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peleshWiktoria Zielinska
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Create README file documenting classical generator model (#122)
* Creat README file for classical generator model.
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# Classical Generator
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An electrical machine model with two differential variables (i.e. second order
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model) is often called classical generator model. While its predicitve ability
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is limited, it is useful for studies of grid network properties. Mathematically,
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it is equivalent to a driven damped pendulum model.
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## Model Parameters
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Symbol | Units | Description | Note
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------------|---------|---------------------------------|----------------------
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$\omega_0$ | [rad/s] | synchronous frequency |
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$H$ | [s] | rotor inertia |
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$D$ | [p.u.] | damping coefficient |
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$R_a$ | [p.u.] | winding resistance |
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$X_{dp}$ | [p.u.] | machine reactance parameter |
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### Model Derived Parameters
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- $g = \dfrac{R_a}{R_a^2 + X_{dp}^2}$
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- $b = \dfrac{-X_{dp}}{R_a^2 + X_{dp}^2}$
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<br>
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## Model Variables
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### Internal Variables
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#### Differential
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Symbol | Units | Description | Note
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------------|---------|---------------------|----------------------
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$\delta$ | [rad] | machine power angle |
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$\omega$ | [p.u] | machine speed | Optionally read by a governor or a stabilizer component
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#### Algebraic
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Symbol | Units | Description | Note
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--------|--------|-------------------------------------|-------------
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$T_{e}$ | [p.u.] | electrical torque |
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$I_r$ | [p.u.] | machine real injection current | read by bus
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$I_i$ | [p.u.] | machine imaginary injection current | read by bus
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Note: All three can be expressed as function called by model equations. We add
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these as variables as they are needed for outputs.
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<br>
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### External Variables
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External variables enter component model equations but are owned by other
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components. The other components also provide equations needed to have a
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balanced system of equations.
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#### Differential
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None.
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#### Algebraic
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Symbol | Units | Description | Note
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-------|---------|-------------------------------|----------------------
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$V_r$ | [p.u.] | machine bus real voltage | owned by a bus object
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$V_i$ | [p.u.] | machine bus imaginary voltage | owned by a bus object
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$P_m$ | [p.u.] | mechanical power input | owned by governor, constant if no governor is connected to the machine
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$E_p$ | [p.u.] | field winding voltage | owned by exciter, constant if no exciter is connected to the machine
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<br>
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## Model Equations
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### Differential Equations
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```math
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\begin{aligned}
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\dot{\delta} &= (\omega - 1) \cdot \omega_0 \\
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\dot{\omega} &= \frac{1}{2H}\left( \frac{P_{m} - D(\omega - 1)}{\omega} - T_{e}\right)
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\end{aligned}
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```
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### Algebraic Equations
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```math
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\begin{aligned}
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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)\\
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0 &= I_r + gV_r - bV_i - E_p(g \cos\delta - b \sin\delta) \\
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0 &= I_i + gV_r + bV_i - E_p(b \cos\delta + g \sin\delta)
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\end{aligned}
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```
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As noted earlier, all three algebraic equations can be expressed as functions
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and substituted directly in the component and bus equations, respectively. We
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use redundant variables for modeling convenience.
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<br>
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## Initialization
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To initialize the model, given bus voltages $V_r$, $V_i$, and initial generator
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injection active and reactive power, $P$ and $Q$, we take following steps to
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initialize the system:
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First compute injection currents from initial power injection power and bus
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voltages:
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```math
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\begin{aligned}
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I_r &= \frac{PV_r + QV_i}{V_r^2 + V_i^2} \\
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I_i &= \frac{PV_i - QV_r}{V_r^2 + V_i^2}
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\end{aligned}
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```
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Next compute field winding voltage and machine angle:
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```math
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\begin{aligned}
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E_r &= \frac{ g(I_r + gV_r - bV_i) + b (I_i + bV_r + gV_i) }{g^2 + b^2} \\
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E_i &= \frac{ -b(I_r + gV_r - bV_i) + g (I_i + bV_r + gV_i) }{g^2 + b^2} \\
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E_p &= \sqrt{E_r^2 + E_i^2} \\
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\delta &= \arctan \dfrac{E_i}{E_r}
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\end{aligned}
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```
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Set machine speed to the synchronous speed:
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```math
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\omega = 1
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```
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Now, we can compute electrical torque and set mechanical torque to be equal
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to the electrical.
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```math
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\begin{aligned}
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T_{elec} &= gE_p^2 - E_p \left[ (gV_r - bV_i ) \cos\delta + (bV_r + gV_i )\sin\delta \right] \\
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P_{mech} &= T_{elec}
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\end{aligned}
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```
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With this, we initialize the machine at a steady state.

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