DPsim – Concepts

Docs: Nodal Analysis

Wed, 18 Mar 2020 00:00:00 +0000

A circuit with $b$ branches has $2b$ unknowns since there are $b$ voltages and $b$ currents. Hence, $2b$ linear independent equations are required to solve the circuit. If the circuit has $n$ nodes and $b$ branches, it has

Kirchoff’s current law (KCL) equations
Kirchoff’s voltage law (KVL) equations
Characteristic equations (Ohm’s Law)

There are only $n-1$ KCLs since the nth equation is a linear combination of the remaining $n-1$. At the same time, it can be demonstrated that if we can imagine a very high number of closed paths in the network, only $b-n+1$ are able to provide independent KVLs. Finally there are $b$ characteristic equations, describing the behavior of the branch, making a total of $2b$ linear independent equations.

The nodal analysis method reduces the number of equations that need to be solved simultaneously. $n-1$ voltage variables are defined and solved, writing $n-1$ KCL based equations. A circuit can be solved using Nodal Analysis as follows

Select a reference node (mathematical ground) and number the remaining $n-1$ nodes, that are the independent voltage variables
Represent every branch current $i$ as a function of node voltage variables $v$ with the general expression $i = g(v)$
Write $n-1$ KCL based equations in terms of node voltage variable.

The resulting equations can be written in matrix form and have to be solved for $v$.

$$\boldsymbol{Y} \boldsymbol{v} = \boldsymbol{i}$$

Assembling the System by Stamping

The matrix $\boldsymbol{Y}$ and the right hand side $\boldsymbol{i}$ are never written out by inspecting the whole circuit at once. Each element contributes its own fixed pattern of entries, and the system is formed by adding those contributions together. A conductance $G$ between nodes $k$ and $l$ adds $G$ to the diagonal entries $Y_{kk}$ and $Y_{ll}$ and subtracts it from the off-diagonal entries $Y_{kl}$ and $Y_{lk}$; if one terminal is the reference node, only the single diagonal entry appears. A current source injecting $I$ into node $k$ from node $l$ adds $I$ to $i_k$ and subtracts it from $i_l$.

This additive assembly is what makes the method practical. An element needs to know only the indices of the nodes it is attached to, never anything about the rest of the network, and the same element contributes the same pattern regardless of what it is connected to. It also explains the structure of the result: each row corresponds to one node and holds a non-zero entry only for nodes reachable through a single element, so $\boldsymbol{Y}$ is symmetric for networks of passive elements and sparse for any network of realistic size.

Dynamic Elements and the Companion Model

The formulation above assumes every branch current can be written as a function of node voltages alone. Inductors and capacitors do not satisfy this, since their currents depend on derivatives. They are brought into the same form by discretising the differential relation over one time step. Applying the trapezoidal rule to a capacitor between the instants $t - \Delta t$ and $t$ gives

$$i(t) = \frac{2C}{\Delta t} v(t) - \left( \frac{2C}{\Delta t} v(t - \Delta t) + i(t - \Delta t) \right),$$

and the same treatment of an inductor gives

$$i(t) = \frac{\Delta t}{2L} v(t) + \left( \frac{\Delta t}{2L} v(t - \Delta t) + i(t - \Delta t) \right).$$

Both have the form of a conductance in parallel with a current source. The conductance depends only on the element value and the time step, so it is a constant contribution to $\boldsymbol{Y}$; the current source depends only on quantities from the previous step, which are known when the current step begins, so it is a contribution to $\boldsymbol{i}$. This pair is called the companion model of the element, and the current source term is called its history term. Once every dynamic element has been replaced by its companion model, the network at each instant is a purely resistive one and the nodal formulation applies unchanged.

Extending the Formulation for Voltage Sources

An ideal voltage source cannot be stamped as a conductance, because its current is not determined by the voltage across it. The formulation is extended by admitting the source current as an additional unknown and adding the equation that constrains its terminal voltage. The system becomes

$$\begin{bmatrix} \boldsymbol{Y} & \boldsymbol{A} \\ \boldsymbol{A}^{\mathsf{T}} & \boldsymbol{0} \end{bmatrix} \begin{bmatrix} \boldsymbol{v} \\ \boldsymbol{j} \end{bmatrix} = \begin{bmatrix} \boldsymbol{i} \\ \boldsymbol{u} \end{bmatrix},$$

where $\boldsymbol{j}$ collects the unknown source currents, $\boldsymbol{u}$ the prescribed source voltages, and $\boldsymbol{A}$ has a single $+1$ and $-1$ per source marking its terminals. This extension is known as modified nodal analysis, and it is the form actually solved. The augmented matrix is no longer positive definite and carries zeros on part of its diagonal, which is why the solver has to be one that tolerates that rather than one specialised to the passive case.

Solving Over Time

Within a simulation the same system is solved once per time step. The left hand side depends only on the element values, the topology and the time step, none of which change from one step to the next in the ordinary case, so the matrix is factorised once and each step reuses that factorisation with a new right hand side. The cost of a step is then a forward and backward substitution rather than a full solve, which is what makes the method viable for large networks and for real time.

Two situations invalidate the factorisation. A switching event changes the topology and therefore the matrix, so the affected configuration has to be factorised again; simulations that switch frequently often pre-compute a factorisation for each configuration instead. A non-linear element makes the entries themselves depend on the solution, which requires iterating within the step until the node voltages and the element operating points agree.

Docs: Dynamic Phasors

Wed, 18 Mar 2020 00:00:00 +0000

In the power systems community, dynamic phasors were initially introduced for power electronics analysis Sanders1991 as a more general approach than state-space averaging. They were used to construct efficient models for the dynamics of switching gate phenomena with a high level of detail as shown in Mattavelli1999. A few years later, dynamic phasors were also employed for power system simulation as described in Demiray2008. In Strunz2006 the authors combine the dynamic phasor approach with the Electromagnetic Transients Program (EMTP) simulator concept which includes Modified Nodal Analysis (MNA). Further research topics include fault and stability analysis under unbalanced conditions as presented in Stankovic2000 and also rotating machine models have been developed in dynamic phasors Zhang 2007.

Bandpass Signals and Baseband Representation

Although here, dynamic phasors are presented as a power system modelling tool, it should be noted that the concept is also known in other domains, for example, microwave and communications engineering Maas2003, Suarez2009, Haykin2009, Proakis2001. In these domains, the approach is often denoted as base band representation or complex envelope. Another common term coming from power electrical engineering is shifted frequency analysis (SFA). In the following, the general approach of dynamic phasors for power system simulation is explained starting from the idea of bandpass signals. This is because the 50 Hz or 60 Hz fundamental and small deviations from it can be seen as such a bandpass signal. Futhermore, higher frequencies, for example, generated by power electronics can be modelled in a similar way.

Two Derivations of the Same Transform

The literature reaches the envelope description along two different routes. One starts from the analytic signal and shifts the spectrum; the other averages a sliding Fourier series. They are usually presented independently, under different names, and it is not always stated that they arrive at the same differential operator. The two routes are set out below so that the equivalence is visible.

Route A: shifting the spectrum

A bandpass signal is one whose spectrum is concentrated in a band around some carrier frequency $\omega_s$, narrow compared with $\omega_s$ itself. The network voltages and currents of a power system are of this kind: energy sits around the 50 Hz or 60 Hz fundamental and spreads only a little either side of it during a transient.

Such a signal can be written exactly as

$$x(t) = \operatorname{Re}\{\, X(t)\, e^{j \omega_s t} \,\},$$

where the complex quantity $X(t)$ is the envelope. It is obtained from the analytic signal $x_a(t) = x(t) + j,\mathcal{H}{x}(t)$, with $\mathcal{H}$ the Hilbert transform, by

$$X(t) = x_a(t)\, e^{-j \omega_s t}.$$

Multiplication by $e^{-j\omega_s t}$ translates the spectrum down by $\omega_s$, so the band that sat around the carrier now sits around zero. That translation is what the name shifted frequency analysis refers to. No information is lost, and nothing is approximated: the operation is invertible and $x(t)$ can be recovered from $X(t)$ at any instant.

The consequence that matters for simulation is what happens to the derivative. Differentiating the expression above gives

$$\frac{dx}{dt} = \operatorname{Re}\left\{ \left( \frac{dX}{dt} + j \omega_s X \right) e^{j \omega_s t} \right\},$$

so in envelope coordinates the time derivative becomes

$$\frac{d}{dt} \;\longrightarrow\; \frac{d}{dt} + j \omega_s .$$

An inductor $v = L,\frac{di}{dt}$ therefore satisfies $V = L,\frac{dI}{dt} + j \omega_s L, I$. Two limiting cases are worth noting. When the envelope is constant, the first term vanishes and what remains is $V = j\omega_s L, I$, the classical steady-state phasor relation. When $\omega_s = 0$, the second term vanishes and the envelope is the instantaneous signal itself, which is the electromagnetic transient description. Both the phasor and the instantaneous representation are special cases of the same expression.

Route B: averaging a Fourier series

The dynamic phasor derivation starts elsewhere. Over a window of length $T = 2\pi/\omega_s$ ending at time $t$, the waveform is expanded in a Fourier series

$$x(t + s) = \sum_{k} X_k(t)\, e^{j k \omega_s s}, \qquad s \in (-T, 0],$$

whose coefficients are themselves functions of time,

$$X_k(t) = \frac{1}{T} \int_{t-T}^{t} x(\tau)\, e^{-j k \omega_s \tau} \, d\tau .$$

These sliding-window coefficients are the dynamic phasors. Each one is constant whenever the waveform is periodic and varies only as the waveform’s shape changes, which is why they are also called generalized averages. Integrating by parts gives the property the method is built on,

$$\frac{dX_k}{dt} = \left\langle \frac{dx}{dt} \right\rangle_k - j k \omega_s X_k ,$$

that is, the same shift as Route A, applied to the $k$-th coefficient.

Where the routes meet

Retaining only $k = 1$ makes Route B’s relation identical to Route A’s: the first coefficient obeys $\frac{d}{dt} + j\omega_s$ and the network equations that follow are the same. The two derivations produce one model.

The difference is emphasis rather than substance. Route A treats a single carrier and is exact for any signal, so the accuracy question is only whether the signal is genuinely bandpass around the carrier that was chosen. Route B offers a whole family of coefficients, so it can carry a DC component in $X_0$, the fundamental in $X_1$ and switching harmonics in higher $X_k$ simultaneously, at the cost of one complex state per coefficient retained. Truncating that series is where the approximation enters: energy at frequencies that were not given a coefficient is discarded, and the sliding window acts as a low-pass filter on what remains, so envelope components varying as fast as $\omega_s$ itself are attenuated.

This explains the split in usage. Work concerned with converter switching, harmonic interaction or unbalanced and fault conditions keeps several coefficients and speaks of dynamic phasors. Work concerned with transient simulation of the fundamental keeps one and speaks of shifted frequency analysis, emphasising the enlarged time step rather than the series. The communications and microwave literature calls the same object the complex envelope or the lowpass equivalent. The three names describe one transform, and the choice between them says more about a paper’s lineage than about its mathematics.

Relationship to Electromagnetic Transient Models

An electromagnetic transient (EMT) model solves for the instantaneous quantities directly. Because the waveform it integrates oscillates at the carrier, the step size has to resolve that oscillation, and the accuracy of the result is bounded by how finely the fundamental period is sampled. A shifted frequency model integrates the envelope instead. In sinusoidal steady state the envelope is constant, so the step size is dictated by the bandwidth of the transient rather than by the carrier, and it can be substantially larger for the same accuracy. This is the reason the approach is attractive for transient stability studies and for real-time simulation of large networks.

The saving is not free, and it is worth being precise about where it comes from. It is not that the envelope model is a coarser description of the same system; for a single carrier the transform is exact. It is that the carrier oscillation has been removed from the quantity being integrated and folded into the coefficient $j\omega_s$, which the model handles analytically rather than numerically. What the envelope model cannot represent is content the retained coefficients do not cover, which for a single-coefficient model means anything outside the band around $\omega_s$.

Because $x(t) = \operatorname{Re}{X(t) e^{j\omega_s t}}$ holds at every instant, converting between the two descriptions is an algebraic operation at a given point in time, not a filtering or a fitting problem. That is what makes mixed-domain and co-simulation setups possible: part of a network can be solved in envelope form and part in instantaneous form, with the interface performing the multiplication in one direction and the extraction of the envelope in the other. Two conditions have to be respected at such an interface. The envelope is defined relative to a chosen $\omega_s$ and a chosen time origin, so both sides must agree on the carrier frequency and share a phase reference; and the extraction direction needs the analytic signal, which for a general instantaneous waveform cannot be formed from a single sample. Any delay introduced between the two sides appears as a phase error on the carrier, and that error grows with $\omega_s$ rather than with the envelope’s own rate of change.

The state-space form of the shift, and how the resulting complex system is solved as a real one, are covered in State-Space Nodal.

Docs: Powerflow

Wed, 18 Mar 2020 00:00:00 +0000

The power flow problem is about the calculation of voltage magnitudes and angles for one set of buses. The solution is obtained from a given set of voltage magnitudes and power levels for a specific model of the network configuration. The power flow solution exhibits the voltages and angles at all buses and real and reactive flows can be deduced from the same.

Power System Model

Power systems are modeled as a network of buses (nodes) and branches (lines). To a network bus, components such a generator, load, and transmission substation can be connected. Each bus in the network is fully described by the following four electrical quantities:

$\vert V_{k} \vert$: the voltage magnitude
$\theta_{k}$: the voltage phase angle
$P_{k}$: the active power
$Q_{k}$: the reactive power

There are three types of networks buses: VD bus, PV bus and PQ bus. Depending on the type of the bus, two of the four electrical quantities are specified as shown in the table below.

Bus Type	Known	Unknown
$VD$	$\vert V_{k} \vert, \theta_{k}$	$P_{k}, Q_{k}$
$PV$	$P_{k}, \vert V_{k} \vert$	$Q_{k}, \theta_{k}$
$PQ$	$P_{k}, Q_{k}$	$\vert V_{k} \vert, \theta_{k}$

Single Phase Power Flow Problem

The power flow problem can be expressed by the goal to bring a mismatch function $\vec{f}$ to zero. The value of the mismatch function depends on a solution vector $\vec{x}$:

$$\vec{f}(\vec{x}) = 0$$

As $\vec{f}(\vec{x})$ will be nonlinear, the equation system will be solved with Newton-Raphson:

$$-\textbf{J}(\vec{x}) \Delta \vec{x} = \vec{f} (\vec{x})$$

where $\Delta \vec{x}$ is the correction of the solution vector and $\textbf{J}(\vec{x})$ is the Jacobian matrix. The solution vector $\vec{x}$ represents the voltage $\vec{V}$ by polar or cartesian quantities. The mismatch function $\vec{f}$ will either represent the power mismatch $\Delta \vec{S}$ in terms of

$$\left [ \begin{array}{c} \Delta \vec{P} \\ \Delta \vec{Q} \end{array} \right ]$$

or the current mismatch $\Delta \vec{I}$ in terms of

$$\left [ \begin{array}{c} \Delta \vec{I_{real}} \\ \Delta \vec{I_{imag}} \end{array} \right ]$$

where the vectors split the complex quantities into real and imaginary parts. Futhermore, the solution vector $\vec{x}$ will represent $\vec{V}$ either by polar coordinates

$$\left [ \begin{array}{c} \vec{\delta} \\ \vert \vec{V} \vert \end{array} \right ]$$

or rectangular coordinates

$$\left [ \begin{array}{c} \vec{V_{real}} \\ \vec{V_{imag}} \end{array} \right ]$$

This results in four different formulations of the powerflow problem:

with power mismatch function and polar coordinates
with power mismatch function and rectangular coordinates
with current mismatch function and polar coordinates
with current mismatch function and rectangular coordinates

To solve the problem using NR, we need to formulate $\textbf{J} (\vec{x})$ and $\vec{f} (\vec{x})$ for each powerflow problem formulation.

Of these four, DPsim currently implements the power mismatch function with polar coordinates, detailed below. It is the formulation used by both the dense (PFSolverPowerPolar) and sparse (PFSolverPowerPolarSparse) solvers; the other three are not implemented.

Powerflow Problem with Power Mismatch Function and Polar Coordinates

Formulation of Mismatch Function

The injected power at a node $k$ is given by:

$$S_{k} = V_{k} I _{k}^{*}$$

The current injection into any bus $k$ may be expressed as:

$$I_{k} = \sum_{j=1}^{N} Y_{kj} V_{j}$$

Substitution yields:

$$\begin{align} S_{k} &= V_{k} \left ( \sum_{j=1}^{N} Y_{kj} V_{j} \right )^{*} \nonumber \\ &= V_{k} \sum_{j=1}^{N} Y_{kj}^{*} V_{j} ^{*} \nonumber \end{align}$$

We may define $G_{kj}$ and $B_{kj}$ as the real and imaginary parts of the admittance matrix element $Y_{kj}$ respectively, so that $Y_{kj} = G_{kj} + jB_{kj}$. Then we may rewrite the last equation:

$$\begin{align} S_{k} &= V_{k} \sum_{j=1}^{N} Y_{kj}^{*} V_{j}^{*} \nonumber \\ &= \vert V_{k} \vert \angle \theta_{k} \sum_{j=1}^{N} (G_{kj} + jB_{kj})^{*} ( \vert V_{j} \vert \angle \theta_{j})^{*} \nonumber \\ &= \vert V_{k} \vert \angle \theta_{k} \sum_{j=1}^{N} (G_{kj} - jB_{kj}) ( \vert V_{j} \vert \angle - \theta_{j}) \nonumber \\ &= \sum_{j=1} ^{N} \vert V_{k} \vert \angle \theta_{k} ( \vert V_{j} \vert \angle - \theta_{j}) (G_{kj} - jB_{kj}) \nonumber \\ &= \sum_{j=1} ^{N} \left ( \vert V_{k} \vert \vert V_{j} \vert \angle (\theta_{k} - \theta_{j}) \right ) (G_{kj} - jB_{kj}) \nonumber \\ &= \sum_{j=1} ^{N} \vert V_{k} \vert \vert V_{j} \vert \left ( cos(\theta_{k} - \theta_{j}) + jsin(\theta_{k} - \theta_{j}) \right ) (G_{kj} - jB_{kj}) \end{align}$$

If we now perform the algebraic multiplication of the two terms inside the parentheses, and collect real and imaginary parts, and recall that $S_{k} = P_{k} + jQ_{k}$, we can express (1) as two equations: one for the real part, $P_{k}$, and one for the imaginary part, $Q_{k}$, according to:

$$\begin{align} {P}_{k} = \sum_{j=1}^{N} \vert V_{k} \vert \vert V_{j} \vert \left ( G_{kj}cos(\theta_{k} - \theta_{j}) + B_{kj} sin(\theta_{k} - \theta_{j}) \right ) \\ {Q}_{k} = \sum_{j=1}^{N} \vert V_{k} \vert \vert V_{j} \vert \left ( G_{kj}sin(\theta_{k} - \theta_{j}) - B_{kj} cos(\theta_{k} - \theta_{j}) \right ) \end{align}$$

These equations are called the power flow equations, and they form the fundamental building block from which we solve the power flow problem.

We consider a power system network having $N$ buses. We assume one VD bus, $N_{PV}-1$ PV buses and $N-N_{PV}$ PQ buses. We assume that the VD bus is numbered bus $1$, the PV buses are numbered $2,…,N_{PV}$, and the PQ buses are numbered $N_{PV}+1,…,N$. We define the vector of unknown as the composite vector of unknown angles $\vec{\theta}$ and voltage magnitudes $\vert \vec{V} \vert$:

$$\begin{align} \vec{x} = \left[ \begin{array}{c} \vec{\theta} \\ \vert \vec{V} \vert \\ \end{array} \right ] = \left[ \begin{array}{c} \theta_{2} \\ \theta_{3} \\ \vdots \\ \theta_{N} \\ \vert V_{N_{PV+1}} \vert \\ \vert V_{N_{PV+2}} \vert \\ \vdots \\ \vert V_{N} \vert \end{array} \right] \end{align}$$

The right-hand sides of equations (2) and (3) depend on the elements of the unknown vector $\vec{x}$. Expressing this dependency more explicitly, we rewrite these equations as:

$$\begin{align} P_{k} = P_{k} (\vec{x}) \Rightarrow P_{k}(\vec{x}) - P_{k} &= 0 \quad \quad k = 2,...,N \\ Q_{k} = Q_{k} (\vec{x}) \Rightarrow Q_{k} (\vec{x}) - Q_{k} &= 0 \quad \quad k = N_{PV}+1,...,N \end{align}$$

We now define the mismatch vector $\vec{f} (\vec{x})$ as:

$$\begin{align} \vec{f} (\vec{x}) = \left [ \begin{array}{c} f_{1}(\vec{x}) \\ \vdots \\ f_{N-1}(\vec{x}) \\ ------ \\ f_{N}(\vec{x}) \\ \vdots \\ f_{2N-N_{PV} -1}(\vec{x}) \end{array} \right ] = \left [ \begin{array}{c} P_{2}(\vec{x}) - P_{2} \\ \vdots \\ P_{N}(\vec{x}) - P_{N} \\ --------- \\ Q_{N_{PV}+1}(\vec{x}) - Q_{N_{PV}+1} \\ \vdots \\ Q_{N}(\vec{x}) - Q_{N} \end{array} \right] = \left [ \begin{array}{c} \Delta P_{2} \\ \vdots \\ \Delta P_{N} \\ ------ \\ \Delta Q_{N_{PV}+1} \\ \vdots \\ \Delta Q_{N} \end{array} \right ] = \vec{0} \end{align}$$

That is a system of nonlinear equations. This nonlinearity comes from the fact that $P_{k}$ and $Q_{k}$ have terms containing products of some of the unknowns and also terms containing trigonometric functions of some the unknowns.

Formulation of Jacobian

As discussed in the previous section, the power flow problem will be solved using the Newton-Raphson method. Here, the Jacobian matrix is obtained by taking all first-order partial derivates of the power mismatch functions with respect to the voltage angles $\theta_{k}$ and magnitudes $\vert V_{k} \vert$ as:

$$\begin{align} J_{jk}^{P \theta} &= \frac{\partial P_{j} (\vec{x} ) } {\partial \theta_{k}} = \vert V_{j} \vert \vert V_{k} \vert \left ( G_{jk} sin(\theta_{j} - \theta_{k}) - B_{jk} cos(\theta_{j} - \theta_{k} ) \right ) \\ J_{jj}^{P \theta} &= \frac{\partial P_{j}(\vec{x})}{\partial \theta_{j}} = -Q_{j} (\vec{x} ) - B_{jj} \vert V_{j} \vert ^{2} \\ J_{jk}^{Q \theta} &= \frac{\partial Q_{j}(\vec{x})}{\partial \theta_{k}} = - \vert V_{j} \vert \vert V_{k} \vert \left ( G_{jk} cos(\theta_{j} - \theta_{k}) + B_{jk} sin(\theta_{j} - \theta_{k}) \right ) \\ J_{jj}^{Q \theta} &= \frac{\partial Q_{j}(\vec{x})}{\partial \theta_{k}} = P_{j} (\vec{x} ) - G_{jj} \vert V_{j} \vert ^{2} \\ J_{jk}^{PV} &= \frac{\partial P_{j} (\vec{x} ) } {\partial \vert V_{k} \vert } = \vert V_{j} \vert \left ( G_{jk} cos(\theta_{j} - \theta_{k}) + B_{jk} sin(\theta_{j} - \theta_{k}) \right ) \\ J_{jj}^{PV} &= \frac{\partial P_{j}(\vec{x})}{\partial \vert V_{j} \vert } = \frac{P_{j} (\vec{x} )}{\vert V_{j} \vert} + G_{jj} \vert V_{j} \vert \\ J_{jk}^{QV} &= \frac{\partial Q_{j} (\vec{x} ) } {\partial \vert V_{k} \vert } = \vert V_{j} \vert \left ( G_{jk} sin(\theta_{j} - \theta_{k}) + B_{jk} cos(\theta_{j} - \theta_{k}) \right ) \\ J_{jj}^{QV} &= \frac{\partial Q_{j}(\vec{x})}{\partial \vert V_{j} \vert } = \frac{Q_{j} (\vec{x} )}{\vert V_{j} \vert} - B_{jj} \vert V_{j} \vert \\ \end{align}$$

The formulas above use the voltage magnitude $\vert V_k \vert$ as the unknown. The DPsim implementation instead uses the relative voltage increment $\Delta \vert V_k \vert / \vert V_k \vert$, which scales every voltage-magnitude column ($J^{PV}$, $J^{QV}$) by $\vert V_k \vert$. For example $J_{jj}^{PV}$ becomes $P_j(\vec{x}) + G_{jj} \vert V_j \vert^2$. This is paired with the multiplicative voltage update $\vert V_k \vert \leftarrow \vert V_k \vert (1 + \Delta \vert V_k \vert / \vert V_k \vert)$, so the solution is identical; only the scaling of the voltage columns differs.

The linear system of equations that is solved in every Newton iteration can be written in matrix form as follows:

$$\begin{align} -\left [ \begin{array}{cccccc} \frac{\partial \Delta P_{2} }{\partial \theta_{2}} & \cdots & \frac{\partial \Delta P_{2} }{\partial \theta_{N}} & \frac{\partial \Delta P_{2} }{\partial \vert V_{N_{G+1}} \vert} & \cdots & \frac{\partial \Delta P_{2} }{\partial \vert V_{N} \vert} \\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ \frac{\partial \Delta P_{N} }{\partial \theta_{2}} & \cdots & \frac{\partial \Delta P_{N}}{\partial \theta_{N}} & \frac{\partial \Delta P_{N}}{\partial \vert V_{N_{G+1}} \vert } & \cdots & \frac{\partial \Delta P_{N}}{\partial \vert V_{N} \vert} \\ \frac{\partial \Delta Q_{N_{G+1}} }{\partial \theta_{2}} & \cdots & \frac{\partial \Delta Q_{N_{G+1}} }{\partial \theta_{N}} & \frac{\partial \Delta Q_{N_{G+1}} }{\partial \vert V_{N_{G+1}} \vert } & \cdots & \frac{\partial \Delta Q_{N_{G+1}} }{\partial \vert V_{N} \vert} \\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ \frac{\partial \Delta Q_{N}}{\partial \theta_{2}} & \cdots & \frac{\partial \Delta Q_{N}}{\partial \theta_{N}} & \frac{\partial \Delta Q_{N}}{\partial \vert V_{N_{G+1}} \vert } & \cdots & \frac{\partial \Delta Q_{N}}{\partial \vert V_{N} \vert} \end{array} \right ] \left [ \begin{array}{c} \Delta \theta_{2} \\ \vdots \\ \Delta \theta_{N} \\ \Delta \vert V_{N_{G+1}} \vert \\ \vdots \\ \Delta \vert V_{N} \vert \end{array} \right ] = \left [ \begin{array}{c} \Delta P_{2} \\ \vdots \\ \Delta P_{N} \\ \Delta Q_{N_{G+1}} \\ \vdots \\ \Delta Q_{N} \end{array} \right ] \end{align}$$

Solution of the Problem

The solution update formula is given by:

$$\begin{align} \vec{x}^{(i+1)} = \vec{x}^{(i)} + \Delta \vec{x}^{(i)} = \vec{x}^{(i)} - \textbf{J}^{-1} \vec{f} (\vec{x}^{(i)}) \end{align}$$

To sum up, the NR algorithm, for application to the power flow problem is:

Set the iteration counter to $i=1$. Use the initial solution $V_{i} = 1 \angle 0^{\circ}$
Compute the mismatch vector $\vec{f}({\vec{x}})$ using the power flow equations
Perform the following stopping criterion tests:
- If $\vert \Delta P_{i} \vert < \epsilon_{P}$ for all type PQ and PV buses and
- If $\vert \Delta Q_{i} \vert < \epsilon_{Q}$ for all type PQ
- Then go to step 6
- Otherwise, go to step 4.
Evaluate the Jacobian matrix $\textbf{J}^{(i)}$ and compute $\Delta \vec{x}^{(i)}$.
Compute the update solution vector $\vec{x}^{(i+1)}$. Return to step 3.
Stop.

Convergence and Step Control

The iteration is governed by two parameters:

Tolerance (default $10^{-8}$): the run is converged once every entry of the mismatch vector satisfies $\vert f_{i}(\vec{x}) \vert <$ tolerance (an infinity-norm test over all $\Delta P$ and $\Delta Q$ components).
Maximum iterations (default $20$): an upper bound on the number of Newton steps per power flow solve.

To improve robustness far from the solution, the full Newton step is scaled by a single factor $\alpha \in (0, 1]$ rather than damped component-wise. The factor is the largest value that keeps the per-step changes within fixed bounds:

$$\alpha = \min \left( 1,\ \frac{\Delta\theta_{max}}{\max_k \vert \Delta\theta_k \vert},\ \frac{\Delta V_{max}}{\max_k \vert \Delta V_k / V_k \vert} \right)$$

with $\Delta\theta_{max} = 0.2\ \text{rad}$ and $\Delta V_{max} = 0.1\ \text{pu}$. Because the whole step is scaled by one factor, the Newton search direction is preserved, so $\alpha = 1$ near the solution and quadratic convergence is retained; $\alpha < 1$ only bounds large early steps. Voltage magnitudes are updated multiplicatively ($V_k \leftarrow V_k (1 + \alpha, \Delta V_k / V_k)$), consistent with the relative voltage increment used in the Jacobian.

The formulation above is independent of DPsim. For the solvers it actually ships, their convergence controls and the reactive power limit handling, see power flow solvers.

Docs: State-Space Nodal

Tue, 23 Jun 2026 00:00:00 +0000

The state-space nodal (SSN) method represents a component by its own continuous state-space model and couples it to the network through the nodal admittance matrix. A component is described by

$$\frac{d\mathbf{x}}{dt} = \mathbf{A}\mathbf{x} + \mathbf{B}\mathbf{u}, \qquad \mathbf{y} = \mathbf{C}\mathbf{x} + \mathbf{D}\mathbf{u},$$

where $\mathbf{x}$ is the internal state, and the input $\mathbf{u}$ and output $\mathbf{y}$ are the terminal quantities exchanged with the network (a voltage and the corresponding current). Trapezoidal discretisation of $(\mathbf{A}, \mathbf{B})$ yields a discrete model $(\mathbf{A}_d, \mathbf{B}_d)$ and a Norton equivalent: a constant conductance $\mathbf{W}$ stamped into the system matrix plus a history current source recomputed each step from the previous state and input. Because the component is solved simultaneously with the network in the same nodal system, SSN is numerically robust without the parasitic snubbers that delayed current-injection schemes require.

This builds directly on Nodal Analysis and is the companion of State-Space Extraction, which recovers a state-space model from an MNA simulation rather than starting from one.

Shift to the Dynamic-Phasor Envelope

The same component model is discretised differently in an envelope domain: the operator becomes $\frac{d}{dt} + j\omega_s$, so what is discretised is $\mathbf{A} - j\omega_s\mathbf{I}$ rather than $\mathbf{A}$, and splitting the envelope into real and imaginary parts turns that into a real system of twice the size. This is derived in full under SSN across domains, together with why the equivalent admittance is complex in an envelope domain and real in an instantaneous one.

The real-augmented form matches how the rest of the dynamic phasor system is already assembled: a complex admittance $g = g_r + j g_i$ is stamped as the real block $\left[\begin{smallmatrix} g_r & -g_i \ g_i & g_r \end{smallmatrix}\right]$, with real and imaginary node parts in separate halves of a real-valued system. The SSN component therefore needs no complex assembly of its own, and the trapezoidal discretisation used by the instantaneous models applies unchanged.

Components

The single-phase dynamic-phasor SSN models are:

Full_Serial_RLC, a series resistor-inductor-capacitor one-port with a hand-derived state-space model, used as the reference component.
GenericTwoTerminalVTypeSSN and GenericTwoTerminalITypeSSN, which accept a user-supplied $(\mathbf{A}, \mathbf{B}, \mathbf{C}, \mathbf{D})$ and build the V-type (voltage input, current output) or I-type (current input, voltage output) stamping accordingly.

All three reproduce the classical dynamic-phasor stamping of the same circuit, and the reconstructed time-domain waveform matches the EMT and EMT-SSN results within discretisation error.

Three-Phase Components

The same real-augmented model extends per phase to DP::Ph3. The $3 \times 3$ $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$, $\mathbf{D}$ matrices are general, so off-diagonal entries can couple the phases together:

Full_Serial_RLC, the three-phase series RLC one-port.
GenericTwoTerminalVTypeSSN and GenericTwoTerminalITypeSSN, the three-phase generic V-type and I-type components.

As in the single-phase case, all three reproduce the classical three-phase dynamic-phasor stamping exactly, and the reconstructed time-domain waveform matches the EMT and EMT-SSN results within discretisation error once corrected for the RMS-to-peak scaling that EMT::Ph3 sources apply and DP::Ph3 sources do not, since the DP envelope already carries the complex amplitude directly. The notebooks below only exercise the symmetrical, diagonal case; coupling between phases is not covered by existing tests.

Variable Components

The components above are fixed-parameter LTI systems: $(\mathbf{A}, \mathbf{B}, \mathbf{C}, \mathbf{D})$ are built once from the component’s parameters and never change. Some SSN components instead depend on the operating point and are relinearized and re-stamped every step: the same simultaneous nodal solve, but around a Jacobian frozen at the previous step’s converged state rather than a constant matrix. The averaged grid-following inverter is one such component, and its three-phase analogue carries a complex envelope per phase while sharing a single positive-sequence dq control frame; see Power Electronics for their state vectors and model equations.

Validation and Examples

Two notebooks accompany the single-phase models, both on a single-carrier series RLC one-port. examples/Notebooks/Circuits/DP_generalizedSSN_RLC.ipynb validates the DP-SSN models against the classical dynamic-phasor stamping and the EMT and EMT-SSN waveforms. examples/Notebooks/Circuits/DP_SSN_RLC_accuracy.ipynb studies the time-step and frequency-dependent accuracy against a small-step EMT reference.

The three-phase analogues, examples/Notebooks/Circuits/DP_Ph3_generalizedSSN_RLC.ipynb and examples/Notebooks/Circuits/DP_Ph3_SSN_RLC_accuracy.ipynb, repeat both studies on DP::Ph3 circuits, including a current-driven network with generic V-type and I-type components and a three-phase fault transient.

Docs: State-Space Nodal Across Domains

Fri, 31 Jul 2026 00:00:00 +0000

A component’s state-space model does not change between simulation domains. The same $\boldsymbol{A}$, $\boldsymbol{B}$, $\boldsymbol{C}$ and $\boldsymbol{D}$ describe the same physics whichever domain solves them. What changes is the operator that is discretised, and that difference propagates all the way to whether the resulting nodal stamp is real or complex.

The construction of the stamp itself is unchanged and is derived under state-space nodal components.

The instantaneous case

Here the state equation is integrated as written,

$$\dot{\boldsymbol{x}} = \boldsymbol{A}\boldsymbol{x} + \boldsymbol{B}\boldsymbol{u},$$

the trapezoidal rule gives a real discrete pair, and the equivalent admittance $\boldsymbol{W} = \boldsymbol{C}\boldsymbol{B}_d + \boldsymbol{D}$ is real.

The envelope case

In an envelope domain the state is a complex envelope $\tilde{\boldsymbol{x}}$ carrying an implicit $e^{j\omega_s t}$. Differentiating that product contributes the carrier term derived under dynamic phasors, so the operator seen by the envelope is

$$\frac{d}{dt} \; \longrightarrow \; \frac{d}{dt} + j\omega_s ,$$

and the system that must be discretised is not $\boldsymbol{A}$ but $\boldsymbol{A} - j\omega_s \boldsymbol{I}$.

The real-augmented form

Rather than integrate a complex system, split the envelope into real and imaginary parts. The shifted operator becomes a real system of twice the size,

$$\begin{bmatrix} \dot{\boldsymbol{x}}_{re} \\ \dot{\boldsymbol{x}}_{im} \end{bmatrix} = \begin{bmatrix} \boldsymbol{A} & \omega_s \boldsymbol{I} \\ -\omega_s \boldsymbol{I} & \boldsymbol{A} \end{bmatrix} \begin{bmatrix} \boldsymbol{x}_{re} \\ \boldsymbol{x}_{im} \end{bmatrix} + \begin{bmatrix} \boldsymbol{B} & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{B} \end{bmatrix} \begin{bmatrix} \boldsymbol{u}_{re} \\ \boldsymbol{u}_{im} \end{bmatrix}.$$

The off-diagonal $\pm\omega_s \boldsymbol{I}$ blocks are the carrier rotation, and the block structure $\begin{bmatrix} P & -Q \ Q & P \end{bmatrix}$ is the real representation of the complex number $P + jQ$. Discretising this real system with the same trapezoidal rule and recombining the blocks recovers the complex discrete pair, from which the equivalent admittance and the history term follow exactly as in the instantaneous case.

Two things follow. The equivalent admittance is complex in an envelope domain and real in an instantaneous one, so the same component stamps differently. And setting $\omega_s = 0$ collapses the augmented system back to the instantaneous one, which is the general statement about the envelope transform applied here: the instantaneous formulation is the zero-carrier special case, not a separate method.

Why this matters for accuracy

The carrier rotation is handled analytically, inside $\boldsymbol{A}$, rather than numerically by resolving the oscillation with small steps. For a component whose envelope varies slowly against a fast carrier, the step size is then set by the envelope’s own bandwidth rather than by the carrier frequency. That is the entire accuracy argument for using an envelope domain here, and it fails for exactly the reason it succeeds: content outside the retained band around $\omega_s$ has no representation at all.

Docs: State-Space Extraction

Thu, 28 May 2026 00:00:00 +0000

A nodal simulation computes a trajectory: given a set of sources and initial conditions it produces the node voltages step by step. It does not, by itself, say anything about the system’s modes, its damping, or how close it is to instability. Those questions are answered by the state-space description, which the extraction recovers from the discretised network that the simulation is already solving. The result is the discrete-time model of exactly what is being simulated, including the effect of the discretisation itself, rather than of an idealised continuous system that the simulation approximates.

Where the Discrete Model Comes From

The starting point is the companion form described under nodal analysis. Each dynamic element has already been turned into a conductance in parallel with a history current source, and its history term is a difference equation in the element’s own quantity: the current for an inductor, the voltage for a capacitor. Those quantities are the states. The network solve then supplies the terminal voltages that drive them.

This gives a system in two parts rather than one. The states advance using their own local recurrence together with the network solution at the new instant, and the network solution at that instant is itself driven by the states carried over from the previous one.

Assembled Form

The extracted model is assembled in the form:

$$\mathbf{x}[k+1] = \mathbf{A}_{d,\mathrm{local}} \mathbf{x}[k] + \mathbf{B}_{d,\mathrm{MNA}} \mathbf{x}_{\mathrm{MNA}}[k+1]$$

$$\mathbf{Y} \mathbf{x}_{\mathrm{MNA}}[k+1] = \mathbf{C}_{d,\mathrm{MNA}} \mathbf{x}[k]$$

where:

$\mathbf{x}$ is the extraction-state vector,
$\mathbf{x}_{\mathrm{MNA}}$ is the full MNA unknown vector,
$\mathbf{Y}$ is the active MNA system matrix,
$\mathbf{A}_{d,\mathrm{local}}$ contains local component state-transition contributions,
$\mathbf{B}_{d,\mathrm{MNA}}$ maps MNA unknowns to the state update,
$\mathbf{C}_{d,\mathrm{MNA}}$ maps extraction states to MNA current injections.

Eliminating the MNA unknown vector gives the global discrete-time state matrix

$$\mathbf{A}_{d} = \mathbf{A}_{d,\mathrm{local}} + \mathbf{B}_{d,\mathrm{MNA}} \operatorname{solve} \left( \mathbf{Y}, \mathbf{C}_{d,\mathrm{MNA}} \right)$$

The resulting matrix $\mathbf{A}_{d}$ describes the homogeneous discrete-time dynamics of the EMT MNA simulation model at the current operating point and system-matrix configuration.

The elimination is written as a solve rather than as an inverse for a reason. $\mathbf{Y}$ is sparse and already factorised for the time stepping, so the term is obtained by one forward and backward substitution per column of $\mathbf{C}_{d,\mathrm{MNA}}$, that is, one per extracted state. Forming $\mathbf{Y}^{-1}$ explicitly would discard the sparsity and cost far more than the extraction itself.

Interpreting the Result

The eigenvalues of $\mathbf{A}_{d}$ are discrete-time modes, so they are read against the unit circle rather than against the imaginary axis. A mode is stable when $|\lambda_d| < 1$, and the closer it sits to the unit circle the more lightly damped it is. A corresponding continuous-time eigenvalue follows from

$$\lambda_c = \frac{\ln \lambda_d}{\Delta t},$$

whose real part gives the damping and whose imaginary part gives the oscillation frequency, up to the ambiguity that any frequency above the Nyquist rate of the time step is indistinguishable from one below it.

It is worth being clear about what this eigenvalue belongs to. It is a mode of the discretised system, not of the underlying continuous one. The two differ by the distortion the integration rule introduces, and for the trapezoidal rule that distortion grows with frequency, so the fastest modes are the least faithful. Reducing the time step reduces the discrepancy; comparing extraction results across two time steps is a practical way to see which modes are trustworthy.

Validity

The extracted model is linear and time invariant, and it describes the system only for the configuration it was taken from. Any event that changes $\mathbf{Y}$, a switch opening or closing above all, produces a different $\mathbf{A}_{d}$, so a switching study means one extraction per configuration rather than one for the simulation. The same holds for non-linear elements, whose contributions are those at the operating point reached when the extraction was performed; moving the operating point requires extracting again.

References

J. A. Hollman and J. R. Marti, Step-by-step eigenvalue analysis with EMTP discrete-time solutions, IEEE Transactions on Power Systems, 2010. https://doi.org/10.1109/TPWRS.2009.2039810
Y. Han, H. Sun, B. Huang, S. Qin, M. Mu, and Y. Yu, “Discrete-Time State-Space Construction Method for SSO Analysis of Renewable Power Generation Integrated AC/DC Hybrid System,” IEEE Transactions on Power Systems, 2022. https://doi.org/10.1109/TPWRS.2021.3115248

Docs: Modal Analysis

Fri, 31 Jul 2026 00:00:00 +0000

Extracting a state-space model, as described under state-space extraction, produces a discrete state matrix. Its eigenvalues describe how the system behaves without simulating it: which oscillations exist, how fast each decays, and which states are involved in each one.

From discrete to continuous eigenvalues

The extracted model is discrete, so its eigenvalues $z$ live in the complex plane where stability means $|z| < 1$. That is awkward to read, because the quantities of interest are a frequency in hertz and a damping ratio, both of which are natural in the continuous plane.

Because the model was discretised with the trapezoidal rule, the mapping back is its inverse, the bilinear transform

$$\lambda = \frac{2}{\Delta t} \, \frac{z - 1}{z + 1}.$$

This maps the interior of the unit disc onto the left half plane exactly, so a mode that is stable in one description is stable in the other, with no threshold effects at the boundary. From $\lambda = \sigma + j\omega$ the damped frequency is $\omega / 2\pi$ and the damping ratio is $-\sigma / |\lambda|$.

The mapping is exact for the discretisation used, not an approximation of it. What it cannot undo is the frequency warping the trapezoidal rule introduced in the first place: a continuous mode at a frequency approaching the Nyquist rate is represented at a shifted frequency in the discrete model, and mapping back returns the shifted value rather than the original. Modes well below Nyquist are unaffected; modes near it should not be read literally.

What the eigenvectors say

The eigenvalues say which modes exist but not which parts of the system take part in them. The right eigenvectors describe how each mode appears in the states, the left eigenvectors describe how strongly each state excites each mode, and the product of the two, element by element,

$$p_{ki} = \phi_{ki} \, \psi_{ik},$$

is the participation factor of state $k$ in mode $i$.

Participation factors are the useful output. A poorly damped oscillation is a number; knowing that two particular machine rotor states dominate it is actionable. They are also dimensionless and normalised in a way that makes them comparable across states with different units, which raw eigenvector entries are not.

The computation requires the eigenvector matrix to be invertible. It is not, when the state matrix is defective, meaning it has a repeated eigenvalue without a full set of independent eigenvectors. This is not a numerical failure but a property of the system, and it is the one case where participation factors are not defined at all.

Choice of frame

The states of the extracted model are in whatever frame each component works in, which for a network containing machines means several rotating frames turning at different speeds plus the network’s own. Eigenvalues of that system are still correct, but the modes mix frames, and a mode’s frequency is then relative to whichever frame its dominant states live in.

Transforming everything into one common frame before the analysis removes that ambiguity, at the price of choosing the frame and its initial angle. The two choices are therefore: analyse in the native frames and read each mode relative to its own states, or transform to a single frame and read every frequency against the same reference. The second is what makes modes from different machines directly comparable.

Limits

The analysis is linear and local. It describes the system as it is at the operating point where the model was extracted, and says nothing about behaviour after a large disturbance moves it elsewhere. A system can be comfortably damped at its nominal point and not at another, so a single modal analysis is evidence about one condition rather than about the system.

Docs: Alternative Solution Methods

Fri, 31 Jul 2026 00:00:00 +0000

Nodal analysis is not the only way to advance a network in time. Three other formulations exist, each answering a different objection to it.

Differential-algebraic formulation

Nodal analysis discretises each dynamic element separately, replacing it with a companion model, and then solves an algebraic system. The step size is chosen in advance and applies to everything.

A differential-algebraic formulation instead writes the whole system as it stands,

$$\boldsymbol{F}\!\left(t, \boldsymbol{x}, \dot{\boldsymbol{x}}\right) = \boldsymbol{0},$$

mixing the differential equations of the dynamic elements with the algebraic constraints of the network, and hands it to an integrator that chooses its own order and step size to meet a requested error tolerance. Each component contributes its residual, the part of $\boldsymbol{F}$ it is responsible for, rather than a companion model.

What this buys is error control. The user asks for an accuracy rather than a step size, and the integrator takes small steps through fast transients and long ones through quiet intervals. What it costs is determinism: the number of steps and the work per step are not known in advance, which rules the method out for real-time execution and makes run times hard to predict. It is a method for producing reference results, not for meeting a deadline.

The formulation also admits systems that nodal analysis handles awkwardly, because a constraint does not have to be rewritten as an admittance to participate.

Ordinary differential formulation for a component

A weaker version of the same idea applies to one component rather than the whole network. The network is still solved by nodal analysis at a fixed step, but a single component whose internal dynamics are stiff or awkward is integrated by a dedicated solver between network steps.

The component exposes its state derivative, the solver advances it over the network step, and the result re-enters the network as an ordinary companion contribution. This keeps the deterministic outer loop while allowing one component to be integrated more carefully than the rest. The limitation is the same one as any staggered coupling: the component sees the network’s terminal conditions from the beginning of the step, so the exchange is only first-order accurate however carefully the inner integration is done.

Tearing a network

The third objection is about size rather than accuracy. Factorising the system matrix costs super-linearly in the number of nodes, so one large network is more expensive than the sum of its parts.

Diakoptics exploits this. A small set of branches is removed, chosen so that what remains falls into independent subnetworks. Each subnetwork is factorised on its own, which is cheap, and the removed branches are restored by solving a small dense system in the currents through them,

$$\boldsymbol{v} = \boldsymbol{Y}^{-1}\boldsymbol{i} - \boldsymbol{Y}^{-1}\boldsymbol{K} \left( \boldsymbol{Z}_{tear} + \boldsymbol{K}^{\mathsf{T}}\boldsymbol{Y}^{-1}\boldsymbol{K} \right)^{-1} \boldsymbol{K}^{\mathsf{T}}\boldsymbol{Y}^{-1}\boldsymbol{i},$$

where $\boldsymbol{K}$ records which nodes each removed branch connected and $\boldsymbol{Z}_{tear}$ holds their impedances. The bracketed matrix has the dimension of the number of torn branches, so the method pays off exactly when a network can be split by cutting few branches.

The result is not an approximation. Unlike the delay-based decoupling described under branches, which introduces a travel time and therefore an error, tearing reproduces the solution of the intact network to within round-off, because the removed branches are restored within the same step. The trade is that the subnetworks cannot be advanced independently: the small system couples them, so they must be solved together each step.

Choosing where to tear is the part with no general answer. Too few cuts leave the subnetworks large, too many make the dense system dominate, and a cut through a strongly coupled part of the network produces subnetworks whose solutions are dominated by the correction.

Docs: Models

Mon, 01 Jan 0001 00:00:00 +0000

Each page in this section derives one component family: what it represents, the equations that describe it, and what those equations assume. None of them names a class or a file. How a model is arranged in code is in the Developer Guide, and which domains implement which model is in model availability, which is generated from the source and is therefore the authoritative answer to that question.

The domains

A model is written per simulation domain and per phase count, so the same component may exist as a single-phase dynamic phasor model, a three-phase electromagnetic transient model, or both.

EMT carries instantaneous waveforms, DP carries complex envelopes of those waveforms around a carrier frequency, and SP carries steady-state phasors. Ph1 is single phase, usually a positive sequence representation, and Ph3 is three phase. The three domains are not separate methods but cases of one envelope description, which is the subject of dynamic phasors.

The practical consequence is that the domain determines what a model can represent, not only how fast it runs. An envelope domain cannot represent content outside the band it retains, whatever step size is used.

How the models group

Passive elements and branches. RLC elements covers the elements every other model is built from. Branches covers the lines connecting two nodes, including the travelling-wave line. Transformer covers the two-winding transformer and the ideal transformer within it.

Sources, switches and loads. Sources explains why a current source costs nothing while a voltage source extends the system matrix. Switches and loads cover the two-resistance switch and the impedance and current representations of demand. These three share one theme: a choice that looks physical is usually a numerical trade-off.

Machines. Synchronous generator covers the full-order and transient-stability machines, and reduced order the voltage-behind-reactance family from third to sixth order. The regulators that drive them, exciters, turbines and governors, sit alongside.

Converters. Power electronics covers the averaged inverter models, and converter control the phase-locked loop, the oscillator and the cascaded control that distinguish grid-following from grid-forming behaviour.

Interfaces and compensation. SSN components covers components solved simultaneously with the network, the ideal transformer model the general way of splitting a network, and network injection and compensation the representation of the grid beyond the boundary.

Signal blocks. Signal processing covers the integrators, filters and generators the other models are assembled from. These carry no current and connect to no node.

DPsim – Concepts

Docs: Nodal Analysis

Assembling the System by Stamping

Dynamic Elements and the Companion Model

Extending the Formulation for Voltage Sources

Solving Over Time

Docs: Dynamic Phasors

Bandpass Signals and Baseband Representation

Two Derivations of the Same Transform

Route A: shifting the spectrum

Route B: averaging a Fourier series

Where the routes meet

Relationship to Electromagnetic Transient Models

Docs: Powerflow

Power System Model

Single Phase Power Flow Problem

Powerflow Problem with Power Mismatch Function and Polar Coordinates

Formulation of Mismatch Function

Formulation of Jacobian

Solution of the Problem

Convergence and Step Control

Docs: State-Space Nodal

Shift to the Dynamic-Phasor Envelope

Components

Three-Phase Components

Variable Components

Validation and Examples

Further Reading

Docs: State-Space Nodal Across Domains

The instantaneous case

The envelope case

The real-augmented form

Why this matters for accuracy

Docs: State-Space Extraction

Where the Discrete Model Comes From

Assembled Form

Interpreting the Result

Validity

References

Docs: Modal Analysis

From discrete to continuous eigenvalues

What the eigenvectors say

Choice of frame

Limits

Docs: Alternative Solution Methods

Differential-algebraic formulation

Ordinary differential formulation for a component

Tearing a network

Docs: Models

The domains

How the models group