Models
Mathematical description of the models implemented in DPsim.
The following models are currently available:
- Dynamic phasors
- inductor, capacitor, resistor
- current and voltage source
- load (PQ and Z type)
- pi-line
- transmission line (Bergeron)
- synchronous generator dq-frame full order (Kundur, Krause)
- inverter averaged
- inverter with harmonics (comparable to switched model)
- switch
- EMT
- inductor, capacitor, resistor
- current and voltage source
- load (Z type)
- pi-line
- transmission line (Bergeron)
- synchronous generator dq-frame full order (Kundur, Krause)
- inverter averaged
- switch
1 - Power Electronics
DPsim provides several averaged power-electronic inverter models for simulation using EMT, DP, and SP network modeling domains.
Three-Phase Averaged Voltage Source Inverter with State-Space Nodal Interface
The EMT::Ph3::AvVoltSourceInverterStateSpace model represents a grid-following averaged voltage source inverter in the EMT domain.
It is implemented as a variable state-space nodal component and can therefore be directly stamped into the MNA system.
The model includes a PLL, filtered active/reactive power measurement, outer power control, inner current control, and an LC filter with coupling resistance to the grid node.
The terminal input is the PCC voltage vector
$$
\mathbf{u} =
\begin{bmatrix}
u_a & u_b & u_c
\end{bmatrix}^\top ,
$$
and the state vector is
$$
\mathbf{x} =
\begin{bmatrix}
\theta_{\mathrm{PLL}} &
\phi_{\mathrm{PLL}} &
P &
Q &
\phi_d &
\phi_q &
\gamma_d &
\gamma_q &
v_{c,a} &
v_{c,b} &
v_{c,c} &
i_{f,a} &
i_{f,b} &
i_{f,c}
\end{bmatrix}^\top .
$$
The model output is the interface current injected into the MNA system,
$$
\mathbf{y} =
\frac{\mathbf{u} - \mathbf{v}_c}{R_c}.
$$
Model equations
The controller uses the opposite current direction, i.e. positive current denotes inverter injection into the grid,
$$
\mathbf{i}_{rc} =
\frac{\mathbf{v}_c - \mathbf{u}}{R_c}.
$$
The Park transformation with PLL angle $\theta_{\mathrm{PLL}}$ is used to obtain dq quantities,
$$
\begin{bmatrix}
v_{c,d} \
v_{c,q}
\end{bmatrix}
\mathbf{T}(\theta_{\mathrm{PLL}})\mathbf{v}c,
\qquad
\begin{bmatrix}
i{rc,d} \
i_{rc,q}
\end{bmatrix}
\mathbf{T}(\theta_{\mathrm{PLL}})\mathbf{i}_{rc}.
$$
The instantaneous active and reactive powers are calculated as
$$
p = v_{c,d} i_{rc,d} + v_{c,q} i_{rc,q},
$$
$$
q = -v_{c,d} i_{rc,q} + v_{c,q} i_{rc,d}.
$$
The PLL and power-filter dynamics are
$$
\dot{\theta}_{\mathrm{PLL}}
\omega_n + K_{p,\mathrm{PLL}} v_{c,q} +
K_{i,\mathrm{PLL}} \phi_{\mathrm{PLL}},
$$
$$
\dot{\phi}{\mathrm{PLL}} = v{c,q},
$$
$$
\dot{P} = \omega_c(p - P),
\qquad
\dot{Q} = \omega_c(q - Q).
$$
The outer power-control integrators and current references are
$$
\dot{\phi}d = P{\mathrm{ref}} - P,
\qquad
\dot{\phi}q = Q - Q{\mathrm{ref}},
$$
$$
i_{d,\mathrm{ref}}
K_{p,P}(P_{\mathrm{ref}} - P) + K_{i,P}\phi_d,
$$
$$
i_{q,\mathrm{ref}}
K_{p,P}(Q - Q_{\mathrm{ref}}) + K_{i,P}\phi_q.
$$
The inner current-control integrators and voltage references are
$$
\dot{\gamma}d = i{d,\mathrm{ref}} - i_{rc,d},
\qquad
\dot{\gamma}q = i{q,\mathrm{ref}} - i_{rc,q},
$$
$$
v_{d,\mathrm{ref}}
K_{p,I}(i_{d,\mathrm{ref}} - i_{rc,d}) +
K_{i,I}\gamma_d,
$$
$$
v_{q,\mathrm{ref}}
K_{p,I}(i_{q,\mathrm{ref}} - i_{rc,q}) +
K_{i,I}\gamma_q.
$$
The reference voltage is transformed back to abc coordinates,
$$
\mathbf{v}_{\mathrm{ref}}
\mathbf{T}^{-1}(\theta_{\mathrm{PLL}})
\begin{bmatrix}
v_{d,\mathrm{ref}} \
v_{q,\mathrm{ref}}
\end{bmatrix}.
$$
The LC filter dynamics are
$$
\dot{\mathbf{v}}_c
\frac{1}{C_f}\mathbf{i}_f
+
\frac{1}{C_f R_c}(\mathbf{u} - \mathbf{v}_c),
$$
$$
\dot{\mathbf{i}}_f
\frac{1}{L_f}
\left(
\mathbf{v}_{\mathrm{ref}}
\mathbf{v}_c
R_f \mathbf{i}_f
\right).
$$
At each simulation step, the nonlinear model is locally linearized into the affine state-space form
$$
\dot{\mathbf{x}}
\approx
\mathbf{A}\mathbf{x}
+
\mathbf{B}\mathbf{u}
+
\mathbf{E},
$$
$$
\mathbf{y}
\approx
\mathbf{C}\mathbf{x}
+
\mathbf{D}\mathbf{u}
+
\mathbf{F},
$$
which is then discretized and stamped into the EMT MNA system.
Source code and examples
2 - Transformer
The transformer model is composed of an RL-segment and an ideal transformer.
The single line diagram is depicted in the figure below.
If node reduction is not applied, two virtual nodes are created to stamp this model into the system matrix.
Furthermore, the ideal transformer has an additional equation, which requires an extension of the system matrix.
The complete matrix stamp for the ideal transformer is
$$\begin{array}{c|c c c}
~ & j & k & l \cr
\hline
j & & & -1 \cr
k & & & T \cr
l & 1 & -T & 0
\end{array}
\begin{pmatrix}
v_j \cr
v_k \cr
i_{l} \cr
\end{pmatrix}
=
\begin{pmatrix}
\cr
\cr
0\cr
\end{pmatrix}$$
The variable $j$ denotes the high voltage node while $k$ is the low voltage node.
$l$ indicates the inserted row and column to accommodate the relation between the two voltages at the ends of the transformer.
The transformer ratio is defined as $T = V_{j} / V_{k}$.
A phase shift can be introduced if $T$ is considered as a complex number.
3 - Branches
RX-Line
PI-Line
4 - Induction Machine
5 - RLC-Elements
EMT Equations and Modified Nodal Analysis
Inductance
An inductance is described by
$$v_j(t) - v_k(t) = v_L(t) = L \frac{\mathrm{d} i_L(t)}{\mathrm{d}t}$$
Integration results in an equation to compute the current at time $t$ from a previous state at $t - \Delta t$.
$$i_L(t) = i_L(t - \Delta t) + \frac{1}{L} \ \int_{t - \Delta t}^{t} v_L(\tau) \ \mathrm{d} \tau$$
There are various methods to discretize this equation in order to solve it numerically.
The trapezoidal rule, an implicit second-order method, is commonly applied for circuit simulation:
$$\int_{t - \Delta t}^{t} f(\tau) \ \mathrm{d} \tau \approx \frac{\Delta t}{2}(f(t) + f(t - \Delta t))$$
Applying the trapezoidal rule to leads to
$$i_L(t) = i_L(t - \Delta t) + \frac{\Delta t}{2L}(v_L(t) + v_L(t - \Delta t))$$
This can be rewritten in terms of an equivalent conductance and current source and the number of time steps $k$ with size $\Delta t$.
$$i_L(k) = g_L v_L(k) + i_{L,equiv}(k-1)$$
$$i_{L,equiv}(k-1) = i_L(k-1) + \frac{\Delta t}{2L} v_L(k-1)$$
$$g_L = \frac{\Delta t}{2L}$$
Hence, components described by differential equations are transformed into a DC equivalent circuit as depicted in the figure below.
Capacitance
The same procedure can be applied to a capacitance.
Integration on both side yields
$$i_C(t) = C \frac{\mathrm{d}}{\mathrm{d}t} \ v_C(t)$$
$$v_C(t) = v_C(t - \Delta t) + \frac{1}{C} \int_{t - \Delta t}^t i_C(\tau) \mathrm{d} \tau$$
Finally, the equivalent circuit is described by a current source and a conductance.
$$i_{C}(k) = g_{C} v_C(k) + i_{C,equiv}(k-1)$$
$$i_{C,equiv}(k-1) = -i_{C}(k-1) - g_C v_C(k-1)$$
$$g_{C} = \frac{2C}{\Delta t}$$
This equation set is visualized in the figure below.
Hence, the vector of unknowns $\bm{x}$ and the source vector $\bm{b}$ become time dependent and this leads to the system description:
$$\bm{A} \bm{x}(t) = \bm{b}(t)$$
To simulate the transient behavior of circuits, this linear equation has to be solved repeatedly.
As long as the system topology and the time step is fixed, the system matrix is constant.
Extension with Dynamic Phasors
The dynamic phasor concept can be integrated with nodal analysis.
The overall procedure does not change but the system equations are rewritten using complex numbers and all variables need to be expressed in terms of dynamic phasors.
Therefore, the resistive companion representations of inductances and capacitances have to be adapted as well.
Inductance
In dynamic phasors the integration of the inductance equation yields
$$\begin{align}
\langle v_L \rangle(t) &= \Big \langle L \frac{\mathrm{d} i_L}{\mathrm{d}t} \Big \rangle(t) \nonumber \\
&= L \frac{\mathrm{d}}{dt} \langle i_L \rangle(t) + j \omega L \ \langle i_L \rangle(t)
\end{align}$$
$$\langle i_L \rangle(t) = \langle i_L \rangle(t - \Delta t) + \int_{t - \Delta t}^t \frac{1}{L} \langle v_L \rangle(\tau) - j \omega \ \langle i_L \rangle(\tau) \mathrm{d} \tau$$
Applying the trapezoidal method leads to the finite difference equation:
$$\begin{split}
\langle i_L \rangle(k) = \langle i_L \rangle(k-1) + \frac{\Delta t}{2} \bigg[ \frac{1}{L} (\langle v_L \rangle(k) + \langle v_L \rangle(k-1))
- j \omega (\langle i_L \rangle(t) + \langle i_L \rangle(k-1) \bigg]
\end{split}$$
Solving this for $\langle i_L \rangle(k)$ results in the \ac{DP} equivalent circuit model:
$$\langle i_L \rangle(k) = \frac{a - jab}{1 + b^2} \langle v_L \rangle(k) + \langle i_{L,equiv} \rangle(k-1)$$
with
$$a = \frac{\Delta t}{2L}, \qquad b = \frac{\Delta t \omega}{2}$$
$$\langle i_{L,equiv} \rangle(k-1) = \frac{1 - b^2 - j2b}{1 + b^2} \langle i_L \rangle(k-1) + \frac{a - jab}{1 + b^2} \langle v_L \rangle(k-1)$$
Capacitance
Similarly, a capacitance is described by as follows
$$\langle i_C \rangle(k) = C \ \frac{\mathrm{d} \langle v_C \rangle}{\mathrm{d} t} + j \omega C \ \langle v_C \rangle(t)$$
$$v_C(t) = v_C(t- \Delta t) + \int_{t- \Delta t}^{t} \frac{1}{C} \ i_C(\tau) -j \omega \ v_C(\tau) \ \mathrm{d} \tau$$
Applying the trapezoidal rule for the capacitance equation leads to the finite difference equation:
$$\begin{split}
\langle v_C \rangle(k) = \langle v_C \rangle(k-1)
+ \frac{\Delta t}{2} \bigg[ \frac{1}{C} \ \langle i_C \rangle(k) - j \omega \ \langle v_C \rangle(k) \\
+ \frac{1}{C} \ \langle i_C \rangle(k-1) - j \omega \ \langle v_C \rangle(k-1) \bigg]
\end{split}$$
The DP model for the capacitance is defined by
$$\langle i_C \rangle(k) = \frac{1+jb}{a} \ \langle v_C \rangle(k) + \langle i_{C,equiv} \rangle(k-1)$$
with
$$a = \frac{\Delta t}{2C}, \qquad
b = \frac{\Delta t \omega}{2}$$
$$\langle i_{C,equiv} \rangle(k-1) = - \frac{1-jb}{a} \ \langle v_C \rangle(k-1) - \langle i_C \rangle(k-1)$$
RL-series element
In dynamic phasors the integration of the inductance equation yields
$$\langle v \rangle(t) = L \frac{\mathrm{d}}{dt} \langle i \rangle(t) + j \omega L \ \langle i \rangle(t) + R \ \langle i \rangle(t)$$
$$\langle i \rangle(t) = \langle i \rangle(t - \Delta t) + \int_{t - \Delta t}^t \frac{1}{L} \langle v \rangle(\tau) - j \omega \ \langle i \rangle(\tau) - \frac{R}{L} \ \langle i \rangle(\tau) \mathrm{d} \tau$$
Applying the trapezoidal method leads to the finite difference equation:
$$\begin{split}
\langle i \rangle(k) = \langle i \rangle(k-1) + \frac{\Delta t}{2} \bigg[ \frac{1}{L} (\langle v \rangle(k) + \langle v \rangle(k-1))
- \left( j \omega + \frac{R}{L} \right) (\langle i \rangle(k) + \langle i \rangle(k-1)) \bigg]
\end{split}$$
Solving this for $\langle i \rangle(k)$ results in the \ac{DP} equivalent circuit model:
$$\langle i \rangle(k) = \frac{a + Ra^2 - jab}{(1+Ra)^2 + b^2} \langle v \rangle(k) + \langle i_{equiv} \rangle(k-1)$$
with
$$a = \frac{\Delta t}{2L}, \qquad b = \frac{\Delta t \omega}{2}$$
$$\langle i_{equiv} \rangle(k-1) = \frac{1 - b^2 - j2b + 2Ra + (Ra)^2 - j2Rab}{(1+Ra^2) + b^2} \langle i \rangle(k-1) + \frac{a + Ra^2 - jab}{(1+Ra)^2 + b^2} \langle v \rangle(k-1)$$
6 - Synchronous Generator
Two different synchronous machine models are currently available:
- the full order dq0 reference frame model (EMT, DP) [Kundur, Power system stability and control, 1994]
- and the much simpler transient stability model (DP) [Eremia, Handbook of Electrical Power System Dynamics, 2003]
The machine model is interfaced to the nodal analysis network solver through a current source, which only affects the source vector and not the system matrix Wang2010.
Basic Equations
The equations of the stator and rotor voltages are
$$\begin{align}
\mathbf{v}_{abcs} &= \mathbf{R}_s \mathbf{i}_{abcs} + \frac{d}{dt} \boldsymbol{\lambda}_{abcs} \\
\mathbf{v}_{dqr} &= \mathbf{R}_r \mathbf{i}_{dqr} + \frac{d}{dt} \boldsymbol{\lambda}_{dqr}
\end{align}$$
where
$$\begin{align}
\mathbf{v}_{abcs} &=
\begin{pmatrix}
v_{as} & v_{bs} & v_{cs}
\end{pmatrix}^T \\
%
\mathbf{v}_{dqr} &=
\begin{pmatrix}
v_{fd} & v_{kd} & v_{kq1} & v_{kq2}
\end{pmatrix}^T \\
%
\mathbf{i}_{abcs} &=
\begin{pmatrix}
i_{as} & i_{bs} & i_{cs}
\end{pmatrix}^T \\
%
\mathbf{i}_{dqr} &=
\begin{pmatrix}
i_{fd} & i_{kd} & i_{kq1} & i_{kq2}
\end{pmatrix}^T \\
%
\boldsymbol{\lambda}_{abcs} &=
\begin{pmatrix}
\lambda_{as} & \lambda_{bs} & \lambda_{cs}
\end{pmatrix}^T \\
%
\boldsymbol{\lambda}_{dqr} &=
\begin{pmatrix}
\lambda_{fd} & \lambda_{kd} & \lambda_{kq1} & \lambda_{kq2}
\end{pmatrix}^T \\
%
\mathbf{R}_s &= diag
\begin{bmatrix}
R_s & R_s & R_s
\end{bmatrix} \\
%
\mathbf{R}_r &= diag
\begin{bmatrix}
R_{fd} & R_{kd} & R_{kq1} & R_{kq2}
\end{bmatrix}
\end{align}$$
The flux linkage equations are defined as
$$\begin{equation}
\begin{bmatrix}
\boldsymbol{\lambda}_{abcs} \\
\boldsymbol{\lambda}_{dqr}
\end{bmatrix}
=
\begin{bmatrix}
\mathbf{L}_s & \mathbf{L}_{rs} \\
{(\mathbf{L}_{rs})}^{T} & \mathbf{L}_r
\end{bmatrix}
\begin{bmatrix}
\mathbf{i}_{abcs} \\
\mathbf{i}_{dqr}
\end{bmatrix}
\end{equation}$$
The inductance matrices are varying with the rotor position $\theta_r$ which varies with time.
The mechanical equations are:
$$\begin{align}
\frac{d\theta_r}{dt} &= \omega_r \\
\frac{d\omega_r}{dt} &= \frac{P}{2J} (T_e-T_m)
\end{align}$$
$\theta_r$ is the rotor position, $\omega_r$ is the angular electrical speed, $P$ is the number of poles, $J$ is the moment of inertia, $T_m$ and $T_e$ are the mechanical and electrical torque, respectively.
Motor convention is used for all models.
dq0 Reference Frame 9th Order Model
For stator referred variables, the base quantities for per unit are chosen as follows:
- $v_{s base}$ peak value of rated line-to-neutral voltage in V
- $i_{s base}$ peak value of rated line current in A
- $f_{base}$ rated frequency in Hz
The synchronous generator equations in terms of per unit values in the rotor reference frame become:
$$\begin{equation}
\begin{bmatrix}
\mathbf{v}_{dq0s} \\
\mathbf{v}_{dqr}
\end{bmatrix}
=
\mathbf{R}_{sr}
\begin{bmatrix}
\mathbf{i}_{dq0s} \\
\mathbf{i}_{dqr}
\end{bmatrix}
+
\frac{d}{dt}
\begin{bmatrix}
\boldsymbol{\lambda}_{dq0s} \\
\boldsymbol{\lambda}_{dqr}
\end{bmatrix}
+ \omega_r
\begin{bmatrix}
\boldsymbol{\lambda}_{qds} \\
0
\end{bmatrix}
\end{equation}$$
where
$$\begin{align}
\mathbf{v}_{dq0s} &=
\begin{pmatrix}
v_{ds} & v_{qs} & v_{0s}
\end{pmatrix}^T \nonumber \\
%
\mathbf{i}_{dq0s} &=
\begin{pmatrix}
i_{ds} & i_{qs} & i_{0s}
\end{pmatrix}^T \nonumber \\
%
\boldsymbol{\lambda}_{dq0s} &=
\begin{pmatrix}
\lambda_{ds} & \lambda_{qs} & \lambda_{0s}
\end{pmatrix}^T \nonumber \\
%
\mathbf{R}_{sr} &= diag
\begin{bmatrix}
R_s & R_s & R_s & R_{fd} & R_{kd} & R_{kq1} & R_{kq2}
\end{bmatrix} \nonumber \\
%
\boldsymbol{\lambda}_{dqs} &=
\begin{pmatrix}
-\lambda_{qs} & \lambda_{ds} & 0
\end{pmatrix}^T.
\end{align}$$
The flux linkages are:
$$\begin{equation}
\begin{pmatrix}
\boldsymbol{\lambda}_{dq0s} \\
\boldsymbol{\lambda}_{dqr}
\end{pmatrix}
=
\begin{bmatrix}
\mathbf{L}_{dqss} & \mathbf{L}_{dqsr} \\
\mathbf{L}_{dqrs} & \mathbf{L}_{dqrr}
\end{bmatrix}
\begin{pmatrix}
\mathbf{i}_{dq0s} \\
\mathbf{i}_{dqr}
\end{pmatrix}
\end{equation}$$
where
$$\begin{align}
\mathbf{L}_{dqss} &=
\begin{bmatrix}
L_{d} & 0 & 0 \\
0 & L_{q} & 0 \\
0 & 0 & L_{ls}
\end{bmatrix} \nonumber \\
\mathbf{L}_{dqsr} &=
\begin{bmatrix}
L_{md} & L_{md} & 0 & 0 \\
0 & 0 & L_{mq} & L_{mq} \\
0 & 0 & 0 & 0
\end{bmatrix} \nonumber \\
\mathbf{L}_{dqrs} &=
\begin{bmatrix}
L_{md} & 0 & 0 \\
L_{md} & 0 & 0 \\
0 & L_{mq} & 0 \\
0 & L_{mq} & 0
\end{bmatrix} \nonumber \\
\mathbf{L}_{rr} &=
\begin{bmatrix}
L_{fd} & L_{md} & 0 & 0 \\
L_{md} & L_{kd} & 0 & 0 \\
0 & 0 & L_{kq1} & L_{mq} \\
0 & 0 & L_{mq} & L_{kq2}
\end{bmatrix} \nonumber \\
\end{align}$$
with
$$\begin{align}
L_{d} &= L_{ls} + L_{md} \nonumber \\
L_{q} &= L_{ls} + L_{mq} \nonumber \\
L_{fd} &= L_{lfd} + L_{md} \nonumber \\
L_{kd} &= L_{lkd} + L_{md} \nonumber \\
L_{kq1} &= L_{lkq1} + L_{mq} \nonumber \\
L_{kq2} &= L_{lkq2} + L_{mq}.
\end{align}$$
The mechanical equations in per unit become:
$$\begin{align}
T_e &= \lambda_{qs} i_{ds} - \lambda_{ds} i_{qs} \\
\frac{d \omega_r}{dt} &= \omega_r \\
\frac{1}{\omega_b} \frac{d \omega_r}{dt} &= \frac{1}{2H} (T_m - T_e).
\end{align}$$
For the simulation, fluxes are chosen as state variables.
To avoid the calculation of currents from fluxes using the inverse of the inductance matrix, the equation set needs to be solved for the fluxes analytically.
To simplify the calculations, dq axis magnetizing flux linkages are defined [Krause, Analysis of electric machinery and drive systems, 2002]:
$$\begin{align}
\lambda_{md} &= L_{md} \left( i_{ds} + i_{fd} + i_{kd} \right) \nonumber \\
\lambda_{mq} &= L_{mq} \left( i_{qs} + i_{kq1} + i_{kq2} \right)
\end{align}$$
Using the flux linkages results in a simpler equation set for the fluxes:
$$\begin{align}
\lambda_{ds} &= L_{ls} i_{ds} + L_{md} \left( i_{ds} + i_{fd} + i_{kd} \right) \nonumber \\
\lambda_{qs} &= L_{ls} i_{qs} + L_{mq} \left( i_{qs} + i_{kq1} + i_{kq2} \right) \nonumber \\
\lambda_{0s} &= L_{ls} i_{0s} \nonumber \\
\lambda_{fd} &= L_{ls} i_{fd} + L_{md} \left( i_{ds} + i_{fd} + i_{kd} \right) \nonumber \\
\lambda_{kd} &= L_{ls} i_{kd} + L_{md} \left( i_{ds} + i_{fd} + i_{kd} \right) \nonumber \\
\lambda_{kq1} &= L_{ls} i_{kq1} + L_{mq} \left( i_{qs} + i_{kq1} + i_{kq2} \right) \nonumber \\
\lambda_{kq2} &= L_{ls} i_{kq2} + L_{mq} \left( i_{qs} + i_{kq1} + i_{kq2} \right)
\end{align}$$
$$\begin{align}
\lambda_{ds} &= L_{ls} i_{ds} + \lambda_{md} \nonumber \\
\lambda_{qs} &= L_{ls} i_{qs} + \lambda_{mq} \nonumber \\
\lambda_{0s} &= L_{ls} i_{0s} \nonumber \\
\lambda_{fd} &= L_{lfd} i_{fd} + \lambda_{md} \nonumber \\
\lambda_{kd} &= L_{lkd} i_{kd} + \lambda_{md} \nonumber \\
\lambda_{kq1} &= L_{lkq1} i_{kq1} + \lambda_{mq} \nonumber \\
\lambda_{kq2} &= L_{lkq2} i_{kq2} + \lambda_{mq}
\end{align}$$
Dynamic Phasor Model
The fundamental dynamic phasors are similar to the dq0 quantities for symmetrical conditions since both yield DC quantities in a rotating reference frame.
The network abc dynamic phasor quantities can be converted to dq0 dynamic phasors by applying the symmetrical components transformation and a rotation.
The angle $\delta$ is the orientation of the dq0 reference frame relative to the abc frame.
$$\begin{align}
\langle i_{ds} \rangle_{0} &= \mathbf{Re} \left\{ \langle i_{p} \rangle_1 \ \mathrm{e}^{-j \delta} \right\} \nonumber \\
\langle i_{qs} \rangle_{0} &= \mathbf{Im} \left\{ \langle i_{p} \rangle_1 \ \mathrm{e}^{-j \delta} \right\} \nonumber \\
\langle i_{ds} \rangle_{2} &= \mathbf{Re} \left\{ \langle i_{n} \rangle_{1}^* \ \mathrm{e}^{-j \delta} \right\} \nonumber \\
\langle i_{qs} \rangle_{2} &= \mathbf{Im} \left\{ \langle i_{n} \rangle_{1}^* \ \mathrm{e}^{-j \delta} \right\} \nonumber \\
\langle i_{0s} \rangle_{1} &= \mathbf{Re} \left\{ \langle i_{z} \rangle_1 \right\}
\end{align}$$
The winding currents for positive and zero sequence components can be expressed as
$$\begin{align}
\langle i_{ds} \rangle_0 &= \frac{\langle \lambda_{ds} \rangle_0 - \langle \lambda_{md} \rangle_0 }{L_{ls}} \nonumber \\
\langle i_{qs} \rangle_0 &= \frac{\langle \lambda_{qs} \rangle_0 - \langle \lambda_{mq} \rangle_0}{L_{ls}} \nonumber \\
\langle i_{0s} \rangle_1 &= \frac{\langle \lambda_{0s} \rangle_1}{L_{ls}} \nonumber \\
\langle i_{fd} \rangle_0 &= \frac{\langle \lambda_{fd} \rangle_0 - \langle \lambda_{md} \rangle_0}{L_{lfd}} \nonumber \\
\langle i_{kd} \rangle_0 &= \frac{\langle \lambda_{kd} \rangle_0 - \langle \lambda_{md} \rangle_0}{L_{lkd}} \nonumber \\
\langle i_{kq1} \rangle_0 &= \frac{\langle \lambda_{kq1} \rangle_0 - \langle \lambda_{mq} \rangle_0}{L_{lkq1}} \nonumber \\
\langle i_{kq2} \rangle_0 &= \frac{\langle \lambda_{kq2} \rangle_0 - \langle \lambda_{mq} \rangle_0}{L_{lkq2}}.
\end{align}$$
$$\begin{align}
\frac{d}{dt} \langle \lambda_{ds} \rangle_0 &= \langle v_{ds} \rangle_0 + \langle \omega_r \rangle_0 \langle \lambda_{qs} \rangle_0 + \frac{R_s}{L_{ls}} \left( \langle \lambda_{md} \rangle_0 - \langle \lambda_{ds} \rangle_0 \right) \nonumber \\
\frac{d}{dt} \langle \lambda_{qs} \rangle_0 &= \langle v_{qs} \rangle_0 - \langle \omega_r \rangle_0 \langle \lambda_{ds} \rangle_0 + \frac{R_s}{L_{ls}} \left( \langle \lambda_{mq} \rangle_0 - \langle \lambda_{qs} \rangle_0 \right) \nonumber \\
\frac{d}{dt} \langle \lambda_{0s} \rangle_1 &= \langle v_{0s} \rangle_1 - \frac{R_s}{L_{ls}} \langle \lambda_{0s} \rangle_1 -j \omega_s \langle \lambda_{0s} \rangle_1 \nonumber \\
\frac{d}{dt} \langle \lambda_{fd} \rangle_0 &= \langle v_{fd} \rangle_0 + \frac{R_{fd}}{L_{lfd}} \left( \langle \lambda_{md} \rangle_0 - \langle \lambda_{fd} \rangle_0 \right) \nonumber \\
\frac{d}{dt} \langle \lambda_{kd} \rangle_0 &= \frac{R_{kd}}{L_{lkd}} \left( \langle \lambda_{md} \rangle_0 - \langle \lambda_{kd} \rangle_0 \right) \nonumber \\
\frac{d}{dt} \langle \lambda_{kq1} \rangle_0 &= \frac{R_{kq1}}{L_{lkq1}} \left( \langle \lambda_{mq} \rangle_0 - \langle \lambda_{kq1} \rangle_0 \right) \nonumber \\
\frac{d}{dt} \langle \lambda_{kq2} \rangle_0 &= \frac{R_{kq2}}{L_{lkq2}} \left( \langle \lambda_{mq} \rangle_0 - \langle \lambda_{kq2} \rangle_0 \right).
\end{align}$$
In the dynamic phasor case, the equation for $\frac{d}{dt} \langle \lambda_{0s} \rangle_1$ has a frequency shift.
To complete the state model, the magnetizing flux linkages are expressed as:
$$\begin{align}
\langle \lambda_{md} \rangle_0 &= L_{ad} \left( \frac{\langle \lambda_{ds} \rangle_0}{L_{ls}} + \frac{\langle \lambda_{fd} \rangle_0}{L_{lfd}} + \frac{\langle \lambda_{kd} \rangle_0}{L_{lkd}} \right) \nonumber \\
\langle \lambda_{mq} \rangle_0 &= L_{aq} \left( \frac{\langle \lambda_{qs} \rangle_0}{L_{ls}} + \frac{\langle \lambda_{kq1} \rangle_0}{L_{lkq1}} + \frac{\langle \lambda_{kq2} \rangle_0}{L_{lkq2}} \right)
\end{align}$$
where
$$\begin{align}
L_{ad} &= \left( \frac{1}{L_{md}} + \frac{1}{L_{ls}} + \frac{1}{L_{lfd}} + \frac{1}{L_{lkd}} \right) \nonumber \\
L_{aq} &= \left( \frac{1}{L_{mq}} + \frac{1}{L_{ls}} + \frac{1}{L_{lkq1}} + \frac{1}{L_{lkq2}} \right).
\end{align}$$
The mechanical equations in dynamic phasors are:
$$\begin{align}
T_e &= \langle \lambda_{qs} \rangle_0 \langle i_{ds} \rangle_0 - \langle \lambda_{ds} \rangle_0 \langle i_{qs} \rangle_0 \\
\frac{1}{\omega_s} \frac{d \delta_r}{dt} &= \omega_r - 1 \\
\frac{d \omega_r}{dt} &= \frac{1}{2H} (T_m - T_e).
\end{align}$$
Transient Stability Model