|
| 1 | +# Classical Generator |
| 2 | + |
| 3 | +An electrical machine model with two differential variables (i.e. second order |
| 4 | +model) is often called classical generator model. While its predicitve ability |
| 5 | +is limited, it is useful for studies of grid network properties. Mathematically, |
| 6 | +it is equivalent to a driven damped pendulum model. |
| 7 | + |
| 8 | +## Model Parameters |
| 9 | + |
| 10 | +Symbol | Units | Description | Note |
| 11 | +------------|---------|---------------------------------|---------------------- |
| 12 | +$\omega_0$ | [rad/s] | synchronous frequency | |
| 13 | +$H$ | [s] | rotor inertia | |
| 14 | +$D$ | [p.u.] | damping coefficient | |
| 15 | +$R_a$ | [p.u.] | winding resistance | |
| 16 | +$X_{dp}$ | [p.u.] | machine reactance parameter | |
| 17 | + |
| 18 | +### Model Derived Parameters |
| 19 | + |
| 20 | +- $g = \dfrac{R_a}{R_a^2 + X_{dp}^2}$ |
| 21 | +- $b = \dfrac{-X_{dp}}{R_a^2 + X_{dp}^2}$ |
| 22 | + |
| 23 | +<br> |
| 24 | + |
| 25 | +## Model Variables |
| 26 | + |
| 27 | +### Internal Variables |
| 28 | + |
| 29 | +#### Differential |
| 30 | + |
| 31 | +Symbol | Units | Description | Note |
| 32 | +------------|---------|---------------------|---------------------- |
| 33 | +$\delta$ | [rad] | machine power angle | |
| 34 | +$\omega$ | [p.u] | machine speed | Optionally read by a governor or a stabilizer component |
| 35 | + |
| 36 | +#### Algebraic |
| 37 | + |
| 38 | +Symbol | Units | Description | Note |
| 39 | +--------|--------|-------------------------------------|------------- |
| 40 | +$T_{e}$ | [p.u.] | electrical torque | |
| 41 | +$I_r$ | [p.u.] | machine real injection current | read by bus |
| 42 | +$I_i$ | [p.u.] | machine imaginary injection current | read by bus |
| 43 | + |
| 44 | +Note: All three can be expressed as function called by model equations. We add |
| 45 | +these as variables as they are needed for outputs. |
| 46 | + |
| 47 | +<br> |
| 48 | + |
| 49 | +### External Variables |
| 50 | + |
| 51 | +External variables enter component model equations but are owned by other |
| 52 | +components. The other components also provide equations needed to have a |
| 53 | +balanced system of equations. |
| 54 | + |
| 55 | +#### Differential |
| 56 | + |
| 57 | +None. |
| 58 | + |
| 59 | +#### Algebraic |
| 60 | + |
| 61 | +Symbol | Units | Description | Note |
| 62 | +-------|---------|-------------------------------|---------------------- |
| 63 | +$V_r$ | [p.u.] | machine bus real voltage | owned by a bus object |
| 64 | +$V_i$ | [p.u.] | machine bus imaginary voltage | owned by a bus object |
| 65 | +$P_m$ | [p.u.] | mechanical power input | owned by governor, constant if no governor is connected to the machine |
| 66 | +$E_p$ | [p.u.] | field winding voltage | owned by exciter, constant if no exciter is connected to the machine |
| 67 | + |
| 68 | +<br> |
| 69 | + |
| 70 | + |
| 71 | +## Model Equations |
| 72 | + |
| 73 | +### Differential Equations |
| 74 | + |
| 75 | +```math |
| 76 | +\begin{aligned} |
| 77 | +\dot{\delta} &= (\omega - 1) \cdot \omega_0 \\ |
| 78 | +\dot{\omega} &= \frac{1}{2H}\left( \frac{P_{m} - D(\omega - 1)}{\omega} - T_{e}\right) |
| 79 | +\end{aligned} |
| 80 | +``` |
| 81 | + |
| 82 | +### Algebraic Equations |
| 83 | + |
| 84 | +```math |
| 85 | +\begin{aligned} |
| 86 | + 0 &= T_{e} - \frac{1}{\omega}\left( g E_p^2 - E_p \left[(gV_r - bV_i)\cos\delta + (bV_r + gV_i)\sin\delta \right]\right)\\ |
| 87 | + 0 &= I_r + gV_r - bV_i - E_p(g \cos\delta - b \sin\delta) \\ |
| 88 | + 0 &= I_i + gV_r + bV_i - E_p(b \cos\delta + g \sin\delta) |
| 89 | +\end{aligned} |
| 90 | +``` |
| 91 | +As noted earlier, all three algebraic equations can be expressed as functions |
| 92 | +and substituted directly in the component and bus equations, respectively. We |
| 93 | +use redundant variables for modeling convenience. |
| 94 | + |
| 95 | +<br> |
| 96 | + |
| 97 | +## Initialization |
| 98 | + |
| 99 | +To initialize the model, given bus voltages $V_r$, $V_i$, and initial generator |
| 100 | +injection active and reactive power, $P$ and $Q$, we take following steps to |
| 101 | +initialize the system: |
| 102 | + |
| 103 | +First compute injection currents from initial power injection power and bus |
| 104 | +voltages: |
| 105 | +```math |
| 106 | +\begin{aligned} |
| 107 | +I_r &= \frac{PV_r + QV_i}{V_r^2 + V_i^2} \\ |
| 108 | +I_i &= \frac{PV_i - QV_r}{V_r^2 + V_i^2} |
| 109 | +\end{aligned} |
| 110 | +``` |
| 111 | + |
| 112 | +Next compute field winding voltage and machine angle: |
| 113 | +```math |
| 114 | +\begin{aligned} |
| 115 | +E_r &= \frac{ g(I_r + gV_r - bV_i) + b (I_i + bV_r + gV_i) }{g^2 + b^2} \\ |
| 116 | +E_i &= \frac{ -b(I_r + gV_r - bV_i) + g (I_i + bV_r + gV_i) }{g^2 + b^2} \\ |
| 117 | +E_p &= \sqrt{E_r^2 + E_i^2} \\ |
| 118 | +\delta &= \arctan \dfrac{E_i}{E_r} |
| 119 | +\end{aligned} |
| 120 | +``` |
| 121 | + |
| 122 | +Set machine speed to the synchronous speed: |
| 123 | +```math |
| 124 | +\omega = 1 |
| 125 | +``` |
| 126 | + |
| 127 | +Now, we can compute electrical torque and set mechanical torque to be equal |
| 128 | +to the electrical. |
| 129 | +```math |
| 130 | +\begin{aligned} |
| 131 | +T_{elec} &= gE_p^2 - E_p \left[ (gV_r - bV_i ) \cos\delta + (bV_r + gV_i )\sin\delta \right] \\ |
| 132 | +P_{mech} &= T_{elec} |
| 133 | +\end{aligned} |
| 134 | +``` |
| 135 | + |
| 136 | +With this, we initialize the machine at a steady state. |
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