Lukel/phasor docs

src/Model/PhasorDynamics/SynchronousMachine/GENROUwS/README.md

@@ -1,116 +1,191 @@
 # GENROU
-## Simplifications
-
-- $`X''_{q}=X''_{d}`$
-- $`X''_{d}`$ does not saturate
-- same relative amount of saturation occurs on both $`d`$ and $`q`$ axis
 
+## Block Diagram
 <div align="center">
    <img align="center" src="../../../../../docs/Figures/GENROU.JPG">
 
-
-  Figure 2: GENROU. Figure courtesy of [PowerWorld](https://www.powerworld.com/WebHelp/)
+  Figure 2: GENROU. Figure courtesy of 
+  [PowerWorld](https://www.powerworld.com/WebHelp/)
 </div>
 
-## Equations
-### Algebraic Equations
-
+## Simplifications
+The GENROU model is a variation of the 
+[General Synchronous Machine Model](../README.md)
+- $`X''_{q}=X''_{d}`$
+- $`X''_{d}`$ does not saturate
+- Same relative amount of saturation occurs on both $`d`$ and $`q`$ axis
 
-- Fluxes
+## Nomenclature
+### Algebraic Variables
+- $V_d$, $V_q$   Machine Internal Voltage on the machine d-q reference frame  
+- $I_d$, $I_q$   Terminal currents on the machine d-q reference frame  
+- $V_r$, $V_i$    Terminal voltages on the network reference frame
+- $I_r$, $I_i$   Terminal currents on the network reference frame
+- $\psi''_q$, $\psi''_d$, $\psi''$    Machine Total Subtransient Flux
+- $T_{elec}$  Electrical Torque 
+- $P_{mech}$    Mechanical power from the prime mover 
+- $E_{fd}$     Field winding voltage from the excitation system 
+- $k_{sat}$   Saturation Coefficient 
+### Differential Variables
+- $\delta$    Machine Internal Angle
+- $\omega$  Machine Relative Speed
+- $\psi'_d$, $\psi'_q$, $E'_d$, $E'_q$  Machine Internal Flux Values
+### Parameters
+- $\omega_{0}$ - Nominal Frequnecy ($2\pi 60$)
+- $H$ - Intertia constant, sec (3)
+- $D$ - Damping factor, pu (0)
+- $R_{a}$ - Stator winding resistance, pu (0)
+- $X_{\ell}$ - Stator leakage reactance, pu (0.15)
+- $X_{d}$ - Direct axis synchronous reactance, (2.1)
+- $X'_{d}$ - Direct axis transient reactance, (0.2)
+- $X''_{d}$ - Direct axis sub-transient reactance, (0.18)
+- $X_{q}$ - Quadrature axis synchronous reactance, (0.5)
+- $X'_{q}$ - Quadrature axis transient reactance, (0.47619)
+- $X''_{q}$ - Quadrature axis sub-transient reactance, (0.18)
+- $T'_{d0}$ - Open circuit direct axis transient time const., (7)
+- $T''_{d0}$ - Open circuit direct axis sub-transient time const., (0.04)
+- $T'_{q0}$ - Open circuit quadrature axis transient time const., (0.75)
+- $T''_{q0}$ - Open circuit quadrature axis sub-transient time const., (0.05) 
+- $S_{10}$ - Saturation factor at 1.0 pu flux, (0) 
+- $S_{12}$ - Saturation factor at 1.2 pu flux, (0) 
 
+### Auxillary Parameters
+Transformed parameters used during implementation and for readability.
 ``` math
-E''_{d}=-\psi''_{q}=+E'_{d}\dfrac{X''_{q}-X_{l}}{X'_{q}-X_{l}}+\psi'_{q}\dfrac{X'_{q}-X''_{q}}{X'_{q}-X_{l}}
+\begin{aligned}
+  G   &=\dfrac{R_a}{R_a^2+(X''_q)^2}&
+  B   &=\dfrac{X''_q}{R_a^2+(X''_q)^2}\\
+  S_A &= \dfrac{1.2\sqrt{S_{10}/S_{12}} +1}{\sqrt{S_{10}/S_{12}} +1} & 
+  S_B &= \dfrac{1.2\sqrt{S_{10}/S_{12}} -1}{\sqrt{S_{10}/S_{12}} -1} \\
+  X_{d1} &= X_d-X_d'      & X_{q1} &= X_q-X_q' \\
+  X_{d2} &= X_d'-X_\ell  & X_{q2} &= X_q'-X_\ell\\
+  X_{d3} &= (X_d'-X_d'')/X_{d2}^2 & X_{q3} &= (X_q'-X_q'')/X_{q2}^2 \\
+  X_{d5} &= (X_d''-X_\ell)/X_{d2}    & X_{q5} &= (X_q''-X_\ell)/X_{q2}\\
+  X_{qd} &= (X_q-X_\ell)/(X_d-X_\ell)
+\end{aligned}
 ```
-``` math
-E''_{q}=\psi''_{d}=+E'_{q}\dfrac{X''_{d}-X_{l}}{X'_{d}-X_{l}}+\psi'_{d}\dfrac{X'_{d}-X''_{d}}{X'_{d}-X_{l}}
-```
-```math
-\psi_{d}=-I_{d}X''_{d}+E'_{q}\dfrac{X''_{d}-X_{l}}{X'_{d}-X_{l}}+\psi'_{d}\dfrac{X'_{d}-X''_{d}}{X'_{d}-X_{l}}=-I_{d}X''_{d}+E''_{q}
-```
-```math
-\psi_{q}=-I_{q}X''_{q}-E'_{d}\dfrac{X''_{q}-X_{l}}{X'_{q}-X_{l}}-\psi'_{q}\dfrac{X'_{q}-X''_{q}}{X'_{q}-X_{l}}=-I_{q}X''_{q}-E''_{d}
-```
-- Stator
-``` math
-V_{dterm}=E''_{d}(1+\Delta\omega_{pu})-R_{s}I_{d}+X''_{q}I_{q}
-```
-``` math
-V_{qterm}=E''_{q}(1+\Delta\omega_{pu})-R_{s}I_{q}-X''_{d}I_{d}
-```
-
-### Differential Equations
 
+## Equations
 
-- Mechanical Dynamic Equations
-``` math
-\dfrac{d\delta}{dt}=\Delta \omega_{pu}*\omega_{s}
-```
+### Differential Equations
 ``` math
-2H\dfrac{d\omega}{dt}=\dfrac{P_{mech}-D\omega}{1+\Delta\omega_{pu}}-(\psi_{d}I_{q}-\psi_{q}I_{d})
-```
-- Rotor Dynamic Equations
-```math
-T'_{d0}\dfrac{dE'_{q}}{dt}=E_{fd}-E'_{q}-(X_{d}-X'_{d})(I_{d}-\dfrac{X'_{d}-X''_{d}}{(X'_{d}-X_{l})^2}(+\psi'_{d}+(X'_{d}-X_{l})I_{d}-E'_{q}))-\psi''_{d}Sat(\psi'')
-```
-```math
-T''_{d0}\dfrac{d\psi'_{d}}{dt}=-\psi'_{d}-(X'_{d}-X_{l})I_{d}+E'_{q}
-```
-```math
-T''_{q0}\dfrac{d\psi'_{q}}{dt}=-\psi'_{q}+(X'_{q}-X_{l})I_{q}+E'_{d}
-```
-```math
-T'_{q0}\dfrac{dE'_{d}}{dt}= -E'_{d}+(X_{q}-X'_{q})(I_{q}-\dfrac{X'_{q}-X''_{q}}{(X'_{q}-X_{l})^2}(-\psi'_{q}+(X'_{q}-X_{l})I_{q}+E'_{d}))+\psi''_{q}(\dfrac{X_{q}-X_{l}}{X_{d}-X_{l}})Sat(\psi'')
+\begin{aligned}
+  \dot\delta      &= \omega\cdot\omega_0 \\
+  \dot\omega      &= \dfrac{1}{2H}\left(\dfrac{P_{mech}-D\omega}{1+\omega}
+                   - T_{elec}\right)\\
+  \dot{\psi}'_{d} &= \dfrac{1}{T''_{d0}}(E'_{q}-\psi'_{d}-X_{d2}I_{d})\\
+  \dot{\psi}'_{q} &= \dfrac{1}{T''_{q0}}(E'_{d}-\psi'_{q}+X_{q2}I_{q})\\
+  \dot{E}'_{d}    &= \dfrac{1}{T'_{q0}}
+    \left( -E'_{d}+X_{q1}
+      (I_{q}-X_{q3}(E'_{d}-\psi'_{q}+X_{q2}I_{q}))
+      + X_{qd}\psi''_{q}k_{sat}
+    \right) \\
+  \dot{E}'_{q} &= \dfrac{1}{T'_{d0}}
+    \left(
+      E_{fd}-E'_{q}-X_{d1}
+      (I_{d}+X_{d3}(E'_{q}-\psi'_{d}-X_{d2}I_{d}))
+      -\psi''_{d}k_{sat}
+    \right)\\
+\end{aligned}
 ```
-## Initialization
-
-From the block diagram it can be written:
 
-```math
--\psi'_{d}-(X'_{d}-X_{l})I_{d}+E'_{q}=0
-```
+### Algebraic Equations
+These algebraic equations define internal variables (7) and the algebraic 
+Network Interface Equations (4)
 ``` math
--\psi''_{d}+E'_{q}\dfrac{X''_{d}-X_{l}}{X'_{d}-X_{l}}+\psi'_{d}\dfrac{X'_{d}-X''_{d}}{X'_{d}-X_{l}}=0
-```
-```math
--\psi'_{q}+(X'_{q}-X_{l})I_{q}+E'_{d}=0
+\begin{aligned}
+  \psi''_{q} &= -E'_{d}X_{q5} - \psi'_{q}X_{q4} \\
+  \psi''_{d} &= +E'_{q}X_{d5} + \psi'_{d}X_{d4}\\
+  \psi''     &= \sqrt{(\psi''_{d})^2+(\psi''_{q})^2} \\
+  V_{d}      &= -\psi''_{q}(1+\omega)\\
+  V_{q}      &= +\psi''_{d}(1+\omega)\\
+  T_{elec}   &= (\psi''_{d} - I_dX_d'')I_q-(\psi''_{q} - I_qX_d'')I_d \\
+\end{aligned}
 ```
+
+#### Network Interface equations
+The network interface equations provide the algebraic relationship the network
+ and internal reference frame.
 ``` math
-\psi''_{q}+E'_{d}\dfrac{X''_{q}-X_{l}}{X'_{q}-X_{l}}+\psi'_{q}\dfrac{X'_{q}-X''_{q}}{X'_{q}-X_{l}}=0
-```
-```math
--E'_{d}+(X_{q}-X'_{q})I_{q}+\psi''_{q}(\dfrac{X_{q}-X_{l}}{X_{d}-X_{l}})Sat(\psi'')=0
+\begin{aligned}
+  \begin{bmatrix}
+    I_d \\ I_q
+  \end{bmatrix}
+  &=
+  \begin{bmatrix}
+    \sin \delta & -\cos\delta \\
+    \cos\delta  &  \sin\delta
+  \end{bmatrix}
+  \begin{bmatrix}
+    I_r \\ I_i
+  \end{bmatrix}
+  \\
+  \begin{bmatrix}
+    I_r \\ I_i
+  \end{bmatrix}
+  &=
+  \begin{bmatrix}
+    G & -B \\
+    B & G
+  \end{bmatrix}
+  \left(
+    \begin{bmatrix}
+      \sin \delta & \cos\delta \\
+      -\cos\delta & \sin\delta
+    \end{bmatrix}
+    \begin{bmatrix}
+      V_d \\ V_q
+    \end{bmatrix}
+    -  
+    \begin{bmatrix}
+      V_r \\V_i
+    \end{bmatrix}
+  \right)
+\end{aligned}
 ```
 
-Internal voltage on the referece frame can be calculated directly:
-```math
-V_{r}=V_{rterm}+R_{a}I_{r}-X''_{d}I_{i}
-```
+## Initialization
+
+### Without Saturation
+Pressume there is no saturation to simplify solution procedure for initial 
+conditions.
+
+Using the power-flow solution, we have explicity solutions for the following 
+variables. The internal variables $I_d$, $I_q$, $V_d$, and $V_q$ are calculated
+from the network interface equations. The remaining are algebraicillay solved 
+from the steady-state initial conditions.
 ``` math
-V_{i}=V_{iterm}+R_{a}I_{i}-X''_{d}I_{r}
-```
-then
-```math
-Sat(\psi'')=Sat(\vert V_{r}+jV_{i} \vert)
+\begin{aligned}
+\omega &= 0 \\
+\delta &= \text{arg} \left[V_r + jV_i + (R_a + jX_q) (I_r + jI_i)\right] \\
+  \psi^{''}_{d} &= V_q \\
+  \psi^{''}_{q} &= -V_d \\
+  \psi^{''} &= \sqrt{(\psi''_{d})^2+(\psi''_{q})^2} \\
+  k_{sat}     &= S_B(\psi^{''}-S_A)^2 \\
+  T_{elec}    &= (\psi''_{d} - I_dX_d^{''})I_q-(\psi''_{q} - I_qX_d^{''})I_d \\
+  P_{mech}    &= T_{elec} \\
+  \psi^{'}_d  &=
+  \dfrac{\psi^{''}_d-X_{d5}X_{d2}I_d}{X_{d5}+1}\\
+  \psi^{'}_q  &=\dfrac{X_{q5}X_{q2}I_q-\psi^{''}_q}{X_{q5}+1}\\
+  E^{'}_d     &=\psi^{'}_q - X_{q2}I_q \\
+  E^{'}_q     &=\psi^{'}_d + X_{d2}I_d \\
+  E_{fd}      &= E'_{q}+X_{d1}I_{d}+\psi^{''}_{d}k_{sat} \\
+\end{aligned}
 ```
 
-It is important to point out that finding the initial value of $`\delta`$ for
+### With Saturation
+It is important to point out that finding the initial value of $\delta$ for
 the model without saturation direct method can be used. In case when saturation
 is considered some "claver" math is needed. Key insight for determining initial
-$`\delta`$ is that the magnitude of the saturation depends upon the magnitude
-of $`\psi''`$, which is independent of $`\delta`$.
-```math
-\delta=\tan^{-1}\left(\dfrac{K_{sat}V_{iterm}+K_{sat}R_{a}I_{i}+(K_{sat}X''_{d}+X_{q}-X''_{q})I_{r}}
-                            {K_{sat}V_{rterm}+K_{sat}R_{a}I_{r}-(K_{sat}X''_{d}+X_{q}-X''_{q})I_{i}} \right)
-```
-where
-```math
-K_{sat}=(1+(\dfrac{X_{q}-X_{l}}{X_{d}-X_{l}})Sat(\psi''))
-```
-Following must be true (if not enforce the corrections):
+$\delta$ is that the magnitude of the saturation depends upon the magnitude
+of $\psi''$, which is independent of $\delta$.
 
-```math
-X_{l}<=X''{q}<=X'{q}<=Xq
-```
-```math
-X_{l}<=X''{d}<=X'{d}<=Xd
+``` math
+\begin{aligned}
+  \delta=\tan^{-1}
+  \left[
+    \dfrac{(V_{i}+R_{a}I_{i})k_{sat}+(k_{sat}X''_{d}+X_{q}-X''_{q})I_{r}}
+          {(V_{r}+R_{a}I_{r})k_{sat}-(k_{sat}X''_{d}+X_{q}-X''_{q})I_{i}}
+  \right]
+\end{aligned}
 ```