses first, and sets the other side as a new operation section. Furthermore, the solution in the next iteration step is corrected to its boundary point. In other words, the Katzenelson method is a method that always stops at the boundary of a section without moving the section in a single step when selecting a new motion section.

As shown in FIG. 9, as in the conventional NR method, the solution at the first iteration step is point C. However, the point C is not adopted as a solution, and the boundary between the motion section to which the initial value belongs and the motion section adjacent in the direction of the point C, that is, the boundary between the section 3 and the section 2 is the updated solution x ⁽¹⁾ . The new motion section is section 2 located on the other side of the solution x ⁽¹⁾ . Since the solution to be obtained exists in the interval 2, this solution is reached in the next iteration step, and the iterative calculation ends.

  In this way, the Katzenelson method is guaranteed to converge mathematically. However, in addition to the update of the solution, the calculation amount increases compared with the conventional NR method because a new solution located at the boundary of the section is calculated and only one section can be moved at each iteration step. . In addition, although the calculation amount of both axes NR is larger than that of the conventional NR method, it is not only possible to move one section at a time as in the Katzenelson method, and it can jump over a plurality of operation sections. The method is less computationally intensive than the Katzenelson method. Table 1 shows the convergence and calculation amount of the conventional NR method, the biaxial NR method, and the Katzenelson method.

  According to the method and the simulation apparatus for obtaining the operating point of the system in which the characteristic of the nonlinear element is approximated by piecewise broken line as described above and the simulation apparatus, the operating point of the nonlinear system in which the characteristic of the nonlinear element is approximated by piecewise broken line is calculated using the Katzenelson method. In contrast to the conventional NR method, it is possible to converge more reliably than the conventional NR method.

<Embodiment 2>
When obtaining the operating point of the system in which the characteristic of the nonlinear element is approximated by piecewise broken line, not only the biaxial NR method but also the conventional NR method and the Katzenelson method may be used in combination. A method for solving the equations representing the nonlinear system using these methods will be described with reference to FIG.

First, the current iterator n is initialized with 0 (step S10). Next, an initial setting process is performed (step S11). Next, the iterator n is incremented (step S12). Then, comparing whether a iterator n is the threshold value _{N B} above (Step S13). Threshold N _B may abort the calculation of the conventional NR method, a predetermined value that defines the number of steps for switching both axes NR method.

If the iterator n is less than the threshold value _{N B} (Step S13: No), using conventional NR method performs one step of the iterative calculation (operation point specifying process-convergence determination process) (step S14). If the calculation result by the conventional NR method has converged (step S15: Yes), the process ends. If the result does not converge (step S15: No), the process returns to step S12.

On the other hand, when the iterator n is the threshold value _{N B} above (Step S13: Yes), it compares whether the iterator n is greater than or equal to the threshold value _{N K} (step S16). Threshold N _K may abort the calculation in both axes NR method, a predetermined value that defines the number of steps for switching to Katzenelson method.

If the iterator n is less than the threshold value _{N K} (step S16: No), using both axes NR method performs one step of the iterative calculation (operation point specifying process-convergence determination process) (step S17). If the calculation result by the biaxial NR method has converged (step S15: Yes), the process ends. If the calculation result does not converge (step S15: No), the process returns to step S12.

On the other hand, when the iterator n is equal to or larger than the threshold _{N K} (step S16: Yes), using Katzenelson method performs one step of the iterative calculation (operation point specifying process-convergence determination process) (step S18). If the calculation result by the Katzenelson method has converged (step S15: Yes), the process ends. If the result does not converge (step S15: No), the process returns to step S12.

  As described above, by using the conventional NR method at first, an attempt is made to find a solution in a short calculation time. When the number of iterations exceeds a certain number of iterations, the solution is solved by a biaxial NR method having higher convergence than the conventional NR method. In addition, if a certain number of iterations is exceeded, the Katzenelson method is used to reliably find a solution. In this way, by switching between the conventional NR method, the biaxial NR method, and the Katzenelson method based on the number of iterations, it is possible to obtain a solution so that the equation representing the nonlinear system converges quickly and reliably.

<Example 1>
Embodiments in which a method and a simulation apparatus for obtaining an operating point of a system in which the characteristics of a nonlinear element according to the present invention are approximated by a piecewise broken line are applied to various circuits will be described below.

In the first embodiment, two electronic circuits will be described. Least two electronic circuits, the circuit configuration is the same as that shown in FIG. 1 (a), the different characteristics of the nonlinear resistor R _N. Figure 11 (a) electronic circuit that includes a non-linear resistance _{R N} shown in II, the ones containing a nonlinear resistor _{R N} shown in FIG. 11 (b) referred to as electronic circuits III.

FIG. 12A shows the transition of the solution x ⁽ⁿ⁾ when the conventional NR method is applied to the electronic circuit II, and FIG. 12B shows the operating point solution method according to the second embodiment for the electronic circuit II. It is a figure which shows transition of the solution x ⁽ⁿ⁾ when applied.

As shown in FIG. 12A, in the conventional NR method, when starting from the initial solution x ⁽⁰⁾ (point 0), iterative calculation is performed following points 1, 2, 3, 4, 5, 6,. Are updated as solutions x ⁽⁰⁾ , x ⁽¹⁾ , x ⁽²⁾ , x ⁽³⁾ ,. After point 6, since it moves to point 3, it falls into an infinite loop of 6 → 3 → 4 → 5 → 6 and is obtained forever by simply taking the values of solutions x ⁽³⁾ and x ⁽²⁾ alternately. Does not converge to the solution.

On the other hand, in the iteration up to N _B −1, the above-mentioned trajectory is traced. However, when switching to the biaxial NR method, the solution converges to a solution obtained in one or two iterations. Depending on whether the solution is in x ⁽⁰⁾ , x ⁽¹⁾ , x ⁽²⁾ , or x ^{(3) in} the N _B −1st iteration step, the subsequent trajectories differ, but these four types of trajectories are As shown in FIG. In the electronic circuit II, the Katzenelson method could not be applied because it converged to a solution obtained by the biaxial NR method.

FIG. 13A shows the transition of the solution x ⁽ⁿ⁾ when the conventional NR method is applied to the electronic circuit III, and FIG. 13B shows the operating point solving method according to the second embodiment for the electronic circuit II. It is a figure which shows transition of the solution x ⁽ⁿ⁾ when applied.

As shown in FIG. 13A, in the conventional NR method, when starting from the initial solution x ⁽⁰⁾ (point 0), the points 1, 2, 3, 4, 5, 6, 7,. An iterative calculation is performed and the solutions x ⁽⁰⁾ , x ⁽¹⁾ , x ⁽²⁾ , x ⁽³⁾ , x ⁽⁴⁾ ,. After the point 7, since it returns to the vicinity of the initial value, it falls into an infinite loop of 0 → 1 → 2 →... → 6 → 7 → 0.

  On the other hand, as shown in FIG. 13B, when switched to the biaxial NR method, the solution converges to a solution obtained in one or two iterations. Even in the electronic circuit III, the Katzenelson method could not be applied because it converged to a solution obtained by the biaxial NR method.

<Example 2>
In this example, the method for obtaining the operating point of the system in which the characteristic of the nonlinear element according to the second embodiment is approximated by a piecewise broken line is applied to the analysis of the single-phase inverter circuit by applying a method for practical analysis. Check the convergence performance of the method in the example. FIG. 14 shows the inverter circuit used for the analysis, and FIG. 15 shows the inside of the diode model and the IGBT model. The diode is an SBD (Schottky barrier diode) using SiC, and as shown in FIG. 15, a static rectification characteristic is a non-linear resistance R _st and a dynamic characteristic due to a change in junction capacitance is a non-linear capacitance C _j . Mock up. R _st and the nonlinear characteristics of the C _j, respectively FIG. 16 (a), shown in (b). The IGBT uses Si and is simulated by the Kraus model shown in FIG. In this model, the nonlinear static characteristic of the portion corresponding to the MOSFET included in the IGBT is expressed by the control voltage source V _mos , and the change in the capacitance of the oxide film is simulated by the nonlinear capacitance C _ox . Further, the dynamic characteristic of the base passing current generated by the change of the base charge is simulated by the control current source _Iq , and the dynamic characteristic of the tail current is simulated by the control current source _Ipc . FIG. 16C shows the nonlinear characteristics of the above _Cox . D ₁ simulating a diode is a non-linear resistance having ideal switch characteristics with an on-resistance of 10 ⁻⁴ Ω and an off-resistance of 10 ¹⁴ Ω, and D ₂ is also a non-linear resistance with an on-resistance of 10 ⁻⁵ Ω and an off-resistance of 10 ¹³ Ω. did. For the calculation of V _mos , I _q , and I _pc , refer to the literatures “Yoshikazu Okada, Toshiaki Kikuma, Masahiro Takasaki, Kiyoshi Takenaka, Kazuya Otani, Toshihiko Kuzumaki, Toshiaki Matsumoto,“ Development of Inverter Simulation Program (Part 2) − “Verification of analysis accuracy by actual measurement comparison-”, Chuo Research Institute R07016, 2008 ”is described in detail. The electrolytic capacitor and the film capacitor for smoothing are constituted by discrete components 2 in parallel. For the wiring of these capacitors and each part of the circuit, the parasitic inductance and parasitic capacitance estimated from the actual measurement results are taken into consideration. The result of analyzing the transient phenomenon of this circuit with XTAP (abbreviation of eXpandable Transient Analysis Program. Refer to Research Report H06002, H07004, H07005, R06017, R07016 of Central Research Institute of Electric Power) agrees well with the actual measurement result.

In FIG. 15, elements having nonlinear characteristics are surrounded by dotted lines. However, as will be understood, since this circuit includes a large number of nonlinear elements, its convergence is not easy. Transient analysis for one cycle was performed by the operating point solution method according to the second embodiment, with the calculation time increment set to Δt = 10 ns, the parameters for switching the iterative calculation method set to N _B = 100, and N _K = 200. Since it is a transient analysis, the operation of obtaining a solution of the nonlinear circuit at every calculation time step (every Δt) is repeated. Here, since the number of iterations for switching from the conventional NR method to the biaxial NR method is set to a very large value of N _B = 100, the conventional NR method falls into an infinite loop when switching to the biaxial NR method. It may be considered that it did not converge.

  FIG. 17 shows waveforms of inverter output voltage and current for one cycle obtained by the operating point solution method according to the second embodiment. The desired sine wave is obtained, and it can be confirmed that the operation of the inverter circuit is normally simulated. FIG. 18 shows a graph of the number of iterations at each calculation time step in this calculation. From the figure, it can be seen that in most calculation time steps, the number of iterations is about 6, but in some places there are more than 100 calculation time steps. As described above, the calculation time step in which the number of iterations exceeds 100 is a case where the conventional NR method does not converge. After the 100th iteration, the iterative calculation method is switched to the biaxial NR method. It converges immediately on iteration. Note that the number of times of 100 is set to a large value so that the difference in the number of iterations is easily discriminated when the conventional NR method is switched to the biaxial NR method in the graph of FIG. 10 to 20 iterations are sufficient.

  In order to investigate when the conventional NR method does not converge, FIG. 19 shows a plot of the voltage waveforms at both ends of the lower left arm and the lower right arm and the number of iterations with the same time. From the figure, it can be seen that the conventional NR method does not converge at the time when the voltage across the IGBT changes suddenly, that is, at the moment of switching. If the IGBT does not switch, the voltage and current in the circuit will not change suddenly, and the operating section of each nonlinear element will not move greatly. However, if the voltage and current change greatly due to switching, the operating section of each nonlinear element will also increase. If the infinite loop is formed in the convergence process, the conventional NR method will not converge. In this circuit, when the conventional NR method does not converge, the iterative calculation method converges at least 6 iterations immediately after switching to the biaxial NR method, and the Katzenelson method is similar to the electronic circuits I to III described above. It was never used. In other words, both the electronic circuits I to III and the inverter circuit converge at the stage of the biaxial NR method, and it can be said that the biaxial NR method has very good convergence characteristics. The Katzenelson method is rarely used, but it can be said that the two-axis NR method is provided in the latter stage in the sense of insurance as long as it is not a mathematically guaranteed method of convergence.

  FIG. 20 shows a comparison between the result of the operating point solution method according to the second embodiment and the result of the conventional NR method for the inverter output voltage waveform. As already shown in FIG. 17, when the operating point solution method according to the second embodiment is used, the iterative calculation normally converges at each time step, and the calculation for one cycle is completed without any problem. On the other hand, when the conventional NR method is used, the iterative calculation does not converge at around 2.3 ms (+ mark in FIG. 20), and the analysis cannot be continued.

  As described above, in the method of obtaining the operating point of the system in which the characteristic of the nonlinear element according to the present invention is approximated by a piecewise broken line, the program, the recording medium recording the program, and the simulation apparatus, the nonlinear element The operating point of a nonlinear system whose characteristics are approximated by piecewise broken lines can be converged quickly and reliably.

<Embodiment 3>
In the first embodiment and the second embodiment described above, the computer 20 is caused to execute the method for obtaining the operating point of the system in which the characteristic of the nonlinear element according to the present invention is approximated by piecewise broken lines with respect to the electric / electronic circuit. It is not limited to such an aspect. An embodiment of the present invention may be, for example, a real-time simulator device for connecting an electric / electronic circuit virtually constructed by the computer 20 and a real machine and verifying the operation of the real machine.

  FIG. 21 shows an example of a simulation apparatus according to this embodiment. In addition, the same code | symbol is attached | subjected to the same thing as Embodiment 1, and the overlapping description is abbreviate | omitted.

  As shown in the figure, the computer 20 is preset with characteristic information and initial solutions for each element of the electric / electronic circuit, and calculates the voltage / current at a predetermined element or node of the electric / electronic circuit. Yes. The calculated voltage and current values are converted into current and voltage by the D / A converter 51, amplified by the voltage / current amplifier 201, and flow to the control device 200. On the other hand, the control signal output from the control device 200 is input to the program 10 as data representing the voltage and current of one element in the electric / electronic circuit by the A / D converter 50.

  In the computer 20, when executing the method of obtaining the operating point of the system in which the characteristic of the nonlinear element according to the present invention is approximated by the piecewise broken line by the electronic computer, the electric / electronic is synchronized with the operation of the control device 200 in real time. Calculate the behavior of the circuit. That is, a part of the electric / electronic circuit can be simulated by the computer 20 and the operation of the control device 200 connected to the electric / electronic circuit can be verified.

<Other embodiments>
In each of the first to third embodiments, an electric / electronic circuit is targeted as a nonlinear system. However, the present invention is not limited to an electric / electronic circuit, and the present invention is applicable to all systems in which characteristics of nonlinear elements are approximated by piecewise broken lines. It can be done.

  In the second embodiment, the conventional NR method, the biaxial NR method, and the Katzenelson method are used together. However, the present invention is not necessarily limited to this configuration, and the conventional NR method and the biaxial NR method are used together. Any two of the three methods may be used in combination, for example, in combination with the Katzenelson method.

  The present invention can be used in industrial fields that require the determination of the operating point of a system in which the characteristics of nonlinear elements are approximated by piecewise broken lines.

DESCRIPTION OF SYMBOLS 1,100 Simulation apparatus 10 Program 11 Initial setting means 12 Operation | movement area identification means 13 Solution means 14 Convergence determination means 20 Computer 21 CPU
22 RAM
23 ROM
24 hard disk 30 input device 40 output device

Claims (7)

A method of obtaining an operating point of a nonlinear system in which the characteristics of the nonlinear element are approximated by piecewise broken lines by a computer,
An initial setting process for inputting characteristic information obtained by approximating the characteristic of the nonlinear element with a straight line for each of a plurality of operation sections and an initial solution x ⁽⁰⁾ of an equation representing the nonlinear system;
An action interval specifying process for specifying an action interval of the nonlinear element to which the solution x ⁽ⁿ⁾ belongs;
Equation (a) is obtained by calculating the slope and intercept of the straight line of the nonlinear element in the motion section, and linearizing an equation representing the nonlinear system from the slope and intercept.

F ^(n-1) value and y ^(n-1) value are calculated, and equation (a) is solved using the F ^(n-1) value and y ^(n-1) value. Solution processing for obtaining a solution x ⁽ⁿ⁾ ;
The operation section of the nonlinear element the solution x ⁽ⁿ⁾ belongs to the solution to obtain a solution x ⁽ⁿ⁾ when the operation period the solution x ^(n-1) belongs matches, if they do not match the operation section identifying And a convergence determination process for re-execution of the process and the solution finding process,
In the motion section specifying process, when the motion section of the solution x ⁽ⁿ⁾ (where n is 1 or more) is specified, the non-linear element of the nonlinear element to which the solution x ⁽ⁿ⁾ relative to the X axis belongs is specified. One motion interval and a second motion interval of the nonlinear element to which the solution x ⁽ⁿ⁾ when the Y axis is used as a reference are respectively obtained, and the solution x ^{(n−1) of} the first motion interval and the second motion interval is obtained. ⁾ a method of determining by said nonlinear element computer the operating point of the characteristics of the nonlinear elements and identifies divided polygonal line approximation systems towards close-operation period as an operation section of the nonlinear element of belonging. In the method for obtaining the operating point of the system in which the characteristic of the nonlinear element according to claim 1 is approximated by a piecewise broken line by an electronic computer,
After execution of the initial setting process, a solution is obtained using the Newton-Raphson method, and when the solution is not obtained within a predetermined number of iterations, the convergence determination process is executed from the action section specifying process. To obtain the operating point of a system in which the characteristics of the nonlinear element to be approximated are piecewise broken lines. In a method for obtaining an operating point of a system in which the characteristic of the nonlinear element according to claim 2 is approximated by a piecewise broken line by an electronic computer,
In the convergence determination process, when convergence does not occur within a predetermined number of times, a solution is obtained by switching to the Katzenelson method, and the operating point of the system in which the characteristic of the nonlinear element is approximated by a piecewise broken line is obtained by an electronic computer.   The program which makes an electronic computer perform the method of calculating | requiring the operating point of the system by which the characteristic of the nonlinear element as described in any one of Claims 1-3 was approximated by the piecewise broken line by the electronic computer.   A recording medium readable by an electronic computer in which the program according to claim 4 is recorded.   It has a computer comprised so that the method of calculating | requiring the operating point of the system by which the characteristic of the nonlinear element as described in any one of Claims 1-3 was approximated by piecewise broken line by an electronic computer may be characterized, Simulation device. The simulation apparatus according to claim 6,
The nonlinear system is a virtual electric / electronic circuit including a nonlinear element,
A control signal from the control device that controls the electric / electronic circuit is input as input data used for simulation of the electric / electronic circuit via an A / D converter,
A calculated value related to the voltage or current of the predetermined part of the electric / electronic circuit is converted to the voltage or current by a D / A converter and output to the control device,
A simulation apparatus configured to calculate the operation of the electric / electronic circuit in synchronization with the operation of the control apparatus in real time.

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