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Modeling and equations

Grid modeling

Open Load Flow computes power flows from IIDM grid model in bus/view topology. From the view, a very simple network, composed of only buses and branches is created. In the graph vision, we rely on a $\Pi$ model for branches (lines, transformers, boundary lines, etc.):

  • $R$ and $X$ are respectively the real part (resistance) and the imaginary part (reactance) of the complex impedance ;
  • $G_1$ and $G_2$ are the real parts (conductance) on respectively side 1 and side 2 of the branch ;
  • $B_1$ and $B_2$ are the imaginary parts (susceptance) on respectively side 1 and side 2 of the branch ;
  • $A_1$ is the angle shifting on side 1, before the series impedance. For classical branches, the default value is zero ;
  • $\rho_1$ is the ratio of voltages between side 2 and side 1, before the series impedance. For classical branches, the default value is $1$.

As the $\Pi$ model is created from IIDM grid modeling that locates its ratio and phase tap changers in side 1, $A_2$ and $\rho_2$ are always equal to zero and $1$. In case of a branch with voltage or phase control, the $\Pi$ model becomes an array. See below our model:

Pi model{class="only-light"} Pi model{class="only-dark"}

HVDC line

Open Load Flow also supports networks with HVDC lines (High Voltage Direct Current lines). An HVDC line is connected to the rest of the AC network through HVDC converter stations, that can be either LCC (Line-Commutated Converter) or VSC (Voltage-Source Converter).

DC detailed model

Additionally, Open Load Flow supports AC-DC load flow formulation with detailed model of DC elements.
However, it is currently restricted to embedded DC components, meaning that all converters of a given DC component must be connected to the same synchronous component. In other words, a DC component cannot be used to connect two different synchronous components. Consequently, a single connected component should only contain one synchronous component. However, the number of DC components within a connected component is not restricted.
Yet, it is possible to run a load flow on a network with several synchronous components (with their own embedded DC components) as long as they do not belong to the same connected component.

(ac-flow-computing)=

AC flows computing

AC flows computing in OpenLoadFLow relies on solving a system of non-linear squared equations, where the unknowns are voltage magnitude and phase angle at each bus of the network, implying that there are $2N$ unknowns where $N$ is the number of buses. There are two equations per network bus, resulting in $2N$ equations. The nature of these $2$ equations depends on the type of the bus:

  • PQ-bus: active and reactive balance are fixed at the bus,
  • PV-bus: active balance and voltage magnitude are fixed at the bus.

Moreover, at the slack bus, the active balance equation is removed and replaced by an equation fixing the voltage phase angle at 0.

Let $v_i$ be the unknown voltage magnitude at bus $i$. Let $\phi_i$ be the unknown voltage phase angle at bus $i$. Equation fixing voltage magnitude to a reference (also called target) is simply written $v_i = V^{ref}_i$. Equation fixing voltage phase angle at slack bus $i$ is: $\phi_i = 0$

To build the active and reactive balance equations, Open Load Flow first expresses active and reactive power flowing from a bus to another through a line:

$$p_{i,j}= \rho_iv_i(G_i\rho_iv_i + Y\rho_iv_i\text{sin}(\Xi) - Y\rho_jv_j\text{sin}(\theta))$$

$$q_{i,j}= \rho_iv_i(-B_i\rho_iv_i + Y\rho_iv_i\text{cos}(\Xi) - Y\rho_jv_j\text{cos}(\theta))$$

Where $Y$ is the magnitude of the line complex admittance $\frac{1}{R+jX}$, and $\Xi$ such that: $R+jX = \frac{1}{Y}e^{j(\frac{\pi}{2}-\Xi)}$. $\theta$ satisfies: $\theta= \Xi - A_i + A_j - \phi_i + \phi_j.$

Beware that $p_{i,j}$ is the power that goes out from the bus $i$.

Therefore, active and reactive balance equations are expressed as:

$$ P_i^{in} = \sum_{j \in \delta(i)} p_{i,j}$$

$$ Q_i^{in} = \sum_{j \in \delta(i)} q_{i,j}$$

where $\delta(i)$ is the set of buses linked to $i$ in the network graph.

The resulting non-linear system of equations is solved by default via the Newton-Raphson algorithm. The underlying principle of the algorithm is the following:

  • It starts at a certain point $x_0 = (v_0, \phi_0)$ as an approximate solution to the system of equations;
  • Then, in an iterative fashion, it generates a series $x_1, x_2,.., x_k$ of better approximate solutions to the system of equations;
  • These iterates $x_k$ are found by solving a system of equations using local Jacobian matrix $J(v,\phi)$ at the previous point $x_{k-1}$;

See acSolverType below for more details.

Other regulation modes

PQ-bus and PV-bus are used to model local voltage magnitude or local reactive power controls. Other controls are supported in OpenLoadFLow:

  • Remote voltage control for generators, static var compensators and two and three windings transformers with ratio tap changer. Control shared over several controllers buses is supported ;
  • Remote reactive power control for generators ;
  • For static var compensator with a voltage set point, the support of a voltage per reactive power control, also called slope, that modifies a bit the local voltage at connection bus. We only support a local control.

Remote voltage control

In our explanation, we have two buses. A generator or more is connected to bus $b_1$, that is called controller bus. The remote bus $b_2$, where voltage should reach the target, is called controlled bus. The bus $b_1$ is no longer a PQ-bus and becomes a P-bus: only active power balance is fixed for that bus. Bus $b_2$ becomes a PQV-bus, where the voltage magnitude is fixed at the value defined by the voltage control. To resume:

  • At controller bus $b_1$:
    • ${b_1}^{in} = \sum_{j \in v(b_1)} p_{b_1,j}$.
  • At controlled bus $b_2$:
    • $P_{b_2}^{in} = \sum_{j \in v(b_2)} p_{b_2,j}$.
    • $Q_{b_2}^{in} = \sum_{j \in v(b_2)} q_{b_2,j}$.
    • $v_{b_2} = V^{c}_{b_1}$.

Remote reactive power control

A bus $b_1$ has, through a generator, a remote reactive power control on a branch $(i,j)$. This controller bus is treated as a P-bus: only active power balance is fixed for that bus. The reactive power flowing at side i on line $(i,j)$ is fixed by the control (it could be at side j too). To resume:

  • At controller bus $b_1$:
    • $P_{b_1}^{in} = \sum_{j \in v(b_1)} p_{b_1,j}$.
  • At controlled branch $(i,j)$:
    • $q_{i,j} = Q^{c}_{b_1}$.

Local voltage control for a static var compensator with a slope

We only support the simple case where:

  • Only one generator controlling voltage is connected to a bus. If other generators are present, they should have a local reactive power control ;
  • The control is local ;
  • No other generators from other controller buses are controlling the bus where the static var compensator is connected. Let's call it $b_1$.

In that case only, the voltage equation at bus $b_1$ is replaced with:

$$v_{b_1} + s \cdot q_{svc} = V^{c}_{b_1}$$

Where $s$ is the slope of the static var compensator.

Computing HVDC power flow

LCC converters

LCC converters can be considered as fixed loads in the load flow. Indeed, on one side of the line is the rectifier station, and on the other side of the line is the inverter station. The active power flows from the rectifier station to the inverter station, is fixed, and equals to a target value $P$ (AC side). The active power flow at each station at AC side is given by:

  • $P_{rectifier}= P$
  • $P_{inverter}= (1 - LossFactor_{inverter}) * ((1 - LossFactor_{rectifier}) * (P - P_{LineLoss}))$

Power flows are in load sign convention, the active power at the rectifier AC terminal is positive and the active power at the inverter AC terminal is negative. The HVDC line losses $P_{LineLoss}$ are described in a dedicated section further below.

The reactive power consumption of each converter on AC side is determined by the configured converter power factor of the converter station in the grid model, representing the ratio between active power $P$ and apparent power $S$. For each converter, its target reactive power is given by:

  • $Q=\mid P*\tan(\cos(powerfactor))\mid$

Note that LCC converters are always absorbing reactive power.

VSC converters

VSC converters are self-commutated converters that can be assimilated to generators in the loadflow. There can be two main modes to control active power flow through the line:

  • In active power setpoint mode, as for LCC converters, on one side of the line is the rectifier station, and on the other side of the line is the inverter station. The active power flow from the rectifier station to the inverter station is fixed and equals to a target value $P$ (AC side). The active power flow at each station at AC side is given by:

    • $P_{rectifier}= P$
    • $P_{inverter}= (1 - LossFactor_{inverter}) * ((1 - LossFactor_{rectifier}) * (P - P_{LineLoss}))$

  • In AC emulation mode, the active power flow between both stations is given by: $P = P_0 + k~(\phi_1 - \phi_2)$ with $\phi_1$ and $\phi_2$ being the voltage angles at the bus connection for each converter station, and $P_0$ and $k$ being fixed parameters for the HVDC line. If $P$ is positive, the converter station 1 is controller, else it is converter station 2. For example, if station 1 is controller, the active power flow at each station is given by the formula below (HVDC line losses are described in the next paragraph):

    • $P_{controller} = P_0 + k~(\phi_1 - \phi_2)$
    • $P_{noncontroller} = (1 - LossFactor_{noncontroller}) * ((1 - LossFactor_{controller}) * (P_0 + k~(\phi_1 - \phi_2) - P_{LineLoss}))$

The HVDC line losses are described in a dedicated section further below.

In both control modes (active power setpoint mode or in AC emulation mode), the target value $P$ is bounded by a maximum active power $P_{max}$ that can be either:

  • the maxP configured for the HVDC line,
  • or alternatively separate limit values for both directions using the HVDC operator active power range iIDM extension In AC Emulation, these boundaries are handled through the HVDC AC emulation limit outer loop. After solving the equation system, if the computed active power flow through the HVDC overpasses the limits, saturation is applied by the outer loop. Note that this HVDC AC emulation outer loop is only available in AC calculation, and normal DC calculation (not available in DC Woodbury Security analysis and DC Woodbury Sensitivity Analysis).

The reactive power flow on each side of the line depends on whether voltage regulation of the converters is enabled. If the voltage regulation is enabled, then the VSC converter behaves like a generator regulating the voltage. Otherwise, reactive power of the converter at AC side is given by its reactive power setpoint.

HVDC line losses

In both cases of LCC and VSC, the power flows are impacted by losses of the converter stations. In addition to the converter losses, Joule effect (due to resistance in cable) implies line losses in the HVDC line. The HVDC line losses are calculated assuming nominal DC voltage: $P_{LineLoss} = Ri^2$ with $i = P_1 / V$ with $R$ being the HVDC line resistance, $P_1$ being the active power at the output of the controller station and $V$ being the HVDC nominal voltage (equals 1 per unit).

DC flows computing

The DC flows computing relies on several classical assumptions to build a model where the active power flowing through a line depends linearly on the voltage angles at its ends. In this simple model, reactive power flows and active power losses are totally neglected. The following assumptions are made to ease and speed the computations:

  • The voltage magnitude is equal to $1$ per unit at each bus,
  • The series conductance $G_{i,j}$ of each line $(i,j)$ is neglected, only the series susceptance $B_{i,j}$ is considered,
  • The voltage angle difference between two adjacent buses is considered as very small.

Therefore, the power flows from bus $i$ to bus $j$ following the linear expression:

$$ P_{i,j} = \frac{\phi_i-\phi_j+A_{i,j}}{X_{i,j}} $$

Where $X_{i,j}$ is the serial reactance of the line $(i,j)$, $\phi_i$ the voltage angle at bus $i$ and $A_{i,j}$ is the phase angle shifting on side $j$.

DC flows computing gives a linear grid constraints system. The variables of the system are, for each bus, the voltage angle $\phi$. The constraints of the system are the active power balance at each bus, except for the slack bus. The voltage angle at slack bus is set to zero. Therefore, the linear system is composed of $N$ variables and $N$ constraints, where $N$ is the number of buses in the network.

We introduce the linear matrix $J$ of this system that satisfies:

If $i$ is the slack bus,

$$J_{i,i} = 1$$

Else:

$$J_{i,i} = \sum_{j \in \delta(i)} \frac{1}{X_{i,j}}$$

$$J_{i,j} = - \frac{1}{X_{i,j}}, \quad \forall j \in \delta(i)$$

where $\delta(i)$ is the set of buses linked to bus $i$ in the network graph. All other entries of $J$ are zero.

The right-hand-side $b$ of the system satisfied:

If $i$ is the slack bus,

$$b_{i} = 0$$

Else:

$$b_{i} = P_i - \sum_{j \in \delta(i)} \frac{A_{i,j}}{X_{i,j}}$$

where $\delta(i)$ is the set of buses linked to bus $i$ in the network graph.

Where $P_i$ is the injection at bus $i$.

This linear system is resumed by:

$$ J\phi = b $$

The grid constraints system takes as variables the voltage angles. Note that the vector $b$ of right-hand sides is linearly computed from the given injections and phase-shifting angles.

To solve this system, we follow the classic approach of the LU matrices decomposition $J = LU$. Hence, by solving the system using LU decomposition, you can compute the voltage angles by giving as data the injections and the phase-shifting angles.

Area Interchange Control

Area Interchange Control consists in having the Load Flow finding a solution where area interchanges are solved to match the input target interchange values. It is supported for both AC and DC Load Flow computations.

The area interchange control feature is optional, can be activated via the parameter areaInterchangeControl and is performed by an outer loop.

Area Interchange Control is performed using an outer loop, similar in principle to the traditional SlackDistribution outer loop. However unlike the SlackDistribution outer loop which distributes imbalance over the entire synchronous component (island), the Area Interchange Control outer loop performs an active power distribution over areas (filtered on areas having their type matching the configured parameter areaInterchangeControlAreaType), in order to have all areas' active power interchanges matching their target interchanges.

The Area Interchange Control outer loop can handle networks where part (or even all) of the buses are not in an area. For networks that have no areas at all, the behaviour is the same as with the distributed slack outer loop - in such case internally the Area Interchange Control outer loop just triggers the Slack Distribution outer loop logic.

Just like other outer loops, the Area Interchange Control outer loop checks whether area imbalance must be distributed:

  • If no, the outer loop is stable
  • If yes, the outer loop is unstable and a new Newton-Raphson is triggered

Area Interchange Control - algorithm description

The active power is distributed separately on injections (as configured in the parameter balanceType) of each area to compensate the area "total mismatch" that is given by:

$$ Area Total Mismatch = Interchange - Interchange Target + Slack Injection $$

Where:

  • "Interchange" is the sum of the power flows at the boundaries of the area (load sign convention i.e. counted positive for imports).
  • "Interchange Target" is the interchange target parameter of the area.
  • "Slack Injection" is the active power mismatch of the slack bus(es) present in the area (see Slack bus mismatch attribution).

The outer loop iterates until the absolute value of this mismatch is below the configured parameter areaInterchangePMaxMismatch for all areas.

When it is the case, "interchange only" mismatch is computed for all areas:

$$ Interchange Mismatch = Interchange - Interchange Target $$

If the absolute value of this mismatch is below the parameter areaInterchangePMaxMismatch for all areas and the absolute value of slack bus active power mismatch is below the parameter slackBusPMaxMismatch, then the outer loop declares a stable status, meaning that the interchanges are correct and the slack bus active power is distributed.

If not, the remaining slack bus mismatch is first distributed over the buses that have no area.

If some slack bus mismatch still remains, it is distributed over all the areas (see Remaining slack bus mismatch distribution).

Areas validation

There are some cases where areas are considered invalid and will not be considered for the area interchange control:

  • Areas without interchange target
  • Areas without boundaries
  • Areas that have boundaries in multiple synchronous/connected components. If all the boundaries are in the same component but some buses are in different components, only the part in the component of the boundaries will be considered.

In such cases the involved areas are not considered in the Area Interchange Control outer loop, however other valid areas will still be considered.

Interchange flow calculation

In iIDM each area defines the boundary points to be considered in the interchange. iIDM supports two ways of modeling area boundaries:

  • either via an equipment terminal,
  • or via a BoundaryLine boundary.

In the BoundaryLine case, the flow at the boundary side is considered as it should be, for both unpaired BoundaryLines and BoundaryLines paired in a TieLine.

Slack bus mismatch attribution

Depending on the location of the slack bus(es), the role of distributing the active power mismatch will be attributed based on the following logic:

  • If the slack bus is part of an area: the slack power is attributed to the area (see "total mismatch" calculation in Algorithm description). Indeed, in this case the slack injection can be seen as an interchange to 'the void' which must be resolved.
  • Slack bus has no area:
    • Connected to other bus(es) without area: treated as the slack mismatch of the buses without area
    • Connected to only buses that have an area:
      • All connected branches are boundaries of those areas: Not attributed to anyone, the mismatch will already be present in the interchange mismatch
      • Some connected branches are not declared as boundaries of the areas: Amount of mismatch to distribute is split equally among the areas (added to their "total mismatch")

Remaining slack bus mismatch distribution

This section covers the case where the "total mismatch" of all areas is in [-areaInterchangePMaxMismatch;areaInterchangePMaxMismatch], but some slack bus active power mismatch remains (even after trying to distribute on buses with no area). This remaining slack bus active power mismatch will be distributed by all areas, each one will get a share of this mismatch to distribute.

This distribution will affect each area's interchange and will not necessarily make it closer to its target. The distribution factor of each area will be computed in a way that minimises chances of having the area increase its interchange mismatch up to more than areaInterchangePMaxMismatch in absolute value.
So the factor is proportional to the "margin" of active power that the area can distribute while keeping $-areaInterchangePMaxMismatch < Area Total Mismatch < areaInterchangePMaxMismatch$.

It is computed like this:
$Factor = sign(Slack Bus Mismatch) * Area Total Mismatch + areaInterchangePMaxMismatch $
Then factors are normalized to have sum of factors equal to 1.

The distribution is iterative (inside the same outer loop iteration). Each area distributes its share, if some areas cannot fully distribute it, they are excluded from this distribution and the remaining slack is distributed among other areas at next iteration. The distribution iterates until all the mismatch has been distributed and fails if all areas cannot distribute anymore but some mismatch remains.

Zero impedance boundary branches

The following applies when the lowImpedanceBranchMode is set to REPLACE_BY_ZERO_IMPEDANCE_LINE. Currently, computations involving zero-impedance branches used as boundary branches are not supported. However, it is still possible to submit network models that include zero-impedance boundary branches.
If a terminal of a zero-impedance branch is designated as a boundary, Open LoadFlow will internally assign the branch an impedance value equal to the lowImpedanceThreshold parameter.

Fast-Decoupled Algorithm

Fast-Decoupled is an algorithm to solve the inner-loop of the load flow problem, like the Newton-Raphson one. It is activated giving the FAST_DECOUPLED value to the acSolverType parameter. The solved equation system is the same as the one solved by Newton-Raphson method. However, the Jacobian matrix used is decomposed into two smaller matrices, decoupling the active power balance equations from voltage magnitudes variations and reactive power balance equations from voltage phases variations.

Note that "Fast-Decoupled" is the academic name for this algorithm and not a statement about its performances versus the Newton-Raphson algorithm.

Method

The Fast-Decoupled method is composed of two parts, one relative to the decoupling, another relative to the speeding of the calculations.

The decoupling is obtained by dividing the Jacobian matrix into two matrices, one relative to voltage phases variables and active power equations, the other relative to voltage magnitudes variables and reactive power equations. Combining those two smaller matrices, one can obtain an approximation of the real Jacobian, where some terms have been discarded.

By approximating more terms on top of the Jacobian structure simplification, the algorithm provides two Jacobian matrices that are constant, with respect to the multiplication by a diagonal matrix. Thus, the LU decomposition of the two Jacobian matrices is done only once, at the start of the first Fast-Decoupled iteration.

Regarding state vector scaling, the Fast-Decoupled uses both personalized max voltage change and line-search routines. Without these routines, the algorithm struggles to converge on realistic large networks, as it has a simplified vision of the impact of the system variables. Note that stateVectorScalingMode is not taken into account.

Limitations

The current implemented version cannot compute when one of the following parameter is activated:

In case where the user has selected both the Fast-Decoupled algorithm and one of this parameter, an exception is triggered.

Users should notice that default parameters are optimized for the Newton-Raphson algorithm, as it is the default one. When the Fast-Decoupled algorithm is used, we recommend these values for some convergence parameters:

AC DC flows computing

AC DC flows computing in OpenLoadFLow is similar to AC flows computing, but with AC and DC equations in the same system. The unknowns are voltage magnitude and phase angle for each AC bus, voltage for each DC bus, and active/reactive power for each voltage source converter.
Concerning AC side, the equations are the same as in AC flows computing, concerning DC side, the equations induced by DC components are the followings:

DC bus

At least one DC bus must be connected to the ground in each DC network, its potential is therefore set to 0. Therefore, symmetrical configuration are currently not supported.
For the others DC buses, each one introduces an equation of current balance: $\sum_{i} I_i = 0$ where $I_i$ are the currents going out of the DC bus. These terms are introduced by the DC components connected to the DC bus.

DC Line

Each DC line adds one term in both of its two connected DC buses current balance:

$\sum_{i} I_i + \frac{V_1 - V_2}{R}= 0$ for dcBus1

$\sum_{i} I_i - \frac{V_1 - V_2}{R}= 0$ for dcBus2

Line Commutated Converter

Line commutated converters are not supported yet by Open Load Flow.

Voltage source converters

Let consider a network that is composed of one AC network, and one DC network. The voltage source converter is the link between AC and DC networks, it is linked to one AC bus at one side, and two DC buses at the other side.
Please note that converters with a second optional AC terminal are not supported by Open Load Flow.

The converter can control either the power received by the AC network (P_PCC control mode) or the voltage between its two DC buses (V_DC control mode). At least one of the voltage source converters of the DC network must be in V_DC mode. Otherwise, an exception will be thrown.

In addition to the control modes P_PCC and V_DC, the voltage source converter can be set in two modes :

  • Reactive power control mode, in which it imposes the reactive power received from AC to DC, which is 0 by default. In this case, the AC voltage is not fixed.
  • Voltage regulator control mode, in which it imposes the voltage at its AC Bus. In this case the reactive power is not fixed.

Warning: At the moment, the active and reactive power control and the AC voltage control are enforced at the converter AC terminal, not at its PCC terminal.

We note $P_{AC}$ the power flow injected by AC network into the converter. So $P_{AC}>0$ if the power flows from AC to DC and $P_{AC}<0$ otherwise.

If the converter is in P_PCC control mode, we add an equation to impose $P_{AC}$ :

$P_{AC}$ = $P_{Ref}$

Else the converter is in V_DC control mode, and we add an equation to impose the voltage between its two DC buses :

$V_{1} - V_{2} = V_{Ref}$

Similarly, if the converter controls reactive power, we add an equation to impose $Q_{AC}$ :

$Q_{AC}$ = $Q_{Ref}$

Else the converter controls the AC voltage, and we add an equation to impose $V_{AC}$:

$V_{AC}= V_{Ref}$

On the AC bus, the active and reactive power injected into the converter is added to its power balance.
On the DC side, we introduce the variable $I_{Conv}$ which is the current flowing in the converter from dcBus1 to dcBus2. It is added to the current balances of dcBus1 and dcBus2

$\sum_{i} I_i + I_{Conv} = 0$ for dcBus1

$\sum_{i} I_i - I_{Conv}= 0$ for dcBus2

Power Equations

The last equation of converters ensures the conservation of power between AC and DC.

$$P_{DC} + P_{AC} = P_{Loss}$$

with:

  • $P_{AC}$ the power injected by the AC network into the converter
  • $P_{Loss}>=0$ the converter losses depending on AC current. Its computation is detailed below.
  • $P_{DC} = I_{Conv}*(V_1-V_2)$ the power injected by the DC network into the converter.

If the converter acts as rectifier, AC injects power in DC, thus $P_{DC}<0$ and $P_{AC}>0$, so we have :

$$ -|P_{DC}| + P_{AC} = P_{Loss} $$ $$ |P_{DC}| = |P_{AC}| - P_{Loss} $$

And if the converter acts as inverter, DC injects power in AC, thus $P_{DC}>0$ and $P_{AC}<0$, so we have :

$$ P_{DC} - |P_{AC}| = P_{Loss} $$ $$ |P_{AC}| = |P_{DC}| - P_{Loss} $$

In both cases, there is a loss of power when passing through the converter.

$P_{Loss}$ is defined as :
$ P_{Loss} = IdleLoss + SwitchingLoss * |I_{Conv}| + ResistiveLoss * I_{Conv}^{2} $

Idle loss, switching loss and resistive loss are loss factors that depend on the converter.

Using the previous equation of power conservation between AC and DC, we have $$ I_{Conv}(V_1-V_2) + P_{AC} = IdleLoss + SwitchingLoss|I_{Conv}| + ResistiveLoss*I_{Conv}^{2} $$