Modelica中关联连接器与方程：解决过定系统问题

我尝试开发一款MathBox，将微分方程描述的数学模型关联到可接入电路的模型中。作为Modelica新手，我先实现了U=R*I的线性模型并验证可行；但在添加DC/DC Buck变换器平均模型的行为/微分方程后，出现过定系统错误（模型含34个方程、32个变量）。以下为相关代码及报错信息，求问如何实现行为方程且避免过定系统？

原模型代码 #

model mathBox_buckConverterAVG
  extends Modelica.Electrical.Analog.Interfaces.FourPin;

  Real D = 0.5; // Duty cycle
  
  parameter Modelica.Units.SI.Inductance L1 = 1e-3; // Inductance (H)
  parameter Modelica.Units.SI.Capacitance C1 = 1e-6; // Capacitance (F)

  Modelica.Units.SI.Current iL1 "Current through inductance L1";
  Modelica.Units.SI.Voltage vC1 "Voltage over capacitor C1";

/*     p1       L1               p2
i1 ->  x------|||||--------------x  <- i2
                    |          
                   |||        
                   |||C1                    
                   |||        
       n1           |            n2
i1 <-  x-------------------------x  -> i2
*/
equation 

  // Behavioural equation
  der(vC1) = -1/((v2/p2.i)*C1)*vC1 + iL1/C1; // Capacitor voltage
  der(iL1) = -1/L1 * vC1 + D * v1 / L1; // Inductor current

  // Voltage connection
  v1 = L1 * der(iL1) + vC1;
  v2 = vC1;  
  n1.v = n2.v;
  
  // Current connection
  p1.i = iL1;
  0 = p1.i + p2.i - C1*der(vC1);
  0 = n1.i + n2.i + C1*der(vC1);

  annotation (uses(Modelica(version="4.0.0")));
end mathBox_buckConverterAVG;

测试电路代码 #

model Test_buckConverterAVG
  Modelica.Electrical.Analog.Sources.ConstantVoltage constantVoltage(V=10)
    annotation (Placement(transformation(
        extent={{-10,-10},{10,10}},
        rotation=270,
        origin={-48,0})));
  Modelica.Electrical.Analog.Basic.Ground ground annotation (Placement(transformation(extent={{-58,-34},{-38,-14}})));
  Modelica.Electrical.Analog.Basic.Resistor resistor(R=1)
    annotation (Placement(transformation(
        extent={{-10,-10},{10,10}},
        rotation=270,
        origin={40,0})));
  mathBox_buckConverterAVG mathBox_buckConverterAVG1 annotation(
    Placement(visible = true, transformation(extent = {{-10, -10}, {10, 10}}, rotation = 0)));
equation
  connect(constantVoltage.n, ground.p) annotation(
    Line(points = {{-48, -10}, {-48, -14}}, color = {0, 0, 255}));
  connect(mathBox_buckConverterAVG1.n2, resistor.n) annotation(
    Line(points = {{10, -10}, {14, -10}, {14, -14}, {40, -14}, {40, -10}}, color = {0, 0, 255}));
  connect(resistor.p, mathBox_buckConverterAVG1.p2) annotation(
    Line(points = {{40, 10}, {40, 14}, {14, 14}, {14, 10}, {10, 10}}, color = {0, 0, 255}));
  connect(constantVoltage.n, mathBox_buckConverterAVG1.n1) annotation(
    Line(points = {{-48, -10}, {-48, -14}, {-14, -14}, {-14, -10}, {-10, -10}}, color = {0, 0, 255}));
  connect(constantVoltage.p, mathBox_buckConverterAVG1.p1) annotation(
    Line(points = {{-48, 10}, {-48, 14}, {-14, 14}, {-14, 10}, {-10, 10}}, color = {0, 0, 255}));
  annotation (
    Icon(coordinateSystem(preserveAspectRatio=false)),
    Diagram(coordinateSystem(preserveAspectRatio=false)),
    uses(Modelica(version="4.0.0")));
end Test_buckConverterAVG;

报错信息 #

[6] 15:27:21 Translation Error
Internal error Transformation Module PFPlusExt index Reduction Method Pantelides failed!

[7] 15:27:21 Translation Error
post-optimization module removeSimpleEquations (simulation) failed.

[1] 15:34:09 Symbolic Error
Too many equations, over-determined system. The model has 34 equation(s) and 32 variable(s).

过定系统的核心原因是重复定义方程+物理逻辑错误，具体修正点如下：

• 移除冗余电压方程：你已经通过der(iL1)的微分方程定义了电感的电压电流关系，后续的v1 = L1 * der(iL1) + vC1属于重复方程，直接删除。
• 修正电容微分方程：原方程中v2/p2.i的逻辑错误，电容电压的微分方程应符合KCL，即der(vC1) = (iL1 - p2.i)/C1（电容电流等于电感电流减去负载电流）。
• 删除冗余电流方程：0 = p1.i + p2.i - C1*der(vC1)和0 = n1.i + n2.i + C1*der(vC1)是重复的KCL约束，且p1.i = iL1已定义电感电流，保留后者即可。
• 利用接口内置约束：n1.v = n2.v是FourPin接口隐含的公共端等电位约束，无需重复定义，直接删除。

修正后的模型代码 #

model mathBox_buckConverterAVG
  extends Modelica.Electrical.Analog.Interfaces.FourPin;

  Real D = 0.5; // Duty cycle
  
  parameter Modelica.Units.SI.Inductance L1 = 1e-3; // Inductance (H)
  parameter Modelica.Units.SI.Capacitance C1 = 1e-6; // Capacitance (F)

  Modelica.Units.SI.Current iL1 "Current through inductance L1";
  Modelica.Units.SI.Voltage vC1 "Voltage over capacitor C1";

/*     p1       L1               p2
i1 ->  x------|||||--------------x  <- i2
                    |          
                   |||        
                   |||C1                    
                   |||        
       n1           |            n2
i1 <-  x-------------------------x  -> i2
*/
equation 

  // Behavioural equation
  der(vC1) = (iL1 - p2.i)/C1; // 修正后的电容电压微分方程
  der(iL1) = -1/L1 * vC1 + D * v1 / L1; // 电感电流微分方程

  // Voltage connection
  v2 = vC1;  // 输出电压等于电容电压
  
  // Current connection
  p1.i = iL1; // 输入电流等于电感电流

  annotation (uses(Modelica(version="4.0.0")));
end mathBox_buckConverterAVG;

测试电路无需修改，直接运行即可完成正常仿真。

内容的提问来源于stack exchange，提问作者Danakil