Linearised Power Flow¶

Kirchhoff's Current Law (KCL)¶

The Kirchhoff's Current Law (KCL) mandates that the sum of power flows into a bus must equal the sum of power flows out of the bus at each time step. It is covered by the more general Energy Balance constraints which is also applied to non-electric buses. For electric buses, KCL is the specific form of the energy balance constraint.

Kirchhoff's Voltage Law (KVL)¶

For lines and transformers, whose power flows according the impedances, the power flow \(p_{l,t}\) in AC networks is governed by the cycle-based linearised formulation of Kirchhoff's Voltage Law (KVL)

\[\sum_l C_{l,c} x_l p_{l,t} = 0 \quad \forall\, c,t\]

where \(C\) is a cycle basis matrix of the network graph, where the independent cycles \(c\) are expressed as directed linear combinations of lines \(l\), and \(x_l\) is the series reactance.

While there are different formulations of KVL, the cycle-based formulation was found to be substantially faster than other formulations due to its sparsity, as shown in Hörsch et al. (2018)¹. This formulation defines the same feasible space as other standard linearised formulations based on voltage angles that is commonly found in textbooks (B-Theta) or the formulation based on Power Transfer Distribution Factors (PTDFs).

These constraints are set in the function define_kirchhoff_voltage_constraints() and carry the name Kirchhoff-Voltage-Law.

Tip: KVL with DC networks

For DC networks, replace the series reactance \(x_l\) by the series resistance \(r_l\).

Note: Retrieving the PTDF matrix

The PTDF matrix of sub-networks (i.e. synchronous zones) can be calculated with sn.calculate_PTDF() method.

Tip: Using the Link component for Net Transfer Capacities (NTCs)

For simplified transmission representation using Net Transfer Capacities (NTCs), use the Link component with controllable power flow like a transport model. The Link component can also be used to represent a point-to-point HVDC link.

Note: Handling impedance changes with transmission expansion

If \(F_l\) is also subject to optimisation (s_nom_extendable=True), the impedance \(x\) of the line is not automatically changed with the capacity (e.g. to represent added parallel lines). However, the extension n.optimize.optimize_transmission_expansion_iteratively() covers this through an iterative process as done Hagspiel et al. (2014)² .

Linear Loss Approximation¶

The AC transmission losses \(\psi_{l,t}\) for line \(l\) at timestep \(t\) are given by the loss parabola

\[ \psi_{l,t} = r_{l} p_{l,t}^2, \]

where \(r_l\) is the resistance of line \(l\), following Neumann et al. (2022)³.

In PyPSA this non-linear function is linearized with piecewise linear constraints:

\[ 0 \leq \psi_{l,t} \leq r_{l} (\bar{p}_{l,t} \overline{P}_{l})^2 \quad \forall l,t \]

\[ \psi_{l,t} \geq m_k \cdot p_{l,t} + a_k \quad \forall l,t,\ k = 1, \dots, n \]

\[ \psi_{l,t} \geq -m_k \cdot p_{l,t} + a_k \quad \forall l,t,\ k = 1, \dots, n \]

where \(\bar{p}_{l,t}\) is the maximum per-unit powerflow at time \(t\) and \(\overline{P}_{l}^2\) is the (maximum) line capacity. The index \(k\) denotes the segments of the linear approximation with a total of \(n\) segments, with offsets \(a_k\) and with slopes \(m_k\).

The losses also modify the power balance by adding the following term to its left-hand side:

\[ -0.5 \cdot \sum_{l} |K_{n,l}| \cdot \psi_{l,t} \quad \forall n,t \]

splitting losses equally between both connection points.

The dispatch limits of \(p_{\ell,t}\) are now subtracted by \(\psi_{l,t}\).

Two linearization methods are available in PyPSA: the tangent-based approximation (define_tangent_loss_constraints()) and the secant-based approximation with error control (define_secant_loss_constraints()).

The transmission loss approximation is not activated by default, but must be enabled in n.optimize().

# Secant-based approximation with error control
n.optimize(transmission_losses=True)
# Or with custom tolerances
n.optimize(transmission_losses={"mode": "secants", "atol": 1, "rtol": 0.1})
# Tangent-based approximation
n.optimize(transmission_losses={"mode": "tangents", "segments": 3})

When passing a dict, the mode key selects the method. Additional keys are forwarded to the corresponding constraint function:

Secant mode ("secants", see define_secant_loss_constraints()): atol, rtol, max_segments.
Tangent mode ("tangents", see define_tangent_loss_constraints()): segments.

Hint: Calculating transmission losses

The losses can be calculated with n.lines_t.p0 + n.lines_t.p1.

Mapping of symbols to component attributes

Symbol	Attribute	Type
\(p_{l,t}\)	`n.lines_t.p0` or `n.transformers_t.p0`	Decision variable
\(\psi_{l,t}\)	`n.lines_t.losses` or `n.transformers_t.losses`	Decision variable
\(\bar{p}_{l,t}\)	`n.lines_t.s_max_pu` or `n.transformers_t.s_max_pu`	Parameter
\(\bar{P}_{l}\)	`n.lines.s_nom_opt` and `n.transformers.s_nom_opt` (if extendable) or `n.lines.s_nom` or `n.transformers.s_nom` (if non-extendable)	Decision variable / Parameter
\(x_l\)	`n.lines.x_pu_eff` or `n.transformers.x_pu_eff`	Parameter
\(r_l\)	`n.lines.r_pu_eff` or `n.transformers.r_pu_eff`	Parameter
\(C_{l,c}\)	Cycle matrix calculated by `find_cycles()`	Parameter
\(K_{n,l}\)	Incidence matrix calculated by `n.incidence_matrix()`	Parameter

Secant-Based Linearization with Error Control¶

The secant-based linearization is designed to overestimate the AC line losses and is guaranteed to stay within the error bounds defined by the specified absolute (atol) and relative (rtol) error tolerances. The slope \(m_k\) and offset \(a_k\) are derived as

\[ m_{k, l} = r_l (\rho_{k-1} + \rho_{k}) \]

\[ a_{k, l} = - r_l \rho_{k-1} \rho_{k} \]

where \(\rho_k\) are suitably chosen points on the loss parabola. For details on how these points are derived see the original PR.

Tangent-Based Linearization¶

Warning: Depending on parameters, some lines can be lossless when the tangent-based linearization is used

The tangent-based approximation is designed to underestimate losses
Its segments depend on s_nom_max
The first segment models zero losses
If a line has s_nom_max > 2 * n_segments * s_nom_opt then that line is lossless, since all flows stay in the first segment of the approximation.

For the tangent-based linearization the slope \(m_k\) and offset \(a_k\) are derived as:

\[ \psi_{l,t}(k) = r_{l} \left(\frac{k}{n} \cdot \bar{p}_{l,t} \overline{P}_{l} \right)^2 \]

\[ m_k = \frac{d \psi_{l,t}(k)}{dk} = 2 r_{l} \left(\frac{k}{n} \cdot \bar{p}_{l,t} \overline{P}_{l} \right) \]

\[ a_k = \psi_{l,t}(k) - m_k \left(\frac{k}{n} \cdot \bar{p}_{l,t} \overline{P}_{l} \right) \]

The higher the number of tangents, the more accurate the approximation, but also the more constraints are added to the optimisation problem. Typically, 2-4 tangents are sufficient for a reasonably accurate approximation.

More details on this implementation can be found in Neumann et al. (2022)³.

Examples¶

Meshed AC-DC Networks

Builds a stylized 3-node AC network coupled via AC-DC converters to a 3-node DC network.

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Negative LMPs from Line Congestion

Explores how Kirchhoff's Voltage Law can lead to negative locational marginal prices when lines are congested.

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J. Hörsch, H. Ronellenfitsch, D. Witthaut, T. Brown (2018), Linear optimal power flow using cycle flows, Electric Power Systems Research, 158, 126-135, doi:10.1016/j.epsr.2017.12.034. ↩
S. Hagspiel, C. Jägemann, D. Lindenberger, T. Brown, S. Cherevatskiy, E. Tröster (2014), Cost-optimal power system extension under flow-based market coupling, Energy, 66, 654-666, doi:10.1016/j.energy.2014.01.025. ↩
F. Neumann, V. Hagenmeyer, T. Brown (2022), Assessments of linear power flow and transmission loss approximations in coordinated capacity expansion problems, Applied Energy, 314, 118859, doi:10.1016/j.apenergy.2022.118859. ↩↩