Alternative current optimal power flow¶

Generalities¶

The goal of the reactive ACOPF is to compute voltage values on each bus, as well as control values for reactive equipment and controllers of the grid. Then, the following values will be variable in the optimization:

\(\boldsymbol{V_i}\) and \(\boldsymbol{\theta_i}\) the voltage magnitude and phase of bus \(i\).
\(\boldsymbol{P_{i,g}}\) (resp. \(\boldsymbol{Q_{i,g}}\)) the active (resp. reactive) power produced by variable generator \(g\) of bus \(i\).
\(\boldsymbol{Q_{i,vsc}}\) the reactive power produced by voltage source converter stations \(vsc\) of bus \(i\).
\(\boldsymbol{Q_{i,b}}\) the reactive power produced by battery \(b\) of bus \(i\) that regulates voltage.
\(\boldsymbol{b_{i,s}}\) (resp. \(\boldsymbol{b_{i,svc}}\)) the susceptance of shunt \(s\) (resp. of static var compensator \(svc\)) of bus \(i\).
\(\boldsymbol{\rho_{ij}}\) the transformer ratio of the ratio tap changer on branch \(ij\), specified as variable by the user (see Configuration of the run).

Please note that:

Units with active power specified in ampl_network_generators.txt that is less than the configurable parameter Pnull are excluded from the optimization, even if the user designates these generators as fixed in the parameter file param_generators_reactive.txt (see Configuration of the run). Therefore, when the optimization results are exported, these generators are exported with a reactive power target of \(0\).
Neither current limits nor power limits on branches are considered in the optimization.
Branches with one side open are considered in optimization.
The voltage controls are not taken into account in the optimization model, as its purpose is to determine them (see OpenReac). However, the remote control of generators, static var compensators and batteries is taken into account in the export of equipment’s voltage target (see Outputs).
The transformation ratios \(\boldsymbol{\rho_{ij}}\) and the shunt susceptances \(\boldsymbol{b_{i,s}}\) are continuous in the optimization. At the end, these variables may differ from the values associated with the discrete taps of the equipment (see Network data), and rounding may be necessary. In the case of transformers, a second optimization can be carried out to adjust the voltage plan to the new transformation ratios after rounding (see Solving).
Transformers connected in parallel are grouped and constrained to a common transformation ratio (see Parallel transformers).
Only batteries that regulate voltage (as flagged in ampl_network_batteries.txt) have a variable reactive power; the reactive power of other batteries is fixed at its input value. The active power of batteries is never optimized.

Constraints¶

The constraints of the optimization problem depend on parameters specified by the user (see Configuration of the run). In particular, the user can indicate which buses will have associated reactive slacks \(\boldsymbol{\sigma_{i}^{Q,+}}\) and \(\boldsymbol{\sigma_{i}^{Q,-}}\) , expressing the excess (resp. shortfall) of reactive power produced in bus \(i\), and used to ensure reactive power balance. To do so, these buses must be specified in parameter file param_buses_with_reactive_slack.txt, and buses_with_reactive_slacks must be set to \(\text{CONFIGURED}\).

The ACOPF involves the following constraints, in addition to the slack constraint \((1)\) introduced in the Slack bus and main connex component part:

\[\sum\limits_{j\in v(i)} \boldsymbol{p_{ij}} = P_i^{in} - \sum\limits_{g}\boldsymbol{P_{i,g}}, \quad i\in\text{BUSCC} \quad (5)\]

\[\sum\limits_{j\in v(i)} \boldsymbol{q_{ij}} = Q_i^{in} - \boldsymbol{\sigma_{i}^{Q,+}} + \boldsymbol{\sigma_{Q_i}^{-}} - \sum\limits_{g}\boldsymbol{Q_{i,g}} - \sum\limits_{b}\boldsymbol{Q_{i,b}} - \sum\limits_{vsc}\boldsymbol{Q_{i,vsc}} - \sum\limits_{s}\boldsymbol{b_{i,s}}{V_i}^2 - \sum\limits_{svc}\boldsymbol{b_{i,svc}}{V_i}^2, \quad i\in\text{BUSCC} \quad (6)\]

where:

\(\boldsymbol{p_{ij}}\) (resp. \(\boldsymbol{q_{ij}}\)) is the active (resp. reactive) power leaving bus \(i\) on branch \(ij\), calculated as defined in the PowSyBl documentation. Those are variables because they depend on \(\boldsymbol{V_i}\), \(\boldsymbol{V_j}\), \(\boldsymbol{\theta_i}\), \(\boldsymbol{\theta_j}\) and \(\boldsymbol{\rho_{ij}}\).
\(P_i^{in}\) is the constant active power injected or consumed in bus \(i\) by batteries, loads, VSC stations and LCC stations.
\(Q_i^{in}\) is the constant reactive power injected or consumed in bus \(i\), by fixed generators and fixed shunts (see Configuration of the run), batteries that do not regulate voltage, loads and LCC stations.

In order to bound the variables described in Generalities, the limits specified in the files of network data (see Network data) are used. We specify the following special treatments:

The voltage magnitude \(\boldsymbol{V_i}\) lies between the corrected voltage limits described in the Voltage level limit consistency section.
The reactive power \(\boldsymbol{Q_{i,g}}\) produced by unit \(g\) lies between the corrected limits described in the P/Q unit domain section.
The active power \(\boldsymbol{P_{i,g}}\) also lies between the corrected limits described in the P/Q unit domain section, but these bounds are only considered when the configurable parameter \(\alpha\) is different than \(1\) (default value). Otherwise, all active powers evolve proportionally to their initial point \(P_{i,g}^t\) (specified in ampl_network_generators.txt): \(\boldsymbol{P_{i,g}} = P_{i,g}^t + \boldsymbol{\gamma} (P_{g}^{max,c} - P_{i,g}^t)\), where \(\boldsymbol{\gamma}\) is optimized and lies in \([-1;1]\).
The reactive power \(\boldsymbol{Q_{i,vsc}}\) produced by voltage source converter station \(vsc\) is included in \([\min(qP_{vsc}, qp_{vsc}, qp_{vsc}^0)\); \(\max(QP_{vsc}, Qp_{vsc}, Qp_{vsc}^0)]\).
The reactive power \(\boldsymbol{Q_{i,b}}\) produced by a voltage-regulating battery \(b\) lies between corrected reactive limits derived from its reactive capability diagram at its (fixed) active power \(P_b^0\), computed as for units (see P/Q unit domain). The bounds are therefore rectangular, not trapezoidal.

Parallel transformers¶

Transformers connected in parallel (sharing the same pair of buses, or forming a closed loop of transformers inside a single substation) should keep the same transformation ratio: letting them diverge would create circulating reactive flows between the parallel branches. The optimizer therefore detects such groups (called bundles) automatically, and constrains each bundle to a single shared ratio. The grouping is enabled by default and can be opted out through the Java API (OpenReacParameters), in which case the detection is skipped and every transformer ratio is optimized independently.

Bundle detection¶

The detection is purely topological — the numeric qualification below (tie / fix / relax and all the ratio bounds) is entirely derived by the AMPL code. Two strategies are combined:

Simple parallels: transformers sharing the exact same pair of buses in the bus view. Closed couplers merge buses in that view, so two transformers connected through closed switches on both sides are detected as well:

B1 (400 kV) ───┬─────────┬───
               T1        T2
B2 (225 kV) ───┴─────────┴───

Loops: transformers belonging to a chordless cycle of size 3 or 4 inside a single substation. The cycle search runs on a graph whose nodes are the buses of the substation and whose edges are its transformers and internal lines — lines close the loop but are never members. Within a cycle, edges are grouped by nominal voltage pair (only transformers performing the same conversion can share a ratio) and, within a pair, must be adjacent (connected components of the per-pair subgraph):

         A (400 kV)
        /          \
      T1            T2
      /              \
B (225 kV) ──line── C (225 kV)

A (225 kV) ──T1── B (90 kV)
     │                │
    T4               T2
     │                │
D (90 kV) ───T3── C (225 kV)

In the triangle, T1 and T2 perform the same 400/225 conversion and share bus A: they form a bundle of two, the 225 kV line closing the loop. In the square, the four transformers alternate direction around the ring and form a single bundle of four.

Only transformers carrying a ratio tap changer are kept, sets sharing at least one transformer are merged transitively (if \(A \parallel B\) and \(B \parallel C\), then \(A \parallel B \parallel C\)), and the result is ordered deterministically (decreasing size, then smallest id) so that bundle numbering is stable across runs.

Bundle qualification¶

The quantity that must be equalized is the effective per-unit ratio \(c_{ij} \boldsymbol{\rho_{ij}}\), where \(c_{ij}\) is the constant (off-tap) per-unit ratio of the transformer (the “cst ratio (pu)” column of the network data): this is the ratio entering the flow equations, whose mismatch drives circulating flows. When all members of a bundle are identical units (\(c_{ij}\) equal), this is equivalent to equalizing the tap ratios themselves.

The effective ratio applies from terminal 1 to terminal 2 as declared in the network, and nothing in the data model forces two physically parallel transformers to be declared in the same direction. Each member is therefore qualified against a canonical direction of its bundle (that of its first member in id order): a member declared in the same direction is direct (orientation +1), a member declared in the opposite direction is reversed (orientation -1). The comparison uses the nominal voltage side of terminal 1, falling back to the terminal 1 bus for equal nominal voltage bundles sharing a single bus pair; an equal nominal voltage bundle that comes from a loop cannot be oriented and is relaxed with a warning instead of being tied blindly.

For a bundle \(B\) whose members are all optimized variable-ratio transformers, all the effective ratios are tied to one shared variable \(\boldsymbol{\rho_B}\), expressed in the canonical direction:

\[c_{ij} \boldsymbol{\rho_{ij}} = \boldsymbol{\rho_B} \;\; (ij \text{ direct}), \qquad c_{ij} \boldsymbol{\rho_{ij}} \, \boldsymbol{\rho_B} = 1 \;\; (ij \text{ reversed}), \quad ij \in B \quad (7)\]

A reversed member transforms in the opposite declared direction, so equalizing raw effective ratios would enforce a physically wrong condition; with \((7)\), the product of the declared transformations around a loop of transformers is identically \(1\), whatever \(\boldsymbol{\rho_B}\). The shared variable is bounded by the intersection of the members’ effective ranges mapped to the canonical direction — a direct member contributes \([c_{ij} \rho_{ij}^{min}, c_{ij} \rho_{ij}^{max}]\), a reversed member the inverse interval \([1 / (c_{ij} \rho_{ij}^{max}), 1 / (c_{ij} \rho_{ij}^{min})]\) — i.e. \(\boldsymbol{\rho_B} \in [\rho_B^{min}, \rho_B^{max}]\) with \(\rho_B^{min}\) the largest of the mapped lower bounds and \(\rho_B^{max}\) the smallest of the mapped upper bounds.

Depending on this intersection, a bundle is handled in one of three ways:

non-empty interval (\(\rho_B^{min} < \rho_B^{max}\)): the ratios are tied through constraint \((7)\).
single point (\(\rho_B^{min} \approx \rho_B^{max}\)): there is essentially one feasible common effective ratio. Each member \(ij\) is fixed at \(\frac{\rho_B^{min} + \rho_B^{max}}{2}\) (its inverse for a reversed member), clamped to its own declared effective range \([c_{ij} \rho_{ij}^{min}, c_{ij} \rho_{ij}^{max}]\) — the same rule as the empty case below. Since both bounds nearly coincide, this value is essentially the shared ratio \(\rho_B^{min} \approx \rho_B^{max}\).
empty interval (\(\rho_B^{min} > \rho_B^{max}\)): the members’ effective ranges are disjoint, which signals inconsistent input data (e.g. disjoint tap ranges, or rated voltages making a common effective ratio unreachable). The optimizer falls back to a best-effort behavior, fixing each member as close as possible to the center of the gap \(\frac{\rho_B^{min} + \rho_B^{max}}{2}\) (its inverse for a reversed member), clamped to its own declared effective range \([c_{ij} \rho_{ij}^{min}, c_{ij} \rho_{ij}^{max}]\).

A bundle is tied through \((7)\) only if all its members are optimized variable-ratio transformers (specified in param_transformers.txt, see Configuration of the run). A member that is not optimized is frozen at its current tap, which pins the shared ratio to that value and collapses the bundle to the single-point or empty case above. As a consequence, declaring only a subset of a parallel group as variable does not optimize that subset freely: those transformers are fixed at the common ratio, again to avoid circulating flows.

This grouping interacts with the second optimization after tap rounding (see Solving): constraint \((7)\) is relaxed before rounding, so each transformer is rounded to the nearest tap of its own table independently. Members of a bundle may therefore end up on different discrete taps if their tap tables differ; the shared ratio is guaranteed only for the continuous solving.

Finally, if at solve time a member of a bundle carries no reactive loop flow at all — one of its sides is opened, or it lies outside the main connex component — the whole bundle is relaxed: its members are optimized independently for that run, and the event is logged. A member that is in service but that the model cannot move (near-zero impedance, single-ratio tap table) behaves instead like a non-optimized member above: its frozen effective ratio pins the bundle.

Objective function¶

The objective function is a weighted sum of penalty terms. Each of the seven weights is configurable by the user (see Configuration of the run), which allows arbitrating between the different terms of the objective (reactive slack activation, active power generation, voltage targeting, reactive power of units and transformer ratio).

The objective function of the ACOPF is:

\[\begin{split} \begin{aligned} \text{minimize} \quad & \sum\limits_{i} \left( w_{\sigma}^{+}\,\boldsymbol{\sigma_{i}^{Q,+}} + w_{\sigma}^{-}\,\boldsymbol{\sigma_{i}^{Q,-}} \right) \\ & + w_{P} \sum\limits_{g} \left( \alpha \boldsymbol{P_{i,g}} + (1-\alpha)\left(\frac{\boldsymbol{P_{i,g}} - P_{i,g}^t}{\max(1, |P_{i,g}^t|)}\right)^2 \right) \\ & + w_{V}^{\rho} \sum\limits_{i} \left( \boldsymbol{V_i} - (1-\rho)V_{i}^{\text{min,c}} + \rho V_{i}^{\text{max,c}} \right)^2 + w_{V}^{0} \sum\limits_{i} (\boldsymbol{V_i} - V_i^t)^2 \\ & + w_{Q} \sum\limits_{g} \left(\frac{\boldsymbol{Q_{i,g}}}{\max(1,Q_{g}^{\text{min,c}}, Q_{g}^{\text{max,c}})}\right)^2 + w_{Q} \sum\limits_{b} \left(\frac{\boldsymbol{Q_{i,b}}}{\max(1,Q_{b}^{\text{min,c}}, Q_{b}^{\text{max,c}})}\right)^2 \\ & + w_{\rho} \sum\limits_{ij} (\boldsymbol{\rho_{ij}} - \rho_{ij})^2 \end{aligned} \end{split}\]

where:

\(P_{i,g}^t\) (resp. \(V_i^t\)) is the active target (resp. voltage initial point) specified in ampl_network_generators.txt (resp. ampl_network_buses.txt).
\(\rho_{ij}\) is the transformer ratio of line \(ij\), specified in ampl_network_tct.txt.
the weights \(w_{\sigma}^{+}\), \(w_{\sigma}^{-}\), \(w_{P}\), \(w_{V}^{\rho}\), \(w_{V}^{0}\), \(w_{Q}\) and \(w_{\rho}\) correspond respectively to the parameters penalty_invest_rea_pos, penalty_invest_rea_neg, penalty_active_power, penalty_voltage_target_ratio, penalty_voltage_target_data, penalty_units_reactive and penalty_transfo_ratio.

Four of these weights have a fixed default value: \(w_{\sigma}^{+} = 10\), \(w_{\sigma}^{-} = 10\), \(w_{Q} = 0.1\) and \(w_{\rho} = 0.1\). The high default weight on the reactive slacks drives their sum towards \(0\), ensuring reactive power balance at each bus of the network.

The three remaining weights — \(w_{P}\), \(w_{V}^{\rho}\) and \(w_{V}^{0}\) — have a default value that depends on the objective_choice parameter when they are left unset: the term matching the selected objective receives a weight of \(1\), while the other two receive \(0.01\). Specifically, if objective_choice takes on:

\(0\) (MIN_GENERATION), the minimization of active power production \(\sum\limits_{i,g}\boldsymbol{P_{i,g}}\) is prioritized (\(w_{P} = 1\)).
\(1\) (BETWEEN_HIGH_AND_LOW_VOLTAGE_LIMIT), the minimization of \(\sum\limits_{i} \boldsymbol{V_i}-(\rho V_i^{c,max} - (1-\rho)V_i^{c,min})^2\) is prioritized (\(w_{V}^{\rho} = 1\)), where \(\rho\) equals the configurable parameter ratio_voltage_target.
\(2\) (SPECIFIC_VOLTAGE_PROFILE), the minimization of \(\sum\limits_{i} (\boldsymbol{V_i} - V_i^t)^2\) is prioritized (\(w_{V}^{0} = 1\)).

Setting an explicit value on any of these three weights overrides this objective-dependent default, regardless of the selected objective. As all weights accept any value \(\geq 0\), a term can be fully neutralized by setting its weight to \(0\).

Solving¶

Before solving the ACOPF, the voltage magnitudes \(\boldsymbol{V_i}\) are warm-started with \(V_i^t\) (specified in ampl_network_buses.txt), as well as the voltage phases \(\boldsymbol{\theta_i}\) with the results of the DCOPF (see DC optimal powerflow). Please also note that a scaling is applied with user-defined values before solving the ACOPF.

A solving is considered as successful if the non-linear solver employed (see Non-linear optimization solver) finds a feasible approximate solution (even if the sum of slacks is important).

At the user’s request (see Configuration of the run), and if at least one transformer is optimized, a second ACOPF optimization can be performed after rounding the transformer ratios (which, as a reminder, are continuous in the solving) to the nearest tap in the input data (see Network data). This allows the voltage plan to be readjusted to the new fixed transformation ratios in the second optimization. Without this optimization, note that power flows can vary significantly before and after rounding the taps, particularly for transformers with low impedance.

If the ACOPF resolution(s) are successfully completed, the script reactiveopfoutput.run is executed (see In case of convergence). Otherwise, the script reactiveopfexit.run is executed (see In case of inconsistency).