# Approximating Power Flow and Transmission Losses
in Coordinated Capacity Expansion Problems

Fabian Neumann<EMAIL_ADDRESS>Veit Hagenmeyer  Tom Brown  Institute
for Automation and Applied Informatics (IAI), Karlsruhe Institute of
Technology (KIT), Hermann-von-Helmholtz-Platz 1, 76344, Eggenstein-
Leopoldshafen, Germany

###### Abstract

With rising shares of renewables and the need to properly assess trade-offs
between transmission, storage and sectoral integration as balancing options,
building a bridge between energy system models and detailed power flow studies
becomes increasingly important, but is computationally challenging.

In this paper, we compare both common and improved approximations for two
nonlinear phenomena, power flow and transmission losses, in linear capacity
expansion problems that co-optimise investments in generation, storage and
transmission infrastructure. We evaluate different flow representations
discussing differences in investment decisions, nodal prices, the deviation of
optimised flows and losses from simulated AC power flows, and the
computational performance. By using the open European power system model
PyPSA-Eur, that combines high spatial and temporal resolution, we obtain
detailed and reproducible results aiming at facilitating the selection of a
suitable power flow model.

Given the differences in complexity, the optimal choice depends on the
application, the user’s available computational resources, and the level of
spatial detail considered. Although the commonly used transport model can
already identify key features of a cost-efficient system while being
computationally performant, deficiencies under high loading conditions arise
due to the lack of a physical grid representation. Moreover, disregarding
transmission losses overestimates optimal grid expansion by 20%. Adding a
convex relaxation of quadratic losses with two or three tangents to the
linearised power flow equations and accounting for changing line impedances as
the network is reinforced suffices to represent power flows and losses
adequately in design studies. We show that the obtained investment and
dispatch decisions are then sufficiently physical to be used in more detailed
nonlinear simulations of AC power flow in order to better assess their
technical feasibility. This includes determining reactive power flows and
voltages, which the initial linear model neglects. Simpler approximations are
less suitable for such ex-post analysis.

###### keywords:

energy system modelling, linear optimal power flow, transmission losses,
capacity expansion planning, grid reinforcement

††journal: RSER

###### Contents

  1. 1 Introduction
  2. 2 Power System Planning Problem
  3. 3 Nonlinear Power Flow and Losses
    1. 3.1 Nonlinear AC Power Flow
    2. 3.2 Nonlinear Active Power Losses
    3. 3.3 Electrical Parameters of Transmission Lines
  4. 4 Linear Power Flow Models
    1. 4.1 Transport Model
    2. 4.2 Transport Model with Loss Approximation
    3. 4.3 Linearised Power Flow
    4. 4.4 Linearised Power Flow with Loss Approximation
    5. 4.5 Iterative Linearised Power Flow (with Loss Approximation)
  5. 5 Simulation Setup
    1. 5.1 Power System Model for Optimisation
    2. 5.2 Nonlinear AC Power Flow Simulation
  6. 6 Results and Discussion
    1. 6.1 Investments, Nodal Prices and Total System Costs
    2. 6.2 Validation of Loss Approximation
    3. 6.3 Validation of Optimised Line Flows
    4. 6.4 Computational Performance
    5. 6.5 Critical Appraisal
  7. 7 Conclusion
  8. Data Availability
  9. A AC Power Flow Problem Solved with Newton-Raphson
    1. A.1 Without Distributed Slack
    2. A.2 With Distributed Slack
  10. B Relations between Electrical Line Parameters
  11. C Additional Figures and Tables

##  1 Introduction

Energy system models seek to answer what infrastructure a future energy system
requires for given policy goals, where and when it should be built, and how
much it costs. For systems with high shares of renewable energy, the question
to what extent the variability of weather-dependent wind and solar energy will
be balanced in space with continent-spanning transmission networks and in time
with storage and coupling to other energy sectors attracts much research.
Because energy system models are frequently used in policy-making, it becomes
crucial to understand their particular limitations.

To find credible answers for highly renewable systems, it has been
demonstrated that models require coordinated expansion planning of generation,
storage and transmission infrastructure because they strongly interact [1, 2];
high temporal resolution and scope to account for extreme weather events,
storage operation, and investments shaped by the characteristic daily,
synoptic and seasonal patterns of renewables and load [3, 4]; high spatial
resolution and scope to also capture the spatio-temporal patterns, such as
correlations of wind speeds across the continent, and to represent
transmission constraints [5, 6]. As higher shares of renewables increase the
frequency of transmission bottlenecks, more detailed grid modelling is needed
that looks beyond import and export capacities but accounts for physical
conditions such as loop flows, transmission losses, and curtailment due to
otherwise overloaded lines [7, 8].

Especially for planning problems with both static investment and time-
dependent dispatch variables spanning across thousands of operational
conditions, a tractable yet sufficiently trustworthy representation of power
flows is essential. Ideally, outputs are detailed enough to be used as inputs
for more accurate analyses, bridging the granularity gap between coarsely-
resolved planning models and more detailed engineering models. Yet, even in
strategic problems speed matters for performing pivotal sensitivity analyses;
e.g. regarding uncertain cost parameters, reference weather years, technology
choices, and resource boundaries. Unfortunately, the first-choice AC power
flow equations are nonlinear and nonconvex, which makes the embedded AC
optimal power flow problem NP-hard [9, 10, 11].

Even for a linear representation of power flows, considerations of
transmission expansion planning result in a bilinear problem because line
impedances change as line capacities are increased. While we can deal with
this challenge through iterative impedance updates in sequential linear
programming [12], the problem would become even more complex if a discrete set
of transmission expansion plans were considered, rather than continuous line
expansion. More generally, we can approach computational challenges from
multiple angles: by improving solving algorithms or by figuring out what model
details can be simplified while retaining accuracy [13]. Examples include the
level of spatial aggregation, temporal aggregation, technology detail and
diversity, or finally the approximation of power flow.

The transport model, that takes account only of power transfer capacities
while ignoring impedances, and the linearised power flow model, which includes
impedances to consider both Kirchhoff laws but no losses, are commonly used in
energy system models. Among the models reviewed by Ringkjøb et al. [14] around
four in five models use a transport model if flows are represented, whereas
only one in five uses a linearised power flow model. Previous work has
compared these two major variants [15, 16, 17, 18], and some performed
simulations of AC power flow after optimisation [18, 19]. The comparisons
indicate little discrepancy regarding total system cost and cross-border
transmission, but also differences in nodal prices and overlooked line
overloadings when checked against AC power flow calculations. However, the
cogency of existing comparisons is limited by the use of low spatial
resolution models with fewer than 25 nodes. Furthermore, the consideration of
losses is underrepresented in design studies, but alongside characteristic
weather patterns shapes the tradeoffs regarding the volume of transmitted
energy because losses increase as more power is transported [20, 21].

In the present contribution, we offer a comprehensive comparison of linear
representations of power flow and losses in theory and practice. We outline
their characteristic benefits and shortcomings in the context of coordinated
capacity expansion problems, where generation, transmission and storage
infrastructure is jointly planned. Given the multitude of modelling
uncertainties, we assess under which circumstances it is worth embedding more
elaborate flow models than a simple transport model. We further extend beyond
previous research by introducing a computationally inexpensive loss
approximation that incorporates an efficient reformulation of the linearised
power flow equations based on a cycle decomposition of the network graph. By
using an open model of the European power system, PyPSA-Eur [22], spanning the
whole continent with hundreds of nodes and hourly temporal coupling due to the
consideration of storage units, we achieve advanced and reproducible
comparisons in systems with high shares of renewables.

The content complements the best practices of energy system modelling,
characterised in Pfenniger et al. [23] and DeCarolis et al. [24], regarding
the choice of suitable power flow models. While we take an investment planning
perspective in this paper, we underline that the way that the transmission of
power is represented is relevant beyond system planning. For instance, it
plays a role in the design of future electricity markets with multiple bidding
zones and flow-based market coupling [25, 26, 17].

We structured this contribution as follows. We begin with an introduction to
the basic long-term power system planning problem in Section 2 and briefly
review the physics of power flow in Section 3. We continue with the different
linear power flow representations in Section 4. Section 5 presents the
experimental setup, the results of which are discussed and critically
appraised in Section 6. Section 7 concludes this paper with a summary and
recommendations.

##  2 Power System Planning Problem

This section presents the full long-term power system planning problem. We
confine the formulation to the power system, but it can also serve to
represent the power system embedded within the full energy system. The
representation of power flows is one decisive constituent component and its
variants are later introduced in the context of this problem in Section 4.

The objective is to minimize the total annual system costs, comprising
annualised111The annuity factor $\frac{1-(1+\tau)^{-n}}{\tau}$converts the
overnight investment of an asset to annual payments considering its lifetime
$n$ and cost of capital $\tau$. capital costs $c_{*}$ for investments at
locations $i$ in generator capacity $G_{i,r}$ of technology $r$, storage power
capacity $H_{i,s}$ of technology $s$, and transmission line capacities
$P_{\ell}$, as well as the variable operating costs $o_{*}$ for generator
dispatch $g_{i,r,t}$:

$\min_{G,H,F,g}\quad\left[\sum_{i,r}c_{i,r}\cdot
G_{i,r}+\sum_{i,s}c_{i,s}\cdot H_{i,s}+\sum_{\ell}c_{\ell}\cdot
P_{\ell}+\sum_{i,r,t}w_{t}\cdot o_{i,r}\cdot g_{i,r,t}\right]$ (1)

where representative time snapshots $t$ are weighted by the time span $w_{t}$
such that their total duration adds up to one year;
$\sum_{t\in\mathcal{T}}w_{t}=365\cdot 24\text{h}=8760\text{h}$. The objective
function is subject to a set of linear constraints, including multi-period
linear optimal power flow (LOPF) equations, resulting in a linear programme
(LP).

The capacities of generation, storage and transmission infrastructure are
constrained above by their installable potentials and below by any existing
components:

$\displaystyle\underline{G}_{i,r}$ $\displaystyle\leq$ $\displaystyle G_{i,r}$
$\displaystyle\leq$ $\displaystyle\overline{G}_{i,r}$
$\displaystyle\qquad\forall i,r$ (2) $\displaystyle\underline{H}_{i,s}$
$\displaystyle\leq$ $\displaystyle H_{i,s}$ $\displaystyle\leq$
$\displaystyle\overline{H}_{i,s}$ $\displaystyle\qquad\forall i,s$ (3)
$\displaystyle\underline{P}_{\ell}$ $\displaystyle\leq$ $\displaystyle
P_{\ell}$ $\displaystyle\leq$ $\displaystyle\overline{P}_{\ell}$
$\displaystyle\qquad\forall\ell$ (4)

The dispatch of a generator may not only be constrained by its rated capacity
but also by the availability of variable renewable energy, which may be
derived from reanalysis weather data. This can be expressed as a time- and
location-dependent availability factor $\overline{g}_{i,r,t}$, given per unit
of the generator’s capacity:

$\displaystyle 0$ $\displaystyle\leq$ $\displaystyle g_{i,r,t}$
$\displaystyle\leq$ $\displaystyle\overline{g}_{i,r,t}G_{i,r}$
$\displaystyle\qquad\forall i,r,t$ (5)

The dispatch of storage units is split into two positive variables; one each
for charging $h_{i,s,t}^{+}$ and discharging $h_{i,s,t}^{-}$. Both are limited
by the power rating $H_{i,s}$ of the storage units.

$\displaystyle 0$ $\displaystyle\leq$ $\displaystyle h_{i,s,t}^{+}$
$\displaystyle\leq$ $\displaystyle H_{i,s}$ $\displaystyle\qquad\forall i,s,t$
(6) $\displaystyle 0$ $\displaystyle\leq$ $\displaystyle h_{i,s,t}^{-}$
$\displaystyle\leq$ $\displaystyle H_{i,s}$ $\displaystyle\qquad\forall i,s,t$
(7)

The energy levels $e_{i,s,t}$ of all storage units have to be consistent with
the dispatch in all hours.

$\displaystyle e_{i,s,t}=\>$ $\displaystyle\eta_{i,s,0}^{w_{t}}\cdot
e_{i,s,t-1}+w_{t}\cdot h_{i,s,t}^{\text{inflow}}-w_{t}\cdot
h_{i,s,t}^{\text{spillage}}$ $\displaystyle\quad\forall i,s,t$
$\displaystyle+\eta_{i,s,+}\cdot w_{t}\cdot
h_{i,s,t}^{+}-\eta_{i,s,-}^{-1}\cdot w_{t}\cdot h_{i,s,t}^{-}$ (8)

Storage units can have a standing loss $\eta_{i,s,0}$, a charging efficiency
$\eta_{i,s,+}$, a discharging efficiency $\eta_{i,s,-}$, natural inflow
$h_{i,s,t}^{\text{inflow}}$ and spillage $h_{i,s,t}^{\text{spillage}}$. The
storage energy levels are assumed to be cyclic

$\displaystyle e_{i,s,0}=e_{i,s,\left|\mathcal{T}\right|}\qquad\forall i,s$
(9)

and are constrained by their energy capacity

$\displaystyle 0$ $\displaystyle\leq$ $\displaystyle e_{i,s,t}$
$\displaystyle\leq$ $\displaystyle\overline{T}_{s}\cdot H_{i,s}$
$\displaystyle\qquad\forall i,s,t.$ (10)

To reduce the number of decisison variables, we link the energy storage volume
to power ratings using a technology-specific parameter $\overline{T}_{s}$ that
describes the maximum duration a storage unit can discharge at full power
rating.

Total CO2 emissions may not exceed a target level $\Gamma_{\text{CO}_{2}}$.
The emissions are determined from the time-weighted generator dispatch
$w_{t}\cdot g_{i,r,t}$ using the specific emissions $\rho_{r}$ of fuel $r$ and
the generator efficiencies $\eta_{i,r}$

$\displaystyle\sum_{i,r,t}\rho_{r}\cdot\eta_{i,r}^{-1}\cdot w_{t}\cdot
g_{i,r,t}\leq\Gamma_{\text{CO}_{2}}.$ (11)

All power flows $p_{\ell,t}$ are also limited by their capacities $P_{\ell}$

$\displaystyle|p_{\ell,t}|\leq\overline{p}_{\ell}P_{\ell}$
$\displaystyle\qquad\forall\ell,t,$ (12)

where $\overline{p}_{\ell}$ acts as a per-unit security margin on the line
capacity to allow a buffer for the failure of single circuits ($N-1$
condition) and reactive power flows.

Ultimately, we need constraints that define the power flows $p_{\ell,t}$ in
the network. In the next Section 3, we briefly set foundations for nonlinear
power flow and losses. The various alternative flow models are then presented
in Section 4. The subsequent descriptions will omit the time index $t$ for
notational simplicity.

##  3 Nonlinear Power Flow and Losses

This section briefly revises the nonlinear AC power flow equations, important
electrical parameters of transmission lines, and how to calculate active power
losses on a line. We do this to set the foundations for derivations of the
covered flow models.

###  3.1 Nonlinear AC Power Flow

The active power flow $p_{\ell(i,j)}$ of a line $\ell\equiv\ell(i,j)$ from bus
$i$ to bus $j$ can be described in voltage-polar coordinates by

$p_{\ell(i,j)}=g_{\ell}\left|V_{i}\right|^{2}+\left|V_{i}\right|\left|V_{j}\right|\left[g_{\ell}\cos(\theta_{i}-\theta_{j})-b_{\ell}\sin(\theta_{i}-\theta_{j})\right]$
(13)

and, analogously, the reactive power flow $q_{\ell(i,j)}$ is given by

$q_{\ell(i,j)}=b_{\ell}\left|V_{i}\right|^{2}+\left|V_{i}\right|\left|V_{j}\right|\left[g_{\ell}\sin(\theta_{i}-\theta_{j})-b_{\ell}\cos(\theta_{i}-\theta_{j})\right],$
(14)

where $\left|V_{i}\right|$ is the per-unit bus voltage magnitude, $\theta_{i}$
is the bus voltage angle, $g_{\ell}$ is the line conductance, and $b_{\ell}$
is the line susceptance [27].

###  3.2 Nonlinear Active Power Losses

To derive an expression for the active power losses in a transmission line, we
apply the convention that departing power flows are positive and arriving
power flows are negative. Consequently, if power flows from bus $i$ to $j$,
$p_{\ell(i,j)}>0$ and $p_{\ell(j,i)}<0$. The losses $\psi_{\ell}$ are the
difference between power sent and power received [27], therefore

$\psi_{\ell}=p_{\ell(i,j)}+p_{\ell(j,i)}.$ (15)

Substituting equation (13) into equation (15) yields

$\displaystyle\psi_{\ell}$ $\displaystyle=$ $\displaystyle
g_{\ell}\left|V_{i}\right|^{2}+\left|V_{i}\right|\left|V_{j}\right|\left[g_{\ell}\cos(\theta_{i}-\theta_{j})-b_{\ell}\sin(\theta_{i}-\theta_{j})\right]$
(16) $\displaystyle+\quad$ $\displaystyle
g_{\ell}\left|V_{j}\right|^{2}+\left|V_{j}\right|\left|V_{i}\right|\left[g_{\ell}\cos(\theta_{j}-\theta_{i})-b_{\ell}\sin(\theta_{j}-\theta_{i})\right]$
(17)

and using the trigonometric identities $\cos(-\alpha)=\cos(\alpha)$ and
$\sin(-\alpha)=-\sin(\alpha)$ translates to

$\displaystyle\psi_{\ell}$ $\displaystyle=$ $\displaystyle
g_{\ell}\left|V_{i}\right|^{2}+\left|V_{i}\right|\left|V_{j}\right|\left[g_{\ell}\cos(\theta_{i}-\theta_{j})-b_{\ell}\sin(\theta_{i}-\theta_{j})\right]$
(18) $\displaystyle+\quad$ $\displaystyle
g_{\ell}\left|V_{j}\right|^{2}+\left|V_{j}\right|\left|V_{i}\right|\left[g_{\ell}\cos(\theta_{i}-\theta_{j})+b_{\ell}\sin(\theta_{i}-\theta_{j})\right].$
(19)

We can further simplify this expression to the loss formula

$\psi_{\ell}=g_{\ell}\left(\left|V_{i}\right|^{2}+\left|V_{j}\right|^{2}\right)-2\left|V_{i}\right|\left|V_{j}\right|g_{\ell}\cos(\theta_{i}-\theta_{j}).$
(20)

We will use this formula in Section 4.4 to derive a linear approximation for
losses.

###  3.3 Electrical Parameters of Transmission Lines

The complex per-unit impedance $z_{\ell}=r_{\ell}+ix_{\ell}$ is composed of
resistance $r_{\ell}$ and reactance $x_{\ell}$. Likewise, the admittance
$y_{\ell}=g_{\ell}+ib_{\ell}$ is composed of conductance $g_{\ell}$ and
susceptance $b_{\ell}$. Impedance and admittance are reciprocals
($y_{\ell}=z_{\ell}^{-1}$). Hence, if we assume a dominance of reactance over
ohmic resistance ($r_{\ell}\ll x_{\ell}$), as applies for high voltage
overhead transmission lines, we obtain the approximations

$\displaystyle g_{\ell}$ $\displaystyle\approx\frac{r_{\ell}}{x_{\ell}^{2}},$
(21) $\displaystyle b_{\ell}$ $\displaystyle\approx\frac{1}{x_{\ell}}.$ (22)

For a derivation, see B. We will use these relations in Section 4.3 and
Section 4.4. In view of the approximation of line losses in later sections,
note that although we assume that resistance is dominated by reactance, we do
not assume resistance to be zero (cf. [28, 29]).

##  4 Linear Power Flow Models

The AC power flow equations (13) are nonlinear and nonconvex. This is
challenging because multiple local minima exist due to the trigonometric
expressions and when directly incorporated in the optimisation problem they
would make the problem NP-hard [9, 10, 11]. To be able to run large
optimisation problems of the continental power system at sufficient spatial
and temporal resolution it is hence inevitable to retain a convex problem that
can be solved in polynomial time and does not possess local minima.

Figure 1: Illustration of the scope of the present paper and its context. It shows the connections between the covered linear power flow models, their main features, and how they are related to other (convexified) nonlinear formulations. Model | Section | Variables | Eq. Constraints | Ineq. Constraints
---|---|---|---|---
Transport | 4.1 | $\left|\mathcal{L}\right|$ | $\left|\mathcal{N}\right|-1$ | 0
Lossy Transport | 4.2 | $2\left|\mathcal{L}\right|$ | $\left|\mathcal{N}\right|-1$ | 0
Linearised Power Flow | 4.3 | $\left|\mathcal{L}\right|$ | $\left|\mathcal{L}\right|$ | 0
Linearised Power Flow | 4.4 | $2\left|\mathcal{L}\right|$ | $\left|\mathcal{L}\right|$ | $2n\left|\mathcal{L}\right|$
with Loss Approximation |  |  |  |
Table 1: Comparison of the number of variables and equality/inequality
constraints related to flow models per snapshot $t\in\mathcal{T}$. The
constraint count excludes variable bounds. $\left|\mathcal{L}\right|$ is the
number of lines, $\left|\mathcal{N}\right|$ is the number of nodes, and $2n$
represents the number of tangents used for the loss approximation.

In this section we describe and develop various linear representations of
power flow. These are introduced in the order from least to most accurate,
progressively increasing the complexity; namely (i) the common transport
model, (ii) a lossy transport model, (iii) the lossless linearised power flow,
and (iv) a lossy linearised power flow model. Figure 1 shows the relations
between the formulations and Table 1 documents differences in the number of
variables and constraints. The scope of this work is deliberately constrained
to:

  * 1.

only linear problems: To avail of powerful, scalable and fast interior-point
solvers, and to guarantee an optimal solution, we only include formulations
that entail linear problems. However, there exist promising second-order cone
or semidefinite convex relaxations of the power flow equations. For excellent
theoretical reviews of convex relaxations and approximations of power flow see
Molzahn et al. [30], Taylor [27], and Coffrin et al. [31].

  * 2.

only active power: We furthermore confine our analysis to formulations that do
not capture reactive power flows or information on bus voltages. Nonetheless,
linear problems that capture selected aspects of this are under active
research; see e.g. Coffrin et al. [32].

  * 3.

only comparison of different feasible spaces: We compare different linear flow
models that define different feasible spaces. We do not compare equivalent
reformulations of identical models, since this has been analysed in Hörsch et
al. [33].

  * 4.

no copper plate model: Although occasionally encountered in generation and
storage capacity expansion models, we do not include the copper plate
relaxation in our comparisons because it does not capture information on power
flows in transmission networks. The copper plate model removes all lines and
aggregates all components to a single node. It is a relaxation because any
transmission of power becomes unconstrained and incurs no cost. For the impact
of spatial clustering – of which the copper plate model is the extreme – on
optimal investments we refer to Hörsch et al. [5].

###  4.1 Transport Model

The transport model is also known as a network flow model, trade model,
transshipment model or net transfer capacity (NTC) model [27]. It ignores the
effect of impedances on flows (including losses) and, besides the capacity
constraints of lines, only requires nodal power balance according to
Kirchhoff’s Current Law (KCL); i.e. the power injected at each bus must equal
the power withdrawn by attached lines

$p_{i}=\sum_{\ell}K_{i\ell}p_{\ell}\qquad\forall i\in\mathcal{N},$ (23)

where $p_{i}$ is the active power injected or consumed at node $i$ and $K$ is
the incidence matrix of the network graph which has non-zero values $+1$ if
line $\ell$ starts at bus $i$ and $-1$ if line $\ell$ ends at bus $i$.

Because the columns of the incidence matrix each sum up to zero, KCL yields
$\left|\mathcal{N}\right|-1$ linearly independent constraints. These are not
sufficient to uniquely determine the $\left|\mathcal{L}\right|$ unknown flows.
The transport model allows arbitrary flows as long as flow conservation is
fulfilled, also because it is free and lossless to transmit power. This makes
the transport model degenerate, which can be detrimental to the solving speed.
Also, of course, this does not adequately reflect the physical behaviour of
power flows in the transmission network.

Despite its drawbacks, the transport model is very popular. In the
comprehensive review by Ringkjøb et al. [14], it is applied in a majority of
models. This minimalistic representation of flows is useful to develop an
understanding for the potential of increased transfer capacity between
regions, rather than assessing specific transmission bottlenecks and
reinforcement needs. It is often applied in investment models where the grid
is highly aggregated to a few nodes (e.g. one node per country in Europe or
federal state in the United States) or analyses of energy markets across
multiple bidding zones. Its main advantages are ease of implementation and
fast solving speed. For pure dispatch problems without investment decisions
one can even utilise specialised network flow algorithms; for instance the
minimum cost flow algorithm [34].

###  4.2 Transport Model with Loss Approximation

Part of the drawbacks and degeneracy of the transport model stems from the
disregard of transmission losses. As partial remedy, we can amend the
transport model with a simple loss approximation which assumes lines to have a
constant transmission efficiency $\eta_{\ell}$ depending on their length. In
this case, the power arriving at the receiving bus is lower than the power
injected at the sending bus. To differentiate between sending bus and
receiving bus, we need to split the bidirectional power flow variable
$p_{\ell}$ into forward flows $p_{\ell}^{+}$ and backward flows $p_{\ell}^{-}$
with capacity limits

$\displaystyle 0\leq
p_{\ell}^{+}\leq\overline{p}_{\ell}P_{\ell}\qquad\forall\ell\in\mathcal{L}$
(24) $\displaystyle 0\leq
p_{\ell}^{-}\leq\overline{p}_{\ell}P_{\ell}\qquad\forall\ell\in\mathcal{L}$
(25)

which substitute the variables $p_{\ell}$ and their bounds given in equation
(12). Furthermore, we need to adjust the nodal balance constraints (23) to
reflect the transmission losses and separated power flow variables to

$p_{i}=\sum_{\ell}K_{i\ell}^{+}p_{\ell}^{+}-\sum_{\ell}K_{i\ell}^{-}p_{\ell}^{-}\qquad\forall
i\in\mathcal{N},$ (26)

where $K^{+}$ is the lossy incidence matrix of the network graph regarding
forward flows $p_{\ell}^{+}$ which has non-zero values $+1$ if line $\ell$
starts at bus $i$ and $-\eta_{\ell}$ if line $\ell$ ends at bus $i$.
Analogously, $K^{-}$ regards backward flows $p_{\ell}^{-}$ with non-zero
values $\eta_{\ell}$ if line $\ell$ starts at bus $i$ and $-1$ if line $\ell$
ends at bus $i$.

The transmission losses alleviate some degeneracy of the transport model since
considering losses yields an incentive to minimise power flows rather than to
distribute them arbitrarily. However, this is paid for with a doubling in the
number of flow variables. Additionally, while the use of a constant
transmission efficiency is an improvement from the plain transport model, it
still ignores the quadratic relationship between power flow and losses [20].
Note, that if all lines have no losses ($\eta_{\ell}=1$), the lossy transport
model is equivalent to the regular transport model.

###  4.3 Linearised Power Flow

The linearised power flow model, which is also known as linearised load flow,
DC power flow or B$\Theta$ model, extends the lossless transport model. In
addition to the nodal power balance constraints (23) from KCL and capacity
limits (12), linear constraints for Kirchhoff’s Voltage Law (KVL) are
included, which define how power flows split in parallel paths. We derive
these by simplifying the nonlinear power flow equations (13) and (14).
Assuming

  * 1.

all per-unit voltage magnitudes are close to one ($\left|V_{i}\right|\approx
1$),

  * 2.

conductances $g_{\ell}$ are negligible relative to susceptances $b_{\ell}$
($b_{\ell}\gg g_{\ell}$),

  * 3.

voltage angle differences are small enough
($\sin(\theta_{i}-\theta_{j})\approx\theta_{i}-\theta_{j}$ and
$\cos(\theta_{i}-\theta_{j})\approx 0$),

  * 4.

reactive power flows $q_{\ell}$ are negligible compared to real power flows
$p_{\ell}$ ($q_{\ell}\approx 0$),

leads to

$p_{\ell}=b_{\ell}(\theta_{i}-\theta_{j}),$ (27)

and when we further assume $r_{\ell}\ll x_{\ell}$, by substituting (22) we
obtain

$p_{\ell}=\frac{\theta_{i}-\theta_{j}}{x_{\ell}}.$ (28)

This angle-based formulation is the most common linear formulation of KVL
[27]. But it is possible to avoid the auxiliary voltage angle variables and
reduce the required number of constraints by using a cycle basis of the
network graph [33]. Namely, KVL states that the sum of voltage angle
differences across lines around all cycles in the network must sum up to zero.
Considering a set of independent cycles $c$ of the network forming a cycle
basis, which are expressed as a directed linear combination of the lines
$\ell$ in a cycle incidence matrix

$C_{\ell c}=\begin{cases}1&\text{if edge $\ell$ is element of cycle $c$,}\\\
-1&\text{if reversed edge $\ell$ is element of cycle $c$,}\\\
0&\text{otherwise,}\end{cases}$ (29)

KVL is formulated by

$\sum_{\ell}C_{\ell c}(\theta_{i}-\theta_{j})=0\qquad\forall c\in\mathcal{C}.$
(30)

Using equation (28), we can express KVL directly in terms of the power flows
and circumvent the auxiliary voltage angle variables

$\sum_{\ell}C_{\ell c}p_{\ell}x_{\ell}=0\qquad\forall c\in\mathcal{C}.$ (31)

Although less common, this cycle-based formulation (31) has been shown to
significantly outperform the angle-based formulation (28) [33, 35]. There are
even further equivalent reformulations of the linearised power flow [33]; for
example the Power Transfer Distribution Factor (PTDF) formulation, which
directly relates nodal power injections to line flows. Because our focus lies
on the comparison of different flow models, not their diverse reformulations,
we only evaluate the computationally performant cycle-based formulation in the
present contribution.

With the cycle-based formulation one can clearly see that the transport model
is equivalent to the linearised power flow in radial networks; i.e. when the
network has no cycles. Also, the absence of auxiliary voltage angle variables
facilitates the insight that the transport model is a relaxation of the
linearised power flow because the latter only adds constraints in the same
variable space.

The linearised power flow model is claimed to be accurate when reactance
dominates ($x_{\ell}\gg r_{\ell}$) and when parallel lines have similar ratios
[36], but very long lines in highly aggregated networks can deteriorate the
quality of the approximation (see Section 6.3). An advantage of this model
over the transport model is that it captures some meaningful physical
characteristics observed in the operation of electrical grids. Namely, it is
capable of revealing loop flows in meshed networks; for instance recurring
spillover effects between Germany and the Czech Republic. Nevertheless, it
still disregards losses.

If we would consider that lines can be built between buses where there are
currently none, another variant is the so-called hybrid model. This version
formulates linearised power flow constraints for existing lines and employs a
transport model for candidate lines.

###  4.4 Linearised Power Flow with Loss Approximation

Neglecting resistive losses is considered to be among the largest sources of
error in the linearised power flow formulation, particularly in large networks
[36]. The following extension of the lossless linearised power flow (Section
4.3) is a mixture of similar variants encountered in the literature with a
focus on computational efficiency. We reference where we follow or deviate
from previous work below. This or similar formulations have rarely been
applied in the co-optimisation of transmission, storage and generation
capacities, but rather in detailed operational optimal power flow (OPF) or
transmission expansion planning (TEP) problems; see overview in [7].

We start by adding a loss variable $\psi_{\ell}$ for each line. Losses reduce
the effective transmission capacity of a line

$\left|p_{\ell}\right|\leq\overline{p}_{\ell}P_{\ell}-\psi_{\ell}$ (32)

and must be accounted for in the nodal balance equation (23)

$p_{i}=\sum_{\ell}K_{i\ell}p_{\ell}+\frac{\left|K_{i\ell}\right|}{2}\psi_{\ell}\qquad\forall
i\in\mathcal{N}.$ (33)

We split the losses $\psi_{\ell}$ equally between both buses (like in [37, 38,
39]) and do not allocate them at the sending bus exclusively (like in [29,
40]). The latter could be modelled with an absolute value function in the
linear problem. However, this would involve splitting flow and loss variables
each into positive and negative segments. Because this adds many auxiliary
decision variables, we decided in favor of distributing the losses evenly.
This choice is paid for with the possibility of overestimating losses due to
an extensive convex relaxation.

Assuming close to nominal per-unit voltage magnitudes
$\left|V_{i}\right|\approx 1$ the loss formula given in equation (20) becomes

$\psi_{\ell}=2g_{\ell}\left[1-\cos(\theta_{i}-\theta_{j})\right].$ (34)

This is the basis for the linearised loss formulation in [39]. We can also
express this in terms of active power flows $p_{\ell}$ by substituting
equation (28) into equation (34)

$\psi_{\ell}=2g_{\ell}\left[1-\cos(p_{\ell}x_{\ell})\right].$ (35)

This makes the loss formulation independent from the voltage angle variables
and we can therefore avail of the speed-up obtained by using the cycle-based
formulation (31).

Using the small-angle approximation $\cos(\alpha)\approx 1-\alpha^{2}/2$,
equation (35) becomes quadratic

$\psi_{\ell}=2g_{\ell}\left[1-\left(1-\frac{(p_{\ell}x_{\ell})^{2}}{2}\right)\right]=g_{\ell}(p_{\ell}x_{\ell})^{2}.$
(36)

By inserting equation (21) we get

$\psi_{\ell}=\frac{r_{\ell}}{x_{\ell}^{2}}(p_{\ell}x_{\ell})^{2}$ (37)

or simply

$\psi_{\ell}=r_{\ell}p_{\ell}^{2}.$ (38)

This is the basis for the linearised loss formulation in [29]. Equation (38)
is still a quadratic equality constraint, and therefore nonconvex. Other works
have discussed or applied a piecewise linearisation of equation (38) [39, 30,
41, 38, 42]. But because the use of integer variables to define the segments
would entail a nonconvex mixed-integer problem (MILP), we choose not to pursue
this approach. Instead, by building a convex envelope around this constraint
from the upper and lower bounds for $\psi_{\ell}$ as well as a number of
tangents as inequality constraints, we can incorporate transmission losses
while retaining a linear optimisation problem. This is illustrated in Figure
2.

Figure 2: Illustration of the feasible space in the flow–loss
($p_{\ell}$–$\psi_{\ell}$) dimensions.

For setting the lower limit, by definition losses are positive

$\psi_{\ell}\geq 0$ (39)

and by substituting maximal line flows

$p_{\ell}\leq\overline{p}_{\ell}P_{\ell}\leq\overline{p}_{\ell}\overline{P}_{\ell}$
(40)

into (38) we obtain the upper limit

$\psi_{\ell}\leq r_{\ell}(\overline{p}_{\ell}\overline{P}_{\ell})^{2}.$ (41)

Next, we derive $2n$ evenly spaced (like in [30]) mirrored tangents which
approximate equation (38) as inequalities from below. These have the form

$\displaystyle\psi_{\ell}$ $\displaystyle\geq$ $\displaystyle m_{k}\cdot
p_{\ell}+a_{k}$ $\displaystyle\qquad\forall k=1,\dots,n$ (42)
$\displaystyle\psi_{\ell}$ $\displaystyle\geq$ $\displaystyle-$ $\displaystyle
m_{k}\cdot p_{\ell}+a_{k}$ $\displaystyle\qquad\forall k=1,\dots,n$ (43)

At segment $k$ we calculate the losses

$\psi_{\ell}(k)=r_{\ell}\left(\frac{k}{n}\cdot\overline{p}_{\ell}\overline{P}_{\ell}\right)^{2}$
(44)

and the corresponding slope

$m_{k}=\frac{\text{d}\psi_{\ell}(k)}{\text{d}k}=2r_{\ell}\left(\frac{k}{n}\cdot\overline{p}_{\ell}\overline{P}_{\ell}\right)$
(45)

and the offset

$a_{k}=\psi_{\ell}(k)-m_{k}\left(\frac{k}{n}\cdot\overline{p}_{\ell}\overline{P}_{\ell}\right).$
(46)

Together, equations (39), (41), (42), and (43) form a convex envelope of
equation (38).

A recurring criticism of this extensive convex relaxation is the possibility
of so-called fictitious or artificial losses [30, 43, 44, 45, 46]. As
illustrated by Figure 2, the model does indeed allow for overestimating
losses. This can be economical if negative locational marginal prices occur.
Overestimating losses is equivalent to dissipating power at a node. Another
component in the problem formulation that already permits this behaviour are
storage units (see equations (6)–(8)). To avoid binary variables, storage
units may charge and discharge at the same time. Power is then lost by cycling
through the conversion efficiencies. We argue that fictitious losses are not
problematic because (i) negative nodal prices are rare, (ii) such behaviour
could be realised in operation by low-cost resistors and demand response, and
(iii) the loss overestimation is bounded. We will substantiate this argument
with experimental results in Section 6.2.

###  4.5 Iterative Linearised Power Flow (with Loss Approximation)

When using the linearised power flow, with and without losses (Sections 4.3
and 4.4), the impedances of transmission lines affect the flows and losses.
The relations of reactances $x_{\ell}$ determine the distribution of flows
(cf. equation (31)). The resistances $r_{\ell}$ set the losses (cf. equation
(38)). Thus, for reactances we are only interested in relative values, whereas
for the resistances the absolute values are decisive.

Line impedances change as line capacities are increased ($x_{\ell}\propto
1/P_{\ell}$ and $r_{\ell}\propto 1/P_{\ell}$). Ignoring this dependency would
result in distorted power flows. Expanded lines would experience less flow
than they should. Losses may also be overestimated as the extension of
parallel lines reduces the effective resistance.

Consequently, the representation of grid physics is improved by taking account
of the relation between line capacities and impedances, yet also complicates
the problem. If we considered discrete expansion options we would use a
big-$M$ disjunctive relaxation to resolve the nonlinearity [35]. But since we
assume continuous line expansion in view of computational performance, we
instead pursue an iterative heuristic approach. In previous works, we have
shown that this is an acceptable approximation [12].

We sequentially solve the optimisation problem from Section 2 and in each
iteration update the line impedances according to their optimised capacities.
We repeat this process until (i) line expansion choices do not change in
subsequent iterations and convergence is reached, or (ii) a predetermined
number of iterations are performed. In the latter case, the final iteration
should be run with fixed line capacities such that impedances do not change
anymore.

##  5 Simulation Setup

Having developed the individual power flow models in theory, this section
outlines the setup we use to test them. First, we introduce the power system
model in Section 5.1 for which we optimise investments. Second, we outline the
methodology we use to validate the resulting approximated line flows in
Section 5.2.

###  5.1 Power System Model for Optimisation

We evaluate the different flow models on the open power system dataset PyPSA-
Eur (v0.1.0), which covers the whole European transmission system [22]. We
choose a spatial resolution of 250 nodes and a temporal resolution of 4380
snapshots, one for every two hours of a full year. This reflects the maximum
for which all flow models presented in Section 4 could be solved.

Targeting an emission reduction of 100% in the power sector, we only consider
renewable resources [47]. Following the problem formulation from Section 2,
solar PV, onshore and offshore wind capacities are co-optimised with battery
storage, hydrogen storage, and transmission infrastructure in a greenfield
planning approach, subject to spatio-temporal capacity factors and geographic
potentials. Exceptions to greenfield planning are existing transmission
infrastructure, which can only be reinforced but not removed, and today’s run-
of-river and hydropower capacities including pumped hydroelectric energy
storage, which are not extendable due to assumed geographical constraints.
HVDC links are assumed to have losses of 3% per 1000 km [48] and can be
expanded continuously up to 20 GW (each composed of several smaller parallel
circuits). Planned projects from the 2018 Ten Year Network Development Plan
(TYNDP) are included [49]. We only apply link losses to flow models which also
account for losses in HVAC lines. HVAC line capacity can also be expanded
continuously; by the maximum of doubled capacity or additional 5 GW. When
using the lossy transport model, HVAC lines are assumed to have constant
losses in the order of 5% per 1000 km [48]. To approximate $N-1$ security,
lines may only be used up to 70% of their nominal rating. More details are
provided in Hörsch et al. [22].

Technically, the optimisation problem is implemented using the free Python
modelling framework PyPSA (v0.17.0) working with the Pyomo interface [50].
Both optimality and feasibility tolerances are set to a value of 0.1%, which
is sufficient given the mentioned approximations made in the model. We use the
cycle-based formulation of Kirchhoff’s Voltage Law for any model that accounts
for it. The code to reproduce the experiments is openly available at
github.com/fneum/power-flow-models.

In accordance with descriptions in Section 4, the following flow models are
evaluated:

  * 1.

lossless transport model as Transport,

  * 2.

lossy transport model as Lossy Transport,

  * 3.

lossless linearised power flow with no iterations as Lossless,

  * 4.

lossless linearised power flow with 3 iterations as Iterative Lossless,

  * 5.

lossy linearised power flow with 6 tangents and no iterations as Lossy, and

  * 6.

lossy linearised power flow with 6 tangents and 3 iterations as Iterative
Lossy.

###  5.2 Nonlinear AC Power Flow Simulation

All presented flow models approximate the AC power flow equations (Section 3).
Thus, to identify possibly overlooked line overloading, and to demonstrate
characteristic features of particular flow models, we use the AC power flow
equations to assess the quality of the respective approximations.

We compare optimised flows to simulated flows which we obtain by solving the
AC power flow equations ex-post based on the optimised dispatch of
controllable system components. Specifically, we do not reoptimise dispatch
decisions subject to the AC power flow model due to computational constraints
given such large multi-period problems, but only check their feasibility. We
use the Newton-Raphson method (see e.g. [51]) and distribute the total slack
power across all buses in proportion to their total generation capacity [52,
21] (details are provided in A). Moreover, we consider PV buses222For PV
buses, the nodal active power injections $p_{i}$ and voltage magnitudes
$\left|V_{i}\right|$ are known (we assume nominal voltage magnitudes
$\left|V_{i}\right|=1$). Bus voltage angles $\theta_{i}$ and reactive power
feedin $q_{i}$ are to be found. Conversely, for PQ buses the nodal active
power injections $p_{i}$ and reactive power injections $q_{i}$ are known. Bus
voltage magnitudes $\left|V_{i}\right|$ and angles $\theta_{i}$ are to be
found. at each node since the reactive power set points are unknown. Hence, we
assume there to be sufficient reactive power control infrastructure to
maintain nominal voltages. We argue that in systems with high shares of
renewables the PV bus assumption is justified in view of a growing number of
distributed power generation units, each capable of contributing to voltage
control by reactive power injection or consumption, and power electronic
devices such as Flexible Alternating Current Transmission Systems (FACTS).
While the linearised power flow approximations neglect the shunt capacitance
of lines, these are taken into account in the subsequent AC power flow
simulation according to the standard equivalent $\Pi$ model [53]. Suitable
short- to medium-length lines between 25km and 250km make up about 85% of all
lines in the model (Figure 15). The remaining 15% of lines, which are longer
than 250km, are modelled identically although more rigorous alternatives
exist. These include partitioning long lines into multiple shorter sections to
model series compensation [53], or using equations specifically for long lines
that include fewer simplifying approximations of impedances than the $\Pi$
model [54].

##  6 Results and Discussion

This section presents and discusses the results from the experiments as
described in Section 5. As evaluation criteria we consider the total system
costs and the optimal system composition (Section 6.1), the error of optimised
losses (Section 6.2), the error of optimised flows compared to simulated flows
(Section 6.3), as well as peak memory consumption and solving time (Section
6.4).

###  6.1 Investments, Nodal Prices and Total System Costs

|  |  | Lossy |  | Iterative |  | Iterative
---|---|---|---|---|---|---|---
Indicator | Unit | Transport | Transport | Lossless | Lossless | Lossy | Lossy
System cost | bn€p.a. | 220.2 | 226.2 | 224.9 | 225.7 | 243.8 | 238.5
| €/MWh | 70.2 | 72.1 | 71.7 | 71.9 | 77.7 | 76.0
Energy transmitted | EWhkm | 1.56 | 1.26 | 1.36 | 1.28 | 0.90 | 0.94
Network expansion | TWkm | 216 | 214 | 206 | 206 | 160 | 170
Transmission losses | % of load | 0 | 2.3 | 0 | 0 | 5.1 | 3.7
Curtailment | % | 2.0 | 1.9 | 2.3 | 2.4 | 2.2 | 2.4
Share of $\left|\theta_{i}-\theta_{j}\right|\geq 30^{\circ}$ | % | 5.1 | 3.7 | 4.6 | 3.9 | 1.4 | 1.5
Table 2: Various statistical indicators compared accross covered flow models.

Figure 3: Comparison of total annual system costs split by system component
for the covered flow models.

Figure 4: Capacity correlation of optimised nodal investments among covered
flow models distinguished by technology. Figure 5: Nodal price duration curves
(snapshots and nodes) for selected flow models. In the omitted section, prices
rise steadily and similarly for all models. Some models allow for negative
nodal prices with occurence below 0.2%.

Transport | Lossy Transport
---|---
|
Lossless | Iterative Lossless
|
Lossy | Iterative Lossy
|

Figure 6: Maps of cost-optimal capacity expansion results for the covered
lossless flow models.

Table 2 presents total transmission losses to sum up to around 4% of the total
load when updated impedances according to line expansion are used. In
comparison to the 1.2% transmission losses reported by the German Federal
Network Agency for the year 2019 [55], this value is higher owing to the
larger volume of power transmission across the whole continent in scenarios
with high shares of renewables. Skipping the update of impedances
overestimates losses (5.5%) because additional parallel lines reduce the total
impedance. The lossy transport model underestimates losses (2.5%) since it
neglects the quadratic relationship between power and losses. Table 2 further
shows low curtailment at around 2% across all flow models due to generous line
expansion allowances.

At first sight, the optimised technology mix appears relatively similar across
all flow models, both in terms of cost composition in Figure 3 and the map of
investments in Figure 6. This is further underlined by the high correlations
of optimised capacities shown in Figure 4. Potentially due to some placement
degeneracies, lowest correlations are found for battery and hydrogen storage.
Further notable differences concern grid reinforcement. The lossless and lossy
transport models feature many new transmission lines in France and
Scandinavia, which disappear as more accurate flow models are applied. The
difference adds up to 20% less network reinforcement. Likewise, the energy
transmitted decreases as more constraints are imposed on power transmission.
In order to avoid grid losses, models that consider transmission losses and
KVL transmitted up to 66% less energy than the transport model. The reduced
spatial transport of power is then compensated by a shift towards hydrogen
storage and controllable HVDC links (e.g. in the West of Germany). Despite the
involved conversion losses, balancing renewables in time through storage
becomes more attractive. Additionally, to offset the energy lost by
transmission but also the reduced amount of power transmission, lossy models
feature more wind and solar generation capacity. This includes both more
localised generation (e.g. more solar panels in Southern Germany and more
onshore wind turbines in Eastern Europe) where previously there were few
production sites, and more concentrated generation in the North Sea region to
pair with the appended storage units. The added capacities raise the system
cost. In total, the annual system costs increase by approximately 5.7%
compared to iterative linearised power flow, or 8% relative to the transport
model.

Besides investments, we also compare electricity prices in an idealised nodal
market by using the dual variables of the nodal balance constraints. The price
duration curves depicted in Figure 5 show that nodal prices are more evenly
distributed in the lossless linear power flow compared compared to the
transport model. The even distribution of prices was also found in Gunkel et
al. [17]. The transport model and lossy transport model do not have the
properties that would allow negative prices. Negative nodal prices are a
consequence of KVL and occur when increasing demand at a bus relieves a
transmission line, allowing power to be exported from somewhere cheap to
somewhere expensive. This lowers the system cost and consequently results in a
negative price at that bus. Other constraints that can generally entail
negative prices are unit commitment constraints, but these are not considered
in this contribution. We find that even for models with KVL and loss
approximations, negative prices are rare ($\leq 0.2\%$). The major differences
regarding nodal prices can be observed in the 10% of highest prices. The
transport model features step-like price profiles, whereas the profiles of the
other models are smoother. The iterative lossy linearised power flow model
possesses the highest yet smoothest price duration curve.

###  6.2 Validation of Loss Approximation

Lossy Transport | Iterative Lossy
---|---
|
Figure 7: Examination of convex envelope relaxation around loss formula $\psi=rp^{2}$ given in equation (38) for lossy transport model and the iterative lossy linearised power flow model in a two-dimensional histogram. The line flows are normalised by their nominal capacity including the $N-1$ security margin ($\nicefrac{{p_{\ell,t}}}{{\overline{p}_{\ell}P_{\ell}}}$) and maximum losses according to security-constrained line capacity respectively, such that lines with different electrical parameters can be mapped onto the same chart. The count refers to a tuple $(\ell,t)$ of line and snapshot. The black line depicts the normalised quadratic loss formula (38). Lossy | Iterative Lossy
---|---
|
Figure 8: Comparison of simulated losses from AC power flow equations and
optimised losses for iterative and non-iterative lossy linearised power flow
in a two-dimensional histogram. Relative losses are shown as
$\nicefrac{{\psi_{\ell,t}}}{{\psi_{\ell}^{\mathrm{max}}}}$ according to
security-constrained line capacity $\overline{p}_{\ell}P_{\ell}$. The count
refers to a tuple $(\ell,t)$ of line and snapshot. The black line indicates
perfect alignment of simulated and optimised losses.

Figure 7 relates optimised line flows $p_{\ell,t}$ to optimised losses
$\psi_{\ell,t}$ for the lossy transport model and the iterative lossy
linearised power flow model. The lossy transport model underestimates losses
under high loading conditions depending on the assumed constant loss factor
and fails to reflect the quadratic relationship between losses and flow. On
the contrary, the results also confirm that approximating losses in linearised
optimal power flow with a convex envelope does not degrade the obtained
solutions. Although the envelope around the loss parabola (38) (cf. Section
4.4, approximates cosine in (35)) allows for losses to take values above the
parabola, the cost associated with losses tends to push losses downwards.
Substantial deviations from the parabola to above only occur when there is no
cost (or even a benefit in the case of negative nodal prices) associated with
higher losses; e.g. when energy is being curtailed, or when there is some
extra consumption of interest to control power flows or some other problem
degeneracy. As previously shown in Figure 5, negative nodal prices and
consequently incentives for loss overestimation are rare ($\leq 0.2\%$). These
circumstances cause the generous convex relaxation to function well.
Underestimating losses is also possible, albeit to a much smaller extent, as a
small fraction of the feasible space lies between the loss parabola and the
tangents that form the convex envelope. Recall that the loss parabola (38) is
already an approximation of the cosine terms in equation (35).

Figure 8 compares transmission losses retrieved from the optimisation problem
to the simulated losses from AC power flow for the iterating and non-iterating
loss approximation. Like in Figure 7, we note that the iterative lossy
formulation manages to sufficiently represent losses observed in the
respective AC power flow simulation. However, when the iteration is skipped
and hence line impedances are not updated according to their optimised
capacities, losses are overestimated.

###  6.3 Validation of Optimised Line Flows

Transport | Lossy Transport
---|---
|
Lossless | Iterative Lossless
|
Lossy | Iterative Lossy
|
Figure 9: Two-dimensional histograms comparing simulated flows (AC power flow) and optimised flows of the indicated flow models. Relative line flows are shown as $p_{\ell,t}/P_{\ell}$. The count refers to a tuple $(\ell,t)$ of line and snapshot. The black line indicates perfect alignment of simulated and optimised flows. Transport | Lossy Transport | Lossless
---|---|---
|  |
Iterative Lossless | Lossy | Iterative Lossy
|  |
Figure 10: Flow duration curves of simulated flows (AC power flow) and optimised flows for the indicated flow models. Relative line loading is shown as $p_{\ell,t}/P_{\ell}$. The count refers to a tuple $(\ell,t)$ of line and snapshot. Figure 11: Duration curves (lines and snapshots) of nodal price differences for lines experiencing no flow, congested lines, and all lines. Figure 12: Share of snapshots where the Newton-Raphson algorithm for solving the AC power flow equations did not converge distinguished by colour-coded synchronous zone. |  |  | Lossy |  | Iterative |  | Iterative
---|---|---|---|---|---|---|---
Indicator | Unit | Transport | Transport | Lossless | Lossless | Lossy | Lossy
Root Mean Squared (RMSE) | MW | 1468 | 1059 | 790 | 679 | 298 | 60
Mean Absolute (MAE) | MW | 775 | 707 | 269 | 207 | 194 | 35
Pearson Correlation (R) | – | 0.91 | 0.94 | 0.97 | 0.98 | 0.99 | 0.998
Coef. of Determination (R2) | – | 0.83 | 0.89 | 0.94 | 0.95 | 0.98 | 0.996
Table 3: Flow errors compared accross covered flow models.

Figure 9 compares line flows from optimisation to simulated line flows from AC
power flow for each of the flow models in a two-dimensional histogram. Figure
10 displays the same information from a different perspective as duration
curves of relative line loading for both simulated and optimised flows (figure
inspired by Brown et al. [19]). Table 3 quantifies the alignment of optimised
and simulated flows with some standard absolute and relative measures of error
that are frequently encountered in the literature (cf. [29]): root mean
squared error (RMSE), mean average error (MAE), Pearson correlation
coefficient ($R$), coefficient of determination ($R^{2}$).

First and foremost, the results reveal that the iterative lossy model matches
simulated flows almost perfectly. Other formulations show deficiencies
particularly under high loading conditions, but generally get the direction of
flow right. The errors become significantly less pronounced and produce less
undesired line overloading, the more physical characteristics of power flow
are considered during optimisation. Limiting the utilisation of line
capacities to 70% prevents abundant overloading. Remarkably, a high Pearson
correlation coefficient of 0.91 is already achieved with the transport model,
indicating that despite its simplicity the model can capture the dominant flow
patterns we observe in the ex-post AC power flow simulation.

Lines with zero flow occur strikingly frequently in the lossy transport model,
causing high deviations from the simulated flows. This can be explained with
the aid of Figure 11. There are many cases where prices are (almost) the same
at two neighbouring buses. In such cases, there is no strict economic need to
move power between them. With a lossless transport model there is no penalty
for moving power between the two nodes, such that the optimisation yields a
random value. However, for the lossy transport there is an incentive to set
the flow to zero to avoid the losses, which is why exactly this phenomenon
frequently occurs when there is no price difference. The physical flow
constraints enforced by KVL make it complicated to realise zero flow on a
line. This is the reason why we do not observe many lines with zero flow for
models that enforce KVL. Conversely, Figure 11 also shows that congested lines
cause high nodal price differences.

In some cases the Newton-Raphson algorithm does not converge. Typical causes
can be high voltage angle differences, voltage drops, and reactive power
flows. The power flow simulation is run separately for each snapshot and each
synchronous zone, so we can check individually what prevalent network
characteristics, in combination with the underlying flow models, cause the
failure to converge. The resulting share of snapshots not converged for each
synchronous zone is presented in Figure 12. Almost exclusively, difficulties
are observed in the Nordic synchronous zone which possesses many long
(aggregated) lines, which lead to high voltage angle differences. With regard
to the whole European system, the number of snapshots where no convergence is
reached is low. We observe better convergence rates for more detailed flow
models and the issue is found to become less problematic as the spatial
resolution of the transmission network is increased.

Given that high voltage angle differences diminish the accuracy of the linear
power flow approximation, a maximum of up to $\pm 30^{\circ}$ is commonly
tolerated in the literature [28, 56, 57]. This domain links to the range
beyond which the relative error of the small-angle approximation of the sine
exceeds 5%. Since the cosine approximation is a second order Taylor series
expansion, unlike the first order sine approximation, it does not reduce the
acceptable range of angle differences further (cf. Table 6). We observe that
across all flow models a majority of voltage angle differences lies within an
uncritically low range where the sine approximation is quite precise (cf.
Figure 14). The share of voltage angle differences outside $\pm 30^{\circ}$
reduces considerably with more physically accurate grid modelling (5% for
transport model versus 1.5% for lossy model, cf. Table 2).

###  6.4 Computational Performance

Peak Memory Consumption | Solving Time
---|---
|
Figure 13: Comparison of computational performance in terms of peak memory
consumption and solving time.

The computational performance of the different flow models, both in terms of
memory and computation time, is shown in Figure 13. More variables and
constraints leads to higher peak memory consumption. The spectrum ranges from
70 GB to 130 GB (around factor 2). Particularly the loss approximation raises
memory requirements significantly in relation to added KVL constraints or
constant efficiencies, also depending on the number of tangents used for the
convex envelope. Solving times range between 5 hours and 50 hours (factor 10).
Lossy and lossless transport model are solved the fastest by far. The lossless
linearised power flow model requires almost twice the time. Iteration has the
biggest impact on solving times, multiplying with the number of iterations.
Finally, we notice that the lossy formulations are more prone to numerical
issues, which could be circumvented by increasing the numeric accuracy
parameter of the solver at the cost of computational speed.

###  6.5 Critical Appraisal

The disregard of voltages and reactive power flows during optimisation ranks
among the severest shortcomings of the presented flow models. The cost and
required capacities for reactive power control are not assessed. The
confinement to linear formulations may also be considered as a weakness in
view of recent developments in convex second-order cone solvers.

Additionally, we consider the high-voltage transmission network only and do
not assess the performance of flow models in low-voltage distribution grids.
This is especially relevant in view of further closing the granularity gap.
Furthermore, losses on the distribution level are not directly modelled but
taken into account only through the electricity demand. Typically, the scale
of losses is higher than at the transmission level, as for instance the German
Federal Network Agency reports [55]. In 2019, losses at the transmission level
amounted to 1.2%, whereas losses at the distribution level were as high as 3%.
Moreover, the relations between ambient temperature, dynamic line rating and
losses are not addressed. Higher ambient temperatures reduce the amount of
power a transmission line can transmit safely but simultaneously increase the
resistance, affecting the losses.

Although the clustered transmission system is of course also simplified due to
computational constraints, we could observe consistent results for spatial
aggregation to 100, 200 and 250 nodes. However, the extent of network
clustering also affects the length of modelled lines and we note that for very
long lines with voltage angle differences beyond $\pm 30^{\circ}$ in highly
aggregated grid models the standard equivalent $\Pi$ model may not be suitable
[53].

##  7 Conclusion

In the present contribution we discuss best practices for incorporating two
inherently nonlinear phenomena, power flow and transmission losses, into
linear capacity expansion problems that co-optimise investments in generation,
storage and transmission infrastructure.

High model fidelity comes at the cost of high computational burden. Given the
cross-disciplinary nature of energy system modelling and differences in
complexity, the selection of a suitable representation of power flows depends
on the application, the user’s availability of computational resources, and
the level of spatial detail considered. A highly aggregated network will not
benefit from detailed power flow modelling, whereas modelling losses is
critical in the presence of continent-spanning power transmission at sub-
national detail. The present paper provides a detailed comparison to
facilitate this choice.

We find that already as little as three tangents are sufficient to accurately
approximate the quadratic losses, which in turn are an approximation of the
trigonometric losses. We do not observe excessive fictitious losses despite
the broad convex relaxation. However, we conclude that accounting for changing
impedances as lines are expanded is essential. Otherwise, losses will be
overestimated.

The literature predominantly employs the lossless transport model in design
studies, which can already capture the main features of a cost-efficient
system, but is too inexact for subsequent nonlinear power flow calculations.
However, a representation of power flows that considers transmission losses as
well as both Kirchhoff laws allows us to bridge between techno-economic models
and more detailed electrotechnical models.

## Acknowledgement

F.N. and T.B. gratefully acknowledge funding from the Helmholtz Association
under grant no. VH-NG-1352. The responsibility for the contents lies with the
authors.

## CRediT Author Statement

Fabian Neumann: Conceptualization, Methodology, Investigation, Software,
Validation, Formal analysis, Visualization, Writing – Original Draft, Writing
– Review & Editing Veit Hagenmeyer: Writing – Review & Editing, Project
administration, Funding acquisition Tom Brown: Conceptualization, Writing –
Review & Editing, Supervision, Project administration, Funding acquisition

## Data Availability

A dataset of the results including networks and graphics is available abstract
doi.org/10.5281/zenodo.3968297. We also refer to the documentation of PyPSA
(pypsa.readthedocs.io), PyPSA-Eur (pypsa-eur.readthedocs.io) and the source
code repository (github.com/fneum/power-flow-models).

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##  Appendix A AC Power Flow Problem Solved with Newton-Raphson

###  A.1 Without Distributed Slack

Given nodal power imbalances $S_{n}$ at any given snapshot for each bus $n$
the AC power flow equations are given by

$S_{n}=P_{n}+iQ_{n}=V_{n}I_{n}^{\ast}=V_{n}\left(\sum_{m}Y_{nm}V_{m}\right)^{\ast},$
(47)

where $V_{n}=\left|V_{n}\right|e^{i\theta_{n}}$ is the complex voltage, whose
rotating angle is taken relative to the slack bus and $Y_{nm}$ is the bus
admittance matrix, based on the branch impedances and shunt admittances
(including those attached to buses).

For the slack bus $n=0$ it is assumed $\left|V_{0}\right|$ is given and that
$\theta_{0}=0$; $P$ and $Q$ are to be found. For the PV buses, $P$ and
$\left|V\right|$ are given; $Q$ and $\theta$ are to be found. For the PQ
buses, $P$ and $Q$ are given; $\left|V\right|$ and $\theta$ are to be found.

Considering PV and PQ as sets of buses, then there are
$\left|\text{PV}\right|+2\left|\text{PQ}\right|$ real-valued equations to
solve:

$\displaystyle\text{Re}\left[V_{n}\left(\sum_{m}Y_{nm}V_{m}\right)^{\ast}\right]-P_{n}$
$\displaystyle=0\quad\forall\>\text{PV}\cup\text{PQ}$ (48)
$\displaystyle\text{Im}\left[V_{n}\left(\sum_{m}Y_{nm}V_{m}\right)^{\ast}\right]-Q_{n}$
$\displaystyle=0\quad\forall\>\text{PQ}$ (49)

We need to find $\theta_{n}$ for all PV and PQ buses and $\left|V_{n}\right|$
for all PQ buses.

These equations $f(x)=0$ are solved using the Newton-Raphson method, with the
Jacobian

$\frac{\partial f}{\partial x}=\left(\begin{matrix}\frac{\partial
P}{\partial\theta}&\frac{\partial P}{\partial\left|V\right|}\\\ \frac{\partial
Q}{\partial\theta}&\frac{\partial
Q}{\partial\left|V\right|}\end{matrix}\right)$ (50)

and the initial guesses $\theta_{n}=0$ and $\left|V_{n}\right|=1$ for unknown
quantities. For more details see for example Grainer and Stevenson [51]. The
total active slack power, which balances remaining mismatches of power
generation and demand resulting from the AC power flow equations, is fully
allocated to the slack bus. This can be a crude assumption, particularly for
large networks with a high penetration of renewables.

###  A.2 With Distributed Slack

A better alternative is to distribute the total active slack power across all
generators in proportion to their capacities (or another distribution scheme)
[52]. The active power flow equations are altered to

$\text{Re}\left[V_{n}\left(\sum_{m}Y_{nm}V_{m}\right)^{\ast}\right]-P_{n}-P_{\text{slack}}\gamma_{n}=0\quad\forall\>\text{PV}\cup\text{PQ}\cup\text{slack}$
(51)

where $P_{\text{slack}}$ is the total slack power and $\gamma_{n}$ is the
share of bus $n$ of the total generation capacity, which is used as
distribution key. We add an additional active power balance equation for the
slack bus since it is now part of the distribution scheme.

The distributed slack approach extends the Jacobian by an additional row for
the derivatives of the slack bus active power balance and by an additional
column for the partial derivatives with respect to $\gamma$

$\frac{\partial f}{\partial x}=\left(\begin{matrix}\frac{\partial
P_{0}}{\partial\theta}&\frac{\partial
P_{0}}{\partial\left|V\right|}&\frac{\partial P_{0}}{\partial\gamma}\\\
\frac{\partial P}{\partial\theta}&\frac{\partial
P}{\partial\left|V\right|}&\frac{\partial P}{\partial\gamma}\\\ \frac{\partial
Q}{\partial\theta}&\frac{\partial Q}{\partial\left|V\right|}&\frac{\partial
Q}{\partial\gamma}\end{matrix}\right).$ (52)

If $\gamma_{n}=0$ for all buses but the slack bus, this is equivalent to a
single slack bus model.

##  Appendix B Relations between Electrical Line Parameters

Following e.g. [51], the complex per-unit impedance
$z_{\ell}=r_{\ell}+ix_{\ell}$ is composed of ohmic resistance $r_{\ell}$ and
reactance $x_{\ell}$ Likewise, the admittance $y_{\ell}=g_{\ell}+ib_{\ell}$ is
composed of conductance $g_{\ell}$ and susceptance $b_{\ell}$ Impedance and
admittance are reciprocals ($y_{\ell}=z_{\ell}^{-1}$), hence we obtain the
relations

$\displaystyle g_{\ell}+ib_{\ell}$
$\displaystyle=\frac{1}{r_{\ell}+ix_{\ell}},$ (53) $\displaystyle
g_{\ell}+ib_{\ell}$
$\displaystyle=\frac{r_{\ell}-ix_{\ell}}{(r_{\ell}+ix_{\ell})(r_{\ell}-ix_{\ell})},$
(54) $\displaystyle g_{\ell}+ib_{\ell}$
$\displaystyle=\frac{r_{\ell}-ix_{\ell}}{r_{\ell}^{2}+x_{\ell}^{2}}.$ (55)

By splitting real and imaginary parts we can express conductance and
susceptance in terms of impedance and reactance:

$\displaystyle g_{\ell}$
$\displaystyle=\text{Re}\left[\frac{r_{\ell}-ix_{\ell}}{r_{\ell}^{2}+x_{\ell}^{2}}\right]=\frac{r_{\ell}}{r_{\ell}^{2}+x_{\ell}^{2}},$
(56) $\displaystyle b_{\ell}$
$\displaystyle=\text{Im}\left[\frac{r_{\ell}-ix_{\ell}}{r_{\ell}^{2}+x_{\ell}^{2}}\right]=\frac{x_{\ell}}{r_{\ell}^{2}+x_{\ell}^{2}}.$
(57)

##  Appendix C Additional Figures and Tables

Labels for used mathematical symbols are included in Table 4. An overview of
applied assumptions in selected equations of the present paper is given in
Table 5. Histograms of voltage angle differences for the evaluated flow models
are plotted in Figure 14. Relative errors of first and second order small-
angle approximations of trigonometric functions are listed in Table 6.
Information on the distribution of line lengths in the considered network are
displayed in Figure 15. Optimised capacities and energy generation by carrier
are summarised in Table 7 and Table 8.

Symbol | Description
---|---
$\mathcal{N}$ | set of buses
$\mathcal{L}$ | set of lines
$\mathcal{C}$ | set of cycles in cycle basis
$g_{\ell}$ | conductance
$b_{\ell}$ | susceptance
$r_{\ell}$ | resistance
$x_{\ell}$ | reactance
$z_{\ell}$ | impedance
$y_{\ell}$ | admittance
$\left|V_{i}\right|$ | voltage magnitude
$\theta_{i}$ | voltage angle
$p_{\ell(i,j)}=p_{\ell}$ | active power flow
$q_{\ell(i,j)}=q_{\ell}$ | reactive power flow
$\psi_{\ell}$ | power loss
$K_{i\ell}$ | incidence matrix
$C_{\ell c}$ | cycle incidence matrix
$p_{i}$ | nodal power injection
$P_{\ell}$ | line capacity
$\overline{P}_{\ell}$ | maximum line capacity
$\overline{p}_{\ell}$ | maximum per-unit flow
$m_{k}$ |  slope of loss tangent for interval $k$
$a_{k}$ |  offset of loss tangent for interval $k$
$n$ | number of intervals for loss tangents
Table 4: Nomenclature Assumptions |  | $\left|V_{i}\right|\approx 1$ | $b_{\ell}\gg g_{\ell}$ | $x_{\ell}\gg r_{\ell}$ | $\sin(\theta_{i}-\theta_{j})$ | $\cos(\theta_{i}-\theta_{j})$ | $q_{\ell}=0$
---|---|---|---|---|---|---|---
Formulas |  |  |  |  | $\approx\theta_{i}-\theta_{j}$ | $\approx 1-\frac{(\theta_{i}-\theta_{j})^{2}}{2}$ |
$g_{\ell}\approx r_{\ell}x_{\ell}^{-2}$ |  (21) |  |  | x |  |  |
$b_{\ell}\approx x_{\ell}^{-1}$ |  (22) |  |  | x |  |  |
$p_{\ell}=b_{\ell}(\theta_{i}-\theta_{j})$ |  (27) | x | x |  | x |  | x
$\sum_{\ell}C_{\ell c}p_{\ell}b_{\ell}^{-1}=0$ |  | x | x |  | x |  | x
$p_{\ell}=x_{\ell}^{-1}(\theta_{i}-\theta_{j})$ |  (28) | x | x | x | x |  | x
$\sum_{\ell}C_{\ell c}p_{\ell}x_{\ell}=0$ |  (31) | x | x | x | x |  | x
$\psi_{\ell}=2g_{\ell}[1-\cos(\theta_{i}-\theta_{j})]$ |  (34) | x |  |  |  |  | x
$\psi_{\ell}=2g_{\ell}[1-\cos(p_{\ell}b_{\ell}^{-1})]$ |  | x | x |  | x |  | x
$\psi_{\ell}=2r_{\ell}x_{\ell}^{-2}[1-\cos(p_{\ell}x_{\ell})]$ |  (35) | x | x | x | x |  | x
$\psi_{\ell}=r_{\ell}p_{\ell}^{2}$ |  (38) | x | x | x | x | x | x
Table 5: Overview of applied assumptions in respective equations. Transport | Lossy Transport | Lossless
---|---|---
|  |
Iterative Lossless | Lossy | Iterative Lossy
|  |
Figure 14: Distribution of voltage angle differences for the indicated flow models. $\theta_{i}-\theta_{j}$ | $\sin(\theta_{i}-\theta_{j})\approx\theta_{i}-\theta_{j}$ | $\cos(\theta_{i}-\theta_{j})\approx 1-\frac{(\theta_{i}-\theta_{j})^{2}}{2}$
---|---|---
$\pm 10^{\circ}$ | 0.5% | 0.0%
$\pm 20^{\circ}$ | 2.1% | 0.1%
$\pm 30^{\circ}$ | 4.7% | 0.4%
$\pm 40^{\circ}$ | 8.6% | 1.3%
Table 6: Relative error of small-angle approximations by angle. Figure 15: Distribution of line lengths in 250 node networks by line classification according to Kundur [53]. Of all lines, 15% fall into the category of long lines, while 6% are classified as short lines. |  |  | Lossy |  | Iterative |  | Iterative
---|---|---|---|---|---|---|---
Carrier | Unit | Transport | Transport | Lossless | Lossless | Lossy | Lossy
Offshore Wind (AC) | GW | 139 | 137 | 139 | 137 | 141 | 135
Offshore Wind (DC) | GW | 209 | 215 | 207 | 208 | 217 | 212
Onshore Wind | GW | 328 | 346 | 343 | 347 | 401 | 393
Run of River | GW | 34 | 34 | 34 | 34 | 34 | 34
Solar | GW | 431 | 461 | 440 | 456 | 535 | 517
Pumped Hydro | GW | 55 | 55 | 55 | 55 | 55 | 55
Hydro Dam | GW | 100 | 100 | 100 | 100 | 100 | 100
Hydrogen Storage | GW | 121 | 128 | 131 | 127 | 150 | 146
Battery Storage | GW | 47 | 45 | 44 | 46 | 42 | 43
HVDC Links | TWkm | 48 | 45 | 60 | 63 | 69 | 67
HVAC Lines | TWkm | 167 | 160 | 146 | 143 | 91 | 103
Table 7: Optimised capacities by technology for different flow models. |  |  | Lossy |  | Iterative |  | Iterative
---|---|---|---|---|---|---|---
Carrier | Unit | Transport | Transport | Lossless | Lossless | Lossy | Lossy
Offshore Wind (AC) | TWh | 525 | 515 | 526 | 515 | 509 | 504
Offshore Wind (DC) | TWh | 935 | 967 | 922 | 927 | 983 | 953
Onshore Wind | TWh | 875 | 908 | 907 | 901 | 988 | 980
Run of River | TWh | 139 | 140 | 139 | 138 | 140 | 140
Solar | TWh | 510 | 539 | 508 | 521 | 586 | 576
Hydro Inflow | TWh | 387 | 387 | 387 | 387 | 387 | 387
Table 8: Energy by carrier for different flow models.