Load flow validation#

A load flow result is considered acceptable if it describes a feasible steady-state of a power system given its physics and its logics. More practically, generations of practitioners have set quasi-standard ways to describe them that makes it possible to define precise rules. They are described below for the different elements of the network.

Buses#

The first law of Kirchhoff must be satisfied for every bus for active and reactive power:

\[\begin{split}\begin{equation} \left| \sum_{branches} P + \sum_{injections} P \right| \leq \epsilon \\ \left| \sum_{branches} Q + \sum_{injections} Q \right| \leq \epsilon \\ \end{equation}\end{split}\]

Branches#

Lines and two-winding transformers are converted into classical PI models:

    V1*exp(j*theta1)     rho1*exp(j*alpha1)             r+j*x              rho2*exp(j*alpha2)   V2*exp(j*theta2)
        (P1,Q1)->      ____O/O__________________________-----__________________________O/O_____     <-(P2,Q2)
                                            |           -----           |
                                  g1+j*b1  |_|                         |_| g2+j*b2
                                            |                           |
                                           _|_                         _|_
                                            _                           _
                                            .                           .

  • Power flow results:

    • \((\|V_1\|, \theta_1)\) and \((\|V_2\|, \theta_2)\): Magnitude (kV) and angle (\(°\)) of the voltage at the sides 1 and 2, respectively.

    • \((P_1, Q_1)\) and \((P_2, Q_2)\): Active power (MW) and reactive power (MVAr) injected in the branch on each side.

  • Characteristics:

    • \((\rho_1, \alpha_1)\) and \((\rho_2, \alpha_2)\): Magnitude (no unit) and angle (\(°\)) of the ideal transformers ratios on each side.

    • \((g_1, b_1)\) and \((g_2, b_2)\): Complex shunt impedance on each side (S).

    • \((r, x)\): Complex series impedance \((\Omega)\).

Thanks to Kirchhoff laws (see the line and 2-winding transformer documentation), estimations of powers are computed according to the voltages and the characteristics of the branch:

\((P_1^{calc}, Q_1^{calc}, P_2^{calc}, Q_2^{calc}) = f(\text{Voltages}, \text{Characteristics})\)

Three-winding transformers#

To be implemented, based on a conversion into 3 two-winding transformers.

Generators#

Active power#

There may be an imbalance between the sum of generator active power setpoints \(\text{targetP}\) on one side and consumption and losses on the other side, after the load flow optimization process. Note that if it is possible to modify the setpoints during the computation (for example, if the results were computed by an Optimal Power Flow and not a Power Flow), there should be no imbalance left.

In case of an imbalance between the sum of generator active power setpoints \(\text{targetP}\) on one side and consumption and losses on the other side, the generation \(P\) of some units has to be adjusted. The adjustment is done by modifying the generation of the generators connected to the slack node of the network. It may also be done by modifying the loads connected to the slack node. The slack node is a computation point designated to be the place where adjustments are done.

This way of performing the adjustment is the simplest solution from a mathematical point of view, but it presents several drawbacks. In particular, it may not be enough in case of a large imbalance. This is why other schemes have been developed, called “distributed slack nodes”.

Generators or loads are usually adjusted proportionally to a shift function to be defined. Three keys have been retained for the validation (\(g\) is a generator): Usual ways of defining this function, for each equipment that may be involved in the compensation (generator or load), read:

  • proportional to \(P_{max}\): \(F = f \times P_{max}\)

  • proportional to \({targetP}\): \(F = f \times targetP\)

  • proportional to \(P_{diff}\): \(F = f (P_{max} - targetP)\)

\(f\) is a participation factor, per unit. For example, a usual definition is: \(f\in\{0,1\}\): either the unit participates or not. The adjustment is then done by doing: \(P <- P \times \hat{K} \times F\) where \(\hat{K}\) is a proportionality factor, usually defined for each unit by \(\dfrac{P_{max}}{\sum{F}}\), \(\dfrac{targetP}{\sum{F}}\) or \(\dfrac{P_{diff}}{\sum{F}}\) depending on the adjustment mode (the sums run over all the units participating in the compensation).

Voltage and reactive power#

If the voltage regulation is deactivated, it is expected that:

\(\left| targetQ - Q \right| < \epsilon\)

If the voltage regulation is activated, the generator is modeled as a \(PV\) node. The voltage target should be reached, except if reactive bounds are hit. Then, the generator is switched to \(PQ\) node and the reactive power should be equal to a limit. Mathematically speaking, one of the following 3 conditions should be met:

\[\begin{align*} |V - targetV| & \leq && \epsilon && \& && minQ & \leq & Q \leq maxQ \\ V - targetV & < & -& \epsilon && \& && |Q-maxQ| & \leq & \epsilon \\ targetV - V & < && \epsilon && \& && |Q-minQ| & \leq & \epsilon \\ \end{align*}\]

Loads#

To be implemented, with tests similar to generators with voltage regulation.

Shunts#

A shunt is expected not to generate or absorb active power:

\(\left| P \right| < \epsilon\)

A shunt is expected to generate reactive power according to the number of activated sections and to the susceptance per section \(B\): \(\left| Q + \text{#sections} * B V^2 \right| < \epsilon\)

Static VAR Compensators#

Static VAR Compensators behave like generators producing zero active power except that their reactive bounds are expressed in susceptance, so that they are voltage dependent.

\(targetP = 0\) MW

  • If the regulation mode is OFF, then \(targetQ\) is constant

  • If the regulation mode is REACTIVE_POWER, it behaves like a generator without voltage regulation

  • If the regulation mode is VOLTAGE, it behaves like a generator with voltage regulation with the following bounds (dependent on the voltage, which is not the case for generators): \(minQ = - Bmax * V^2\) and \(maxQ = - Bmin V^2\)

HVDC lines#

To be done.

VSC#

VSC converter stations behave like generators with the additional constraints that the sum of active power on converter stations paired by a cable is equal to the losses on the converter stations plus the losses on the cable.

LCC#

To be done.

Transformers with a ratio tap changer#

Transformers with a ratio tap changer have a tap with a finite discrete number of positions that allows to change their transformer ratio. Let’s assume that the logic is based on deadband: if the deviation between the measurement and the setpoint is higher than the deadband width, the tap position is increased or decreased by one unit.

As a result, a state is a steady state only if the regulated value is within the deadband or if the tap position is at minimum or maximum: this corresponds to a valid load flow result for the ratio tap changers tap positions.