# Modeling and Simulation in Multibond Graphs Applied to Three-Phase Electrical Systems
## 2 Problem Statement
In the same form, it is established that  
\[v_{R_{e}}^{abc} = R_{e}^{abc}i_{e}^{abc} \tag{6}\] \[v_{I_{e}}^{abc} = L_{e}^{abc}\frac{d\dot{i}_{e}^{abc}}{dt}\] (7) \[v_{R_{2}}^{abc} = R_{2}^{abc}i_{2}^{abc}\] (8) \[v_{L_{2}}^{abc} = L_{2}^{abc}\frac{d\dot{i}_{2}^{abc}}{dt} \tag{9}\]  
where \(v_{R_{e}}^{abc}\) and \(i_{e}^{abc}\) denote the three-phase voltages and currents in \(R_{e}^{abc}\), respectively; \(v_{R_{2}}^{abc}\) and \(i_{2}^{abc}\) denote the three-phase voltages and currents in \(R_{2}^{abc}\), respectively; and \(v_{L_{e}}^{abc}\) and \(v_{L_{2}}^{abc}\) denote the three-phase voltages in \(L_{e}^{abc}\) and \(L_{2}^{abc}\), respectively. \(R_{e}^{abc}\) and \(R_{2}^{abc}\) are the resistance matrices of \(e\) and 2 of the three phases \((a,b,c)\), respectively, and \(L_{e}^{abc}\) and \(L_{2}^{abc}\) are the inductance matrices of \(e\) and 2 of the three phases \((a,b,c)\), respectively. The relationships for the capacitors are  
\[i_{C_{1}}^{abc} = \left[ \begin{array}{ccc}c_{1}^{a}&0&0\\ 0&c_{1}^{b}&0\\ 0&0&c_{1}^{c}\end{array} \right]\frac{d}{dt}\left[ \begin{array}{c}v_{C_{1}} ^{a}\\ v_{C_{1}}^{b}\\ v_{C_{1}}^{c}\end{array} \right]=C_{1}^{abc}\frac{dv_{C_{1}}^{abc}}{dt} \tag{10}\] \[i_{C_{2}}^{abc} = C_{2}^{abc}\frac{dv_{C_{2}}^{abc}}{dt} \tag{11}\]  
where \(i_{C_{1}}^{abc}\) and \(v_{C_{1}}^{abc}\) are the three-phase currents and voltages in \(C_{1}^{abc}\), respectively; \(i_{C_{2}}^{abc}\) and \(v_{C_{2}}^{abc}\) are the three-phase currents and voltages in \(C_{2}^{abc}\), respectively, \(C_{1}^{abc}\) and \(C_{2}^{abc}\) are the capacitance matrices 1 and 2 of the three phases \((a,b,c)\), respectively, and \(c_{1}^{a}\), \(c_{1}^{b}\) and \(c_{1}^{c}\) are the capacitors 1 in each phase \((a,b,c)\). By substituting from (4) and (9) into (1), (2) and (3)  
\[v_{1}^{abc} = R_{1}^{abc}i_{1}^{abc}+L_{1}^{abc}\frac{d\dot{i}_{1}^{abc}}{dt}+v _{C_{1}}^{abc} \tag{12}\] \[v_{C_{1}}^{abc} = R_{e}^{abc}i_{e}^{abc}+L_{e}^{abc}\frac{d\dot{i}_{e}^{abc}}{dt}+v _{C_{2}}^{abc}\] (13) \[v_{2}^{abc} = R_{2}^{abc}i_{2}^{abc}+L_{2}^{abc}\frac{d\dot{i}_{2}^{abc}}{dt}+v _{C_{2}}^{abc} \tag{14}\]  
in addition, \(i_{C_{1}}^{abc}=i_{1}^{abc}-i_{e}^{abc}\) and \(i_{C_{2}}^{abc}=i_{e}^{abc}-i_{2}^{abc}\), and the equations for the capacitors are  
\[C_{1}^{abc}\frac{dv_{C_{1}}^{abc}}{dt} = i_{1}^{abc}-i_{e}^{abc} \tag{15}\] \[C_{2}^{abc}\frac{dv_{C_{2}}^{abc}}{dt} = i_{e}^{abc}-i_{2}^{abc} \tag{16}\]From (12) to (16) represent a fifteenth-order dynamic system for a simple three-phase circuit. However, \(v_{1}^{abc}\) and \(v_{2}^{abc}\) are three-phase voltage sources, these sources are typically balanced and given by  
\[\left[ \begin{array}{c}v^{a}\\ v^{b}\\ v^{c}\end{array} \right]=\left[ \begin{array}{c}V_{m}\sin(wt)\\ V_{m}\sin(wt-2\pi/3)\\ V_{m}\sin(wt+2\pi/3)\end{array} \right] \tag{17}\]  
where \(V_{m}\) is the maximum value of \(v\) and \(w\) is the angular frequency such that \(w=2\pi f\) being \(f\) the frequency in \(Hz\). Hence, considering (17) with (12) to (16), the dynamic system is a linear time-varying (LTV) system of 15 dimensions. A great simplification in the mathematical description of a three-phase electrical system is obtained from [PERSON]'s transformation. The effect of [PERSON]'s transformation is simply to transform all stator quantities from phases \(a\), \(b\) and \(c\) into new variables, the frame of reference of which moves with the rotor [28; 29; 31]. Electrical systems that have three-phase voltages and currents are complicated to model, analyze and control to determine time-varying systems, so the Park transformation has been applied, which allows obtaining equivalent systems without dependence on time. In order to remove the time dependence, [PERSON]'s transformation can be used. This transformation for voltages is defined by  
\[t^{dq0}=Pi^{abc} \tag{18}\]  
where  
\[P=\sqrt{\frac{2}{3}}\left[