Notation: $I_{a,b}$ represents the current flowing into Bus b from the branch connecting Bus a and Bus b, $I_{G,a}$ the current from generator a, $I_{L,a}$ the current from load a, and $V_a$ the voltage at Bus a.

Bus equations:

$$I_{G,1}+I_{2,1}+I_{3,1}+I_{L,1} = 0$$ $$I_{L,2}+I_{1,2}+I_{3,2} = 0$$ $$I_{G,3}+I_{1,3}+I_{2,3} = 0$$

Branch equations:

(branch connecting Bus 1 and Bus 2)

$$\left[ \begin{array}{c} I_{2,1} \\\ I_{1,2} \end{array}\right] = Y^{[1]}\left[ \begin{array}{c} V_1 \\\ V_2 \end{array}\right]$$

(branch connecting Bus 2 and Bus 3)

$$\left[ \begin{array}{c} I_{3,2} \\\ I_{2,3} \end{array}\right] = Y^{[2]}\left[ \begin{array}{c} V_2 \\\ V_3 \end{array}\right]$$

(branch connecting Bus 1 and Bus 3)

$$\left[ \begin{array}{c} I_{3,1} \\\ I_{1,3} \end{array}\right] = Y^{[3]}\left[ \begin{array}{c} V_1 \\\ V_3 \end{array}\right]$$

Load equations:

$$I_{L,1} = \frac{V_1}{R_1}$$ $$I_{L,2} = \frac{V_2}{R_2}$$

Generator 1

Differential equations:

$$\dot{\delta} = \omega \cdot \omega_0$$ $$\dot{\omega} = \frac{1}{2H}\left( \frac{P_{mech} - D\omega}{1+\omega} - T_{elec}\right)$$ $$\dot{E_{qp}} = \frac{1}{T_{dop}}\left( E_{fd} - \left( E_{qp} + X_{d1}\left( I_d + X_{d3}\left( E_{qp} - \psi_{dp} - X_{d2}I_d \right) \right) + \psi_{dpp}\cdot k_{sat} \right) \right)$$ $$\dot{\psi}_{dp} = \frac{1}{T_{dopp}} \left( E_{qp} - \psi_{dp} - X_{d2} I_d \right)$$ $$\dot{\psi}_{qp} = \frac{1}{T_{qopp}} \left( E_{dp} - \psi_{qp} + X_{q2}I_q\right)$$ $$\dot{E}_{dp} = \frac{1}{T_{qop}}\left(-E_{dp} + X_{qd}\psi_{qpp}\cdot k_{sat} + X_{q1} \left(I_q - X_{q3} \left(E_{dp} + I_qX_{q2} - \psi_{qp}\right)\right)\right)$$

Algebraic equations:

$$V_d = -\psi_{qpp}(\psi_{qp^{\prime}}E_{dp})(1+\omega)$$ $$V_q = \psi_{dpp}(\psi_{dp^{\prime}}E_{qp})(1+\omega)$$ $$I_d = I_{G,1, r}\sin\,\sin(\delta) - I_{G,1,i}\cos\,\cos(\delta)$$ $$I_q = I_{G,1, r}\cos\,\cos(\delta) + I_{G,1,i}\sin(\delta)$$

Network interfaces:

$$V_d = V_{1,r}\sin\,\sin(\delta) - V_{1,i}\cos\,\cos(\delta) + I_d R_a -I_q X_{qpp}$$ $$V_q = V_{1,r}\cos\,\cos(\delta) + V_{1,i} \sin\, \sin(\delta) + I_{d}X_{qpp} + I_{q}R_a$$

Generator 3

Differential equations:

$$\dot{\delta} = \omega \cdot \omega_0$$ $$\dot{\omega} = \frac{1}{2H}\left( \frac{P_{mech} - D\omega}{1+\omega} - T_{elec}\right)$$ $$\dot{E_{qp}} = \frac{1}{T_{dop}}\left( E_{fd} - \left( E_{qp} + X_{d1}\left( I_d + X_{d3}\left( E_{qp} - \psi_{dp} - X_{d2}I_d \right) \right) + \psi_{dpp}\cdot k_{sat} \right) \right)$$ $$\dot{\psi}_{dp} = \frac{1}{T_{dopp}} \left( E_{qp} - \psi_{dp} - X_{d2} I_d \right)$$ $$\dot{\psi}_{qp} = \frac{1}{T_{qopp}} \left( E_{dp} - \psi_{qp} + X_{q2}I_q\right)$$ $$\dot{E}_{dp} = \frac{1}{T_{qop}}\left(-E_{dp} + X_{qd}\psi_{qpp}\cdot k_{sat} + X_{q1} \left(I_q - X_{q3} \left(E_{dp} + I_qX_{q2} - \psi_{qp}\right)\right)\right)$$

Algebraic equations:

$$V_d = -\psi_{qpp}(\psi_{qp^{\prime}}E_{dp})(1+\omega)$$ $$V_q = \psi_{dpp}(\psi_{dp^{\prime}}E_{qp})(1+\omega)$$ $$I_d = I_{G,3, r}\sin\,\sin(\delta) - I_{G,3,i}\cos\,\cos(\delta)$$ $$I_q = I_{G,3, r}\cos\,\cos(\delta) + I_{G,3,i}\sin(\delta)$$

Network interfaces:

$$V_d = V_{3,r}\sin\,\sin(\delta) - V_{3,i}\cos\,\cos(\delta) + I_d R_a -I_q X_{qpp}$$ $$V_q = V_{3,r}\cos\,\cos(\delta) + V_{3,i} \sin\, \sin(\delta) + I_{d}X_{qpp} + I_{q}R_a$$