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)³ .

Loss Approximation

The AC transmission losses \(\psi_{l,t}\) are approximated using a tangent-based linearization of the loss parabola:

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

where \(r_l\) is the resistance, following Neumann et al. (2022)².

The approximation uses piecewise linear constraints:

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

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

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

For each segment \(k\) of the total \(n\) segments, 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 losses also modify the power balance by adding the 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 \(\psi_{l,t}\).

These constraints are set in the function define_loss_constraints().

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

n.optimize(transmission_losses=3)

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.

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

Examples

• Meshed AC-DC Networks

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  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

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

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1. 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. ↩

2. 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. ↩

3. 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. ↩