Approximating Power Flow and Transmission Losses in Coordinated Capacity Expansion Problems
AApproximating Power Flow and Transmission Lossesin Coordinated Capacity Expansion Problems
Fabian Neumann a, ∗ , Veit Hagenmeyer a , Tom Brown a a 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 transmis-sion, storage and sectoral integration as balancing options, building a bridge between energysystem models and detailed power flow studies becomes increasingly important, but is compu-tationally challenging.In this paper, we compare both common and improved approximations for two nonlinearphenomena, power flow and transmission losses, in linear capacity expansion problems thatco-optimise investments in generation, storage and transmission infrastructure. We evaluatedifferent flow representations discussing differences in investment decisions, nodal prices, thedeviation of optimised flows and losses from simulated AC power flows, and the computationalperformance. By using the open European power system model PyPSA-Eur, that combineshigh spatial and temporal resolution, we obtain detailed and reproducible results aiming atfacilitating the selection of a suitable power flow model.Given the differences in complexity, the optimal choice depends on the application, theuser’s available computational resources, and the level of spatial detail considered. Althoughthe commonly used transport model can already identify key features of a cost-efficient sys-tem while being computationally performant, deficiencies under high loading conditions arisedue to the lack of a physical grid representation. Moreover, disregarding transmission lossesoverestimates optimal grid expansion by 20%. Adding a convex relaxation of quadratic losseswith two or three tangents to the linearised power flow equations and accounting for chang-ing line impedances as the network is reinforced suffices to represent power flows and lossesadequately in design studies. We show that the obtained investment and dispatch decisions arethen sufficiently physical to be used in more detailed nonlinear simulations of AC power flow inorder to better assess their technical feasibility. This includes determining reactive power flowsand voltages, which the initial linear model neglects. Simpler approximations are less suitablefor such ex-post analysis.
Keywords: energy system modelling, linear optimal power flow, transmission losses, capacityexpansion planning, grid reinforcement ∗ Corresponding author
Email address: [email protected] (Fabian Neumann)
Preprint submitted to RSER August 27, 2020 a r X i v : . [ phy s i c s . s o c - ph ] A ug ontents Appendix A.1 Without Distributed Slack . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Appendix A.2 With Distributed Slack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Appendix B Relations between Electrical Line Parameters 30Appendix C Additional Figures and Tables 30 . Introduction Energy system models seek to answer what infrastructure a future energy system requiresfor given policy goals, where and when it should be built, and how much it costs. For systemswith 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 transmis-sion networks and in time with storage and coupling to other energy sectors attracts muchresearch. Because energy system models are frequently used in policy-making, it becomes cru-cial to understand their particular limitations.To find credible answers for highly renewable systems, it has been demonstrated that mod-els require coordinated expansion planning of generation, storage and transmission infrastruc-ture because they strongly interact [1, 2]; high temporal resolution and scope to account forextreme 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 scopeto also capture the spatio-temporal patterns, such as correlations of wind speeds across the con-tinent, and to represent transmission constraints [5, 6]. As higher shares of renewables increasethe frequency of transmission bottlenecks, more detailed grid modelling is needed that looksbeyond 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 dispatchvariables spanning across thousands of operational conditions, a tractable yet sufficiently trust-worthy representation of power flows is essential. Ideally, outputs are detailed enough to beused as inputs for more accurate analyses, bridging the granularity gap between coarsely-resolved planning models and more detailed engineering models. Yet, even in strategic prob-lems speed matters for performing pivotal sensitivity analyses; e.g. regarding uncertain costparameters, reference weather years, technology choices, and resource boundaries. Unfortu-nately, the first-choice AC power flow equations are nonlinear and nonconvex, which makesthe embedded AC optimal power flow problem NP-hard [9–11].Even for a linear representation of power flows, considerations of transmission expansionplanning result in a bilinear problem because line impedances change as line capacities areincreased. While we can deal with this challenge through iterative impedance updates in se-quential linear programming [12], the problem would become even more complex if a discreteset of transmission expansion plans were considered, rather than continuous line expansion.More generally, we can approach computational challenges from multiple angles: by improvingsolving algorithms or by figuring out what model details can be simplified while retaining accu-racy [13]. Examples include the level of spatial aggregation, temporal aggregation, technologydetail and diversity, or finally the approximation of power flow.The transport model, that takes account only of power transfer capacities while ignoringimpedances, and the linearised power flow model, which includes impedances to consider bothKirchhoff laws but no losses, are commonly used in energy system models. Among the mod-els reviewed by Ringkjøb et al. [14] around four in five models use a transport model if flowsare represented, whereas only one in five uses a linearised power flow model. Previous workhas compared these two major variants [15–18], and some performed simulations of AC powerflow after optimisation [18, 19]. The comparisons indicate little discrepancy regarding totalsystem cost and cross-border transmission, but also differences in nodal prices and overlookedline overloadings when checked against AC power flow calculations. However, the cogencyof existing comparisons is limited by the use of low spatial resolution models with fewer than35 nodes. Furthermore, the consideration of losses is underrepresented in design studies, butalongside characteristic weather patterns shapes the tradeoffs regarding the volume of trans-mitted energy because losses increase as more power is transported [20, 21].In the present contribution, we offer a comprehensive comparison of linear representationsof power flow and losses in theory and practice. We outline their characteristic benefits andshortcomings in the context of coordinated capacity expansion problems, where generation,transmission and storage infrastructure is jointly planned. Given the multitude of modellinguncertainties, we assess under which circumstances it is worth embedding more elaborate flowmodels than a simple transport model. We further extend beyond previous research by intro-ducing a computationally inexpensive loss approximation that incorporates an efficient refor-mulation of the linearised power flow equations based on a cycle decomposition of the networkgraph. By using an open model of the European power system, PyPSA-Eur [22], spanning thewhole continent with hundreds of nodes and hourly temporal coupling due to the considera-tion of storage units, we achieve advanced and reproducible comparisons in systems with highshares of renewables.The content complements the best practices of energy system modelling, characterised inPfenniger et al. [23] and DeCarolis et al. [24], regarding the choice of suitable power flow mod-els. While we take an investment planning perspective in this paper, we underline that theway that the transmission of power is represented is relevant beyond system planning. Forinstance, it plays a role in the design of future electricity markets with multiple bidding zonesand flow-based market coupling [17, 25, 26].We structured this contribution as follows. We begin with an introduction to the basiclong-term power system planning problem in Section 2 and briefly review the physics of powerflow in Section 3. We continue with the different linear power flow representations in Section4. Section 5 presents the experimental setup, the results of which are discussed and criticallyappraised 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 theformulation to the power system, but it can also serve to represent the power system embeddedwithin the full energy system. The representation of power flows is one decisive constituentcomponent 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 annualised capi-tal costs 𝑐 ∗ for investments at locations 𝑖 in generator capacity 𝐺 𝑖,𝑟 of technology 𝑟 , storagepower capacity 𝐻 𝑖,𝑠 of technology 𝑠 , and transmission line capacities 𝑃 𝓁 , as well as the variableoperating costs 𝑜 ∗ for generator dispatch 𝑔 𝑖,𝑟,𝑡 : min 𝐺,𝐻,𝐹,𝑔 [∑ 𝑖,𝑟 𝑐 𝑖,𝑟 ⋅ 𝐺 𝑖,𝑟 + ∑ 𝑖,𝑠 𝑐 𝑖,𝑠 ⋅ 𝐻 𝑖,𝑠 + ∑ 𝓁 𝑐 𝓁 ⋅ 𝑃 𝓁 + ∑ 𝑖,𝑟,𝑡 𝑤 𝑡 ⋅ 𝑜 𝑖,𝑟 ⋅ 𝑔 𝑖,𝑟,𝑡 ] (1)where representative time snapshots 𝑡 are weighted by the time span 𝑤 𝑡 such that their totalduration adds up to one year; ∑ 𝑡∈ 𝑤 𝑡 = 365 ⋅ 24 h = 8760 h. The objective function is subject to The annuity factor −𝑛 𝜏 converts the overnight investment of an asset to annual payments considering its life-time 𝑛 and cost of capital 𝜏 . 𝐺 𝑖,𝑟 ≤ 𝐺 𝑖,𝑟 ≤ 𝐺 𝑖,𝑟 ∀𝑖, 𝑟 (2) 𝐻 𝑖,𝑠 ≤ 𝐻 𝑖,𝑠 ≤ 𝐻 𝑖,𝑠 ∀𝑖, 𝑠 (3) 𝑃 𝓁 ≤ 𝑃 𝓁 ≤ 𝑃 𝓁 ∀𝓁 (4)The dispatch of a generator may not only be constrained by its rated capacity but also bythe availability of variable renewable energy, which may be derived from reanalysis weatherdata. This can be expressed as a time- and location-dependent availability factor 𝑔 𝑖,𝑟,𝑡 , givenper unit of the generator’s capacity: 𝑖,𝑟,𝑡 ≤ 𝑔 𝑖,𝑟,𝑡 𝐺 𝑖,𝑟 ∀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. +𝑖,𝑠,𝑡 ≤ 𝐻 𝑖,𝑠 ∀𝑖, 𝑠, 𝑡 (6) −𝑖,𝑠,𝑡 ≤ 𝐻 𝑖,𝑠 ∀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. 𝑒 𝑖,𝑠,𝑡 = 𝜂 𝑤 𝑡 𝑖,𝑠,0 ⋅ 𝑒 𝑖,𝑠,𝑡−1 + 𝑤 𝑡 ⋅ ℎ inflow 𝑖,𝑠,𝑡 − 𝑤 𝑡 ⋅ ℎ spillage 𝑖,𝑠,𝑡 ∀𝑖, 𝑠, 𝑡+ 𝜂 𝑖,𝑠,+ ⋅ 𝑤 𝑡 ⋅ ℎ +𝑖,𝑠,𝑡 − 𝜂 −1𝑖,𝑠,− ⋅ 𝑤 𝑡 ⋅ ℎ −𝑖,𝑠,𝑡 (8)Storage units can have a standing loss 𝜂 𝑖,𝑠,0 , a charging efficiency 𝜂 𝑖,𝑠,+ , a discharging efficiency 𝜂 𝑖,𝑠,− , natural inflow ℎ inflow 𝑖,𝑠,𝑡 and spillage ℎ spillage 𝑖,𝑠,𝑡 . The storage energy levels are assumed to becyclic 𝑒 𝑖,𝑠,0 = 𝑒 𝑖,𝑠,| | ∀𝑖, 𝑠 (9)and are constrained by their energy capacity 𝑖,𝑠,𝑡 ≤ 𝑇 𝑠 ⋅ 𝐻 𝑖,𝑠 ∀𝑖, 𝑠, 𝑡. (10)To reduce the number of decisison variables, we link the energy storage volume to power ratingsusing a technology-specific parameter 𝑇 𝑠 that describes the maximum duration a storage unitcan discharge at full power rating.Total CO emissions may not exceed a target level Γ CO . The emissions are determined fromthe time-weighted generator dispatch 𝑤 𝑡 ⋅ 𝑔 𝑖,𝑟,𝑡 using the specific emissions 𝜌 𝑟 of fuel 𝑟 and thegenerator efficiencies 𝜂 𝑖,𝑟 ∑ 𝑖,𝑟,𝑡 𝜌 𝑟 ⋅ 𝜂 −1𝑖,𝑟 ⋅ 𝑤 𝑡 ⋅ 𝑔 𝑖,𝑟,𝑡 ≤ Γ CO . (11)All power flows 𝑝 𝓁,𝑡 are also limited by their capacities 𝑃 𝓁 |𝑝 𝓁,𝑡 | ≤ 𝑝 𝓁 𝑃 𝓁 ∀𝓁 , 𝑡, (12)5here 𝑝 𝓁 acts as a per-unit security margin on the line capacity to allow a buffer for the failureof single circuits ( 𝑁 − 1 condition) and reactive power flows.Ultimately, we need constraints that define the power flows 𝑝 𝓁,𝑡 in the network. In thenext Section 3, we briefly set foundations for nonlinear power flow and losses. The variousalternative flow models are then presented in Section 4. The subsequent descriptions will omitthe time index 𝑡 for notational simplicity.
3. Nonlinear Power Flow and Losses
This section briefly revises the nonlinear AC power flow equations, important electricalparameters of transmission lines, and how to calculate active power losses on a line. We do thisto set the foundations for derivations of the covered flow models.
The active power flow 𝑝 𝓁(𝑖,𝑗) of a line 𝓁 ≡ 𝓁 (𝑖, 𝑗) from bus 𝑖 to bus 𝑗 can be described involtage-polar coordinates by 𝑝 𝓁(𝑖,𝑗) = 𝑔 𝓁 |𝑉 𝑖 | + |𝑉 𝑖 | ||𝑉 𝑗 || [𝑔 𝓁 cos(𝜃 𝑖 − 𝜃 𝑗 ) − 𝑏 𝓁 sin(𝜃 𝑖 − 𝜃 𝑗 )] (13)and, analogously, the reactive power flow 𝑞 𝓁(𝑖,𝑗) is given by 𝑞 𝓁(𝑖,𝑗) = 𝑏 𝓁 |𝑉 𝑖 | + |𝑉 𝑖 | ||𝑉 𝑗 || [𝑔 𝓁 sin(𝜃 𝑖 − 𝜃 𝑗 ) − 𝑏 𝓁 cos(𝜃 𝑖 − 𝜃 𝑗 )] , (14)where |𝑉 𝑖 | is the per-unit bus voltage magnitude, 𝜃 𝑖 is the bus voltage angle, 𝑔 𝓁 is the lineconductance, and 𝑏 𝓁 is the line susceptance [27]. To derive an expression for the active power losses in a transmission line, we apply the con-vention that departing power flows are positive and arriving power flows are negative. Conse-quently, if power flows from bus 𝑖 to 𝑗 , 𝑝 𝓁(𝑖,𝑗) > 0 and 𝑝 𝓁(𝑗,𝑖) < 0 . The losses 𝜓 𝓁 are the differencebetween power sent and power received [27], therefore 𝜓 𝓁 = 𝑝 𝓁(𝑖,𝑗) + 𝑝 𝓁(𝑗,𝑖) . (15)Substituting equation (13) into equation (15) yields 𝜓 𝓁 = 𝑔 𝓁 |𝑉 𝑖 | + |𝑉 𝑖 | ||𝑉 𝑗 || [𝑔 𝓁 cos(𝜃 𝑖 − 𝜃 𝑗 ) − 𝑏 𝓁 sin(𝜃 𝑖 − 𝜃 𝑗 )] (16) + 𝑔 𝓁 ||𝑉 𝑗 || + ||𝑉 𝑗 || |𝑉 𝑖 | [𝑔 𝓁 cos(𝜃 𝑗 − 𝜃 𝑖 ) − 𝑏 𝓁 sin(𝜃 𝑗 − 𝜃 𝑖 )] (17)and using the trigonometric identities cos(−𝛼) = cos(𝛼) and sin(−𝛼) = − sin(𝛼) translates to 𝜓 𝓁 = 𝑔 𝓁 |𝑉 𝑖 | + |𝑉 𝑖 | ||𝑉 𝑗 || [𝑔 𝓁 cos(𝜃 𝑖 − 𝜃 𝑗 ) − 𝑏 𝓁 sin(𝜃 𝑖 − 𝜃 𝑗 )] (18) + 𝑔 𝓁 ||𝑉 𝑗 || + ||𝑉 𝑗 || |𝑉 𝑖 | [𝑔 𝓁 cos(𝜃 𝑖 − 𝜃 𝑗 ) + 𝑏 𝓁 sin(𝜃 𝑖 − 𝜃 𝑗 )] . (19)We can further simplify this expression to the loss formula 𝜓 𝓁 = 𝑔 𝓁 (|𝑉 𝑖 | + ||𝑉 𝑗 || ) − 2 |𝑉 𝑖 | ||𝑉 𝑗 || 𝑔 𝓁 cos(𝜃 𝑖 − 𝜃 𝑗 ). (20)We will use this formula in Section 4.4 to derive a linear approximation for losses.6odel Section Variables Eq. Constraints Ineq. ConstraintsTransport 4.1 | | || || − 1 | || || − 1 | | | | | | | 2𝑛 | | with Loss Approximation Table 1: Comparison of the number of variables and equality/inequality constraints related to flow models per snapshot 𝑡 ∈ . The constraint count excludes variable bounds. | | is the number of lines, || || is the number of nodes, and represents the number of tangents used for the loss approximation. The complex per-unit impedance 𝑧 𝓁 = 𝑟 𝓁 + 𝑖𝑥 𝓁 is composed of resistance 𝑟 𝓁 and reactance 𝑥 𝓁 . Likewise, the admittance 𝑦 𝓁 = 𝑔 𝓁 + 𝑖𝑏 𝓁 is composed of conductance 𝑔 𝓁 and susceptance 𝑏 𝓁 . Impedance and admittance are reciprocals ( 𝑦 𝓁 = 𝑧 −1𝓁 ). Hence, if we assume a dominance ofreactance over ohmic resistance ( 𝑟 𝓁 ≪ 𝑥 𝓁 ), as applies for high voltage overhead transmissionlines, we obtain the approximations 𝑔 𝓁 ≈ 𝑟 𝓁 𝑥 , (21) 𝑏 𝓁 ≈ 1𝑥 𝓁 . (22)For a derivation, see Appendix B. We will use these relations in Section 4.3 and Section 4.4. Inview of the approximation of line losses in later sections, note that although we assume thatresistance 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 becausemultiple local minima exist due to the trigonometric expressions and when directly incorpo-rated in the optimisation problem they would make the problem NP-hard [9–11]. To be ableto run large optimisation problems of the continental power system at sufficient spatial andtemporal resolution it is hence inevitable to retain a convex problem that can be solved in poly-nomial time and does not possess local minima.In this section we describe and develop various linear representations of power flow. Theseare 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 linearisedpower flow, and (iv) a lossy linearised power flow model. Figure 4 shows the relations betweenthe formulations and Table 4 documents differences in the number of variables and constraints.The scope of this work is deliberately constrained to:• only linear problems : To avail of powerful, scalable and fast interior-point solvers, and toguarantee an optimal solution, we only include formulations that entail linear problems.However, there exist promising second-order cone or semidefinite convex relaxations ofthe power flow equations. For excellent theoretical reviews of convex relaxations andapproximations of power flow see Molzahn et al. [30], Taylor [27], and Coffrin et al. [31].7 inearised Power Flowwith Loss ApproxmationTransport Model Lossy Transport ModelLinearised Power Flow IterativeLinearised Power Flowwith Loss ApproxmationIterativeLinearised Power Flow η ℓ = ∀ ℓ No Losses Losses ψ ℓ = ∀ ℓ ψ ℓ = ∀ ℓ TransferCapacityPhysicalFlowCorrectImpedances impedanceupdate impedanceupdateaddKVL
AC Power Flow
Convex Approximations / RelaxationsLinear Approximations / Relaxations
SOCPSDPQPLPNLP
Coverage ofthis paper: simplifyingassumptions
Figure 1: Illustration of the scope of the present paper and its context. It shows the connections between the coveredlinear power flow models, their main features, and how they are related to other (convexified) nonlinear formulations. only active power : We furthermore confine our analysis to formulations that do not cap-ture reactive power flows or information on bus voltages. Nonetheless, linear problemsthat capture selected aspects of this are under active research; see e.g. Coffrin et al. [32].• only comparison of different feasible spaces : We compare different linear flow models thatdefine different feasible spaces. We do not compare equivalent reformulations of identicalmodels, since this has been analysed in Hrsch et al. [33].• no copper plate model : Although occasionally encountered in generation and storage ca-pacity expansion models, we do not include the copper plate relaxation in our compar-isons 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 incursno cost. For the impact of spatial clustering – of which the copper plate model is theextreme – on optimal investments we refer to Hrsch et al. [5]. The transport model is also known as a network flow model, trade model, transshipmentmodel 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 bal-ance according to Kirchhoff’s Current Law (KCL); i.e. the power injected at each bus must equalthe power withdrawn by attached lines 𝑝 𝑖 = ∑ 𝓁 𝐾 𝑖𝓁 𝑝 𝓁 ∀𝑖 ∈ , (23)where 𝑝 𝑖 is the active power injected or consumed at node 𝑖 and 𝐾 is the incidence matrix ofthe network graph which has non-zero values +1 if line 𝓁 starts at bus 𝑖 and −1 if line 𝓁 ends atbus 𝑖 .Because the columns of the incidence matrix each sum up to zero, KCL yields || ||−1 linearlyindependent constraints. These are not sufficient to uniquely determine the | | unknown flows.The transport model allows arbitrary flows as long as flow conservation is fulfilled, also becauseit is free and lossless to transmit power. This makes the transport model degenerate, which canbe detrimental to the solving speed. Also, of course, this does not adequately reflect the physicalbehaviour of power flows in the transmission network.Despite its drawbacks, the transport model is very popular. In the comprehensive reviewby Ringkjøb et al. [14], it is applied in a majority of models. This minimalistic representationof flows is useful to develop an understanding for the potential of increased transfer capacitybetween regions, rather than assessing specific transmission bottlenecks and reinforcementneeds. It is often applied in investment models where the grid is highly aggregated to a fewnodes (e.g. one node per country in Europe or federal state in the United States) or analyses ofenergy markets across multiple bidding zones. Its main advantages are ease of implementationand fast solving speed. For pure dispatch problems without investment decisions one can evenutilise specialised network flow algorithms; for instance the minimum cost flow algorithm [34]. Part of the drawbacks and degeneracy of the transport model stems from the disregard oftransmission losses. As partial remedy, we can amend the transport model with a simple loss9pproximation which assumes lines to have a constant transmission efficiency 𝜂 𝓁 dependingon their length. In this case, the power arriving at the receiving bus is lower than the powerinjected at the sending bus. To differentiate between sending bus and receiving bus, we needto split the bidirectional power flow variable 𝑝 𝓁 into forward flows 𝑝 +𝓁 and backward flows 𝑝 −𝓁 with capacity limits +𝓁 ≤ 𝑝 𝓁 𝑃 𝓁 ∀𝓁 ∈ (24) −𝓁 ≤ 𝑝 𝓁 𝑃 𝓁 ∀𝓁 ∈ (25)which substitute the variables 𝑝 𝓁 and their bounds given in equation (12). Furthermore, weneed to adjust the nodal balance constraints (23) to reflect the transmission losses and separatedpower flow variables to 𝑝 𝑖 = ∑ 𝓁 𝐾 +𝑖𝓁 𝑝 +𝓁 − ∑ 𝓁 𝐾 −𝑖𝓁 𝑝 −𝓁 ∀𝑖 ∈ , (26)where 𝐾 + is the lossy incidence matrix of the network graph regarding forward flows 𝑝 +𝓁 whichhas non-zero values +1 if line 𝓁 starts at bus 𝑖 and −𝜂 𝓁 if line 𝓁 ends at bus 𝑖 . Analogously, 𝐾 − regards backward flows 𝑝 −𝓁 with non-zero values 𝜂 𝓁 if line 𝓁 starts at bus 𝑖 and −1 if line 𝓁 endsat bus 𝑖 .The transmission losses alleviate some degeneracy of the transport model since consideringlosses 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, whilethe 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 alllines have no losses ( 𝜂 𝓁 = 1 ), the lossy transport model is equivalent to the regular transportmodel. The linearised power flow model, which is also known as linearised load flow, DC powerflow or B Θ model, extends the lossless transport model. In addition to the nodal power balanceconstraints (23) from KCL and capacity limits (12), linear constraints for Kirchhoff’s VoltageLaw (KVL) are included, which define how power flows split in parallel paths. We derive theseby simplifying the nonlinear power flow equations (13) and (14). Assuming• all per-unit voltage magnitudes are close to one ( |𝑉 𝑖 | ≈ 1 ),• conductances 𝑔 𝓁 are negligible relative to susceptances 𝑏 𝓁 ( 𝑏 𝓁 ≫ 𝑔 𝓁 ),• voltage angle differences are small enough ( sin(𝜃 𝑖 − 𝜃 𝑗 ) ≈ 𝜃 𝑖 − 𝜃 𝑗 and cos(𝜃 𝑖 − 𝜃 𝑗 ) ≈ 0 ),• reactive power flows 𝑞 𝓁 are negligible compared to real power flows 𝑝 𝓁 ( 𝑞 𝓁 ≈ 0 ),leads to 𝑝 𝓁 = 𝑏 𝓁 (𝜃 𝑖 − 𝜃 𝑗 ), (27)and when we further assume 𝑟 𝓁 ≪ 𝑥 𝓁 , by substituting (22) we obtain 𝑝 𝓁 = 𝜃 𝑖 − 𝜃 𝑗 𝑥 𝓁 . (28)10is angle-based formulation is the most common linear formulation of KVL [27]. But it ispossible to avoid the auxiliary voltage angle variables and reduce the required number of con-straints by using a cycle basis of the network graph [33]. Namely, KVL states that the sum ofvoltage 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 ex-pressed as a directed linear combination of the lines 𝓁 in a cycle incidence matrix 𝐶 𝓁𝑐 = ⎧⎪⎪⎪⎨⎪⎪⎪⎩1 if edge 𝓁 is element of cycle 𝑐 , −1 if reversed edge 𝓁 is element of cycle 𝑐 , otherwise, (29)KVL is formulated by ∑ 𝓁 𝐶 𝓁𝑐 (𝜃 𝑖 − 𝜃 𝑗 ) = 0 ∀𝑐 ∈ . (30)Using equation (28), we can express KVL directly in terms of the power flows and circumventthe auxiliary voltage angle variables ∑ 𝓁 𝐶 𝓁𝑐 𝑝 𝓁 𝑥 𝓁 = 0 ∀𝑐 ∈ . (31)Although less common, this cycle-based formulation (31) has been shown to significantly out-perform the angle-based formulation (28) [33, 35]. There are even further equivalent reformu-lations 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 ourfocus lies on the comparison of different flow models, not their diverse reformulations, we onlyevaluate the computationally performant cycle-based formulation in the present contribution.With the cycle-based formulation one can clearly see that the transport model is equivalentto the linearised power flow in radial networks; i.e. when the network has no cycles. Also, theabsence of auxiliary voltage angle variables facilitates the insight that the transport model isa relaxation of the linearised power flow because the latter only adds constraints in the samevariable space.The linearised power flow model is claimed to be accurate when reactance dominates ( 𝑥 𝓁 ≫ 𝑟 𝓁 )and when parallel lines have similar ratios [36], but very long lines in highly aggregated net-works can deteriorate the quality of the approximation (see Section 6.3). An advantage of thismodel over the transport model is that it captures some meaningful physical characteristicsobserved in the operation of electrical grids. Namely, it is capable of revealing loop flows inmeshed networks; for instance recurring spillover effects between Germany and the Czech Re-public. 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 flowconstraints for existing lines and employs a transport model for candidate lines. Neglecting resistive losses is considered to be among the largest sources of error in thelinearised power flow formulation, particularly in large networks [36]. The following extensionof the lossless linearised power flow (Section 4.3) is a mixture of similar variants encounteredin the literature with a focus on computational efficiency. We reference where we follow or11eviate from previous work below. This or similar formulations have rarely been applied inthe co-optimisation of transmission, storage and generation capacities, but rather in detailedoperational optimal power flow (OPF) or transmission expansion planning (TEP) problems; seeoverview in [7].We start by adding a loss variable 𝜓 𝓁 for each line. Losses reduce the effective transmissioncapacity of a line |𝑝 𝓁 | ≤ 𝑝 𝓁 𝑃 𝓁 − 𝜓 𝓁 (32)and must be accounted for in the nodal balance equation (23) 𝑝 𝑖 = ∑ 𝓁 𝐾 𝑖𝓁 𝑝 𝓁 + |𝐾 𝑖𝓁 |2 𝜓 𝓁 ∀𝑖 ∈ . (33)We split the losses 𝜓 𝓁 equally between both buses (like in [37–39]) and do not allocate them atthe sending bus exclusively (like in [29, 40]). The latter could be modelled with an absolute valuefunction in the linear problem. However, this would involve splitting flow and loss variableseach 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 possibilityof overestimating losses due to an extensive convex relaxation.Assuming close to nominal per-unit voltage magnitudes |𝑉 𝑖 | ≈ 1 the loss formula given inequation (20) becomes 𝜓 𝓁 = 2𝑔 𝓁 [1 − cos(𝜃 𝑖 − 𝜃 𝑗 )] . (34)This is the basis for the linearised loss formulation in [39]. We can also express this in terms ofactive power flows 𝑝 𝓁 by substituting equation (28) into equation (34) 𝜓 𝓁 = 2𝑔 𝓁 [1 − cos(𝑝 𝓁 𝑥 𝓁 )] . (35)This makes the loss formulation independent from the voltage angle variables and we can there-fore avail of the speed-up obtained by using the cycle-based formulation (31).Using the small-angle approximation cos(𝛼) ≈ 1 − 𝛼 /2 , equation (35) becomes quadratic 𝜓 𝓁 = 2𝑔 𝓁 [1 − (1 − (𝑝 𝓁 𝑥 𝓁 ) 𝓁 (𝑝 𝓁 𝑥 𝓁 ) . (36)By inserting equation (21) we get 𝜓 𝓁 = 𝑟 𝓁 𝑥 (𝑝 𝓁 𝑥 𝓁 ) (37)or simply 𝜓 𝓁 = 𝑟 𝓁 𝑝 . (38)This is the basis for the linearised loss formulation in [29]. Equation (38) is still a quadraticequality constraint, and therefore nonconvex. Other works have discussed or applied a piece-wise linearisation of equation (38) [30, 38, 39, 41, 42]. But because the use of integer variablesto define the segments would entail a nonconvex mixed-integer problem (MILP), we choose notto pursue this approach. Instead, by building a convex envelope around this constraint fromthe upper and lower bounds for 𝜓 𝓁 as well as a number of tangents as inequality constraints,we can incorporate transmission losses while retaining a linear optimisation problem. This isillustrated in Figure 2. For setting the lower limit, by definition losses are positive12
00 400 200 0 200 400 600
Line Flow [MW] L i n e L o ss e s [ M W ] ψ = rp ψ ≥ ψ r (¯ pP ) ψ + | p | ¯ pP Feasible SpaceTangents
Figure 2: Illustration of the feasible space in the flow–loss ( 𝑝 𝓁 – 𝜓 𝓁 ) dimensions. 𝜓 𝓁 ≥ 0 (39)and by substituting maximal line flows 𝑝 𝓁 ≤ 𝑝 𝓁 𝑃 𝓁 ≤ 𝑝 𝓁 𝑃 𝓁 (40)into (38) we obtain the upper limit 𝜓 𝓁 ≤ 𝑟 𝓁 (𝑝 𝓁 𝑃 𝓁 ) . (41)Next, we derive evenly spaced (like in [30]) mirrored tangents which approximate equation(38) as inequalities from below. These have the form 𝜓 𝓁 ≥ 𝑚 𝑘 ⋅ 𝑝 𝓁 + 𝑎 𝑘 ∀𝑘 = 1, … , 𝑛 (42) 𝜓 𝓁 ≥ −𝑚 𝑘 ⋅ 𝑝 𝓁 + 𝑎 𝑘 ∀𝑘 = 1, … , 𝑛 (43)At segment 𝑘 we calculate the losses 𝜓 𝓁 (𝑘) = 𝑟 𝓁 ( 𝑘𝑛 ⋅ 𝑝 𝓁 𝑃 𝓁 ) (44)and the corresponding slope 𝑚 𝑘 = d 𝜓 𝓁 (𝑘) d 𝑘 = 2𝑟 𝓁 ( 𝑘𝑛 ⋅ 𝑝 𝓁 𝑃 𝓁 ) (45)and the offset 𝑎 𝑘 = 𝜓 𝓁 (𝑘) − 𝑚 𝑘 ( 𝑘𝑛 ⋅ 𝑝 𝓁 𝑃 𝓁 ) . (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 fic-titious or artificial losses [30, 43–46]. As illustrated by Figure 2, the model does indeed allow13or 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 theproblem 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. Poweris then lost by cycling through the conversion efficiencies. We argue that fictitious losses arenot problematic because (i) negative nodal prices are rare, (ii) such behaviour could be realisedin operation by low-cost resistors and demand response, and (iii) the loss overestimation isbounded. We will substantiate this argument with experimental results in Section 6.2. When using the linearised power flow, with and without losses (Sections 4.3 and 4.4), theimpedances of transmission lines affect the flows and losses. The relations of reactances 𝑥 𝓁 de-termine the distribution of flows (cf. equation (31)). The resistances 𝑟 𝓁 set the losses (cf. equation(38)). Thus, for reactances we are only interested in relative values, whereas for the resistancesthe absolute values are decisive.Line impedances change as line capacities are increased ( 𝑥 𝓁 ∝ 1/𝑃 𝓁 and 𝑟 𝓁 ∝ 1/𝑃 𝓁 ). Ignoringthis dependency would result in distorted power flows. Expanded lines would experience lessflow than they should. Losses may also be overestimated as the extension of parallel linesreduces the effective resistance.Consequently, the representation of grid physics is improved by taking account of the re-lation between line capacities and impedances, yet also complicates the problem. If we con-sidered discrete expansion options we would use a big- 𝑀 disjunctive relaxation to resolve thenonlinearity [35]. But since we assume continuous line expansion in view of computationalperformance, we instead pursue an iterative heuristic approach. In previous works, we haveshown that this is an acceptable approximation [12].We sequentially solve the optimisation problem from Section 2 and in each iteration updatethe line impedances according to their optimised capacities. We repeat this process until (i) lineexpansion choices do not change in subsequent iterations and convergence is reached, or (ii) apredetermined number of iterations are performed. In the latter case, the final iteration shouldbe 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 thesetup we use to test them. First, we introduce the power system model in Section 5.1 for whichwe optimise investments. Second, we outline the methodology we use to validate the resultingapproximated line flows in Section 5.2.
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 of250 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 re-sources [47]. Following the problem formulation from Section 2, solar PV, onshore and offshore14ind capacities are co-optimised with battery storage, hydrogen storage, and transmission in-frastructure in a greenfield planning approach, subject to spatio-temporal capacity factors andgeographic potentials. Exceptions to greenfield planning are existing transmission infrastruc-ture, which can only be reinforced but not removed, and today’s run-of-river and hydropowercapacities including pumped hydroelectric energy storage, which are not extendable due to as-sumed 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 parallelcircuits). Planned projects from the 2018 Ten Year Network Development Plan (TYNDP) areincluded [49]. We only apply link losses to flow models which also account for losses in HVAClines. HVAC line capacity can also be expanded continuously; by the maximum of doubled ca-pacity or additional 5 GW. When using the lossy transport model, HVAC lines are assumed tohave constant losses in the order of 5% per 1000 km [48]. To approximate
𝑁 − 1 security, linesmay only be used up to 70% of their nominal rating. More details are provided in Hrsch et al.[22].Technically, the optimisation problem is implemented using the free Python modellingframework PyPSA (v0.17.0) working with the Pyomo interface [50]. Both optimality and feasi-bility tolerances are set to a value of 0.1%, which is sufficient given the mentioned approxima-tions made in the model. We use the cycle-based formulation of Kirchhoff’s Voltage Law forany model that accounts for it. The code to reproduce the experiments is openly available atgithub.com/fneum/power-flow-models.In accordance with descriptions in Section 4, the following flow models are evaluated:• lossless transport model as Transport,• lossy transport model as Lossy Transport,• lossless linearised power flow with no iterations as Lossless,• lossless linearised power flow with 3 iterations as Iterative Lossless,• lossy linearised power flow with 6 tangents and no iterations as Lossy, and• lossy linearised power flow with 6 tangents and 3 iterations as Iterative Lossy.
All presented flow models approximate the AC power flow equations (Section 3). Thus,to identify possibly overlooked line overloading, and to demonstrate characteristic featuresof particular flow models, we use the AC power flow equations to assess the quality of therespective approximations.We compare optimised flows to simulated flows which we obtain by solving the AC powerflow 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 dueto computational constraints given such large multi-period problems, but only check their fea-sibility. We use the Newton-Raphson method (see e.g. [51]) and distribute the total slack poweracross all buses in proportion to their total generation capacity [21, 52] (details are provided inAppendix A). Moreover, we consider PV buses at each node since the reactive power set points For PV buses, the nodal active power injections 𝑝 𝑖 and voltage magnitudes |𝑉 𝑖 | are known (we assume nominalvoltage magnitudes |𝑉 𝑖 | = 1 ). Bus voltage angles 𝜃 𝑖 and reactive power feedin 𝑞 𝑖 are to be found. Conversely, for PQbuses the nodal active power injections 𝑝 𝑖 and reactive power injections 𝑞 𝑖 are known. Bus voltage magnitudes |𝑉 𝑖 | andangles 𝜃 𝑖 are to be found. ossy Iterative IterativeIndicator Unit Transport Transport Lossless Lossless Lossy LossySystem cost bn e p.a. 220.2 226.2 224.9 225.7 243.8 238.5 e /MWh 70.2 72.1 71.7 71.9 77.7 76.0Energy transmitted EWhkm 1.56 1.26 1.36 1.28 0.90 0.94Network expansion TWkm 216 214 206 206 160 170Transmission losses % of load 0 2.3 0 0 5.1 3.7Curtailment % 2.0 1.9 2.3 2.4 2.2 2.4Share of ||𝜃 𝑖 − 𝜃 𝑗 || ≥ 30 ◦ % 5.1 3.7 4.6 3.9 1.4 1.5Table 2: Various statistical indicators compared accross covered flow models. are unknown. Hence, we assume there to be sufficient reactive power control infrastructure tomaintain nominal voltages. We argue that in systems with high shares of renewables the PVbus 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, andpower electronic devices such as Flexible Alternating Current Transmission Systems (FACTS).While the linearised power flow approximations neglect the shunt capacitance of lines, theseare taken into account in the subsequent AC power flow simulation according to the standardequivalent Π model [53]. Suitable short- to medium-length lines between 25km and 250kmmake up about 85% of all lines in the model (Figure C.15). The remaining 15% of lines, which arelonger than 250km, are modelled identically although more rigorous alternatives exist. Theseinclude partitioning long lines into multiple shorter sections to model series compensation [53],or using equations specifically for long lines that include fewer simplifying approximations ofimpedances than the Π model [54].
6. Results and Discussion
This section presents and discusses the results from the experiments as described in Section5. 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 comparedto simulated flows (Section 6.3), as well as peak memory consumption and solving time (Section6.4).
Table 2 presents total transmission losses to sum up to around 4% of the total load whenupdated impedances according to line expansion are used. In comparison to the 1.2% trans-mission losses reported by the German Federal Network Agency for the year 2019 [55], thisvalue is higher owing to the larger volume of power transmission across the whole continentin scenarios with high shares of renewables. Skipping the update of impedances overestimateslosses (5.5%) because additional parallel lines reduce the total impedance. The lossy transportmodel underestimates losses (2.5%) since it neglects the quadratic relationship between powerand losses. Table 2 further shows low curtailment at around 2% across all flow models due togenerous line expansion allowances.At first sight, the optimised technology mix appears relatively similar across all flow mod-els, both in terms of cost composition in Figure 3 and the map of investments in Figure 6. This16 ransport LossyTransport Lossless IterativeLossless Lossy IterativeLossy050100150200250 T o t a l S y s t e m C o s t s [ b n E u r o / a ] SolarRun of riverOnshore WindOffshore Wind (DC)Offshore Wind (AC)Reservoir & DamBattery StoragePumped Hydro StorageHydrogen StorageHVDC LinkHVAC Line
Transport LossyTransport Lossless IterativeLossless Lossy IterativeLossy050100150200250 T o t a l S y s t e m C o s t s [ b n E u r o / a ] SolarRun of riverOnshore WindOffshore Wind (DC)Offshore Wind (AC)Reservoir & DamBattery StoragePumped Hydro StorageHydrogen StorageHVDC LinkHVAC Line
Figure 3: Comparison of total annual system costs split by system component for the covered flow models. T r a n s p o r t L o ss y T r a n s p o r t L o ss l e ss I t e r a t i v e L o ss l e ss L o ss y I t e r a t i v e L o ss y TransportLossyTransportLosslessIterativeLosslessLossyIterativeLossy 1.0 0.99 0.93 0.9 0.69 0.730.99 1.0 0.92 0.88 0.71 0.750.93 0.92 1.0 0.97 0.75 0.790.9 0.88 0.97 1.0 0.8 0.850.69 0.71 0.75 0.8 1.0 0.960.73 0.75 0.79 0.85 0.96 1.0
Offshore Wind (AC) T r a n s p o r t L o ss y T r a n s p o r t L o ss l e ss I t e r a t i v e L o ss l e ss L o ss y I t e r a t i v e L o ss y TransportLossyTransportLosslessIterativeLosslessLossyIterativeLossy 1.0 0.99 0.94 0.97 0.89 0.930.99 1.0 0.96 0.98 0.93 0.960.94 0.96 1.0 0.99 0.94 0.950.97 0.98 0.99 1.0 0.93 0.960.89 0.93 0.94 0.93 1.0 0.990.93 0.96 0.95 0.96 0.99 1.0
Offshore Wind (DC) T r a n s p o r t L o ss y T r a n s p o r t L o ss l e ss I t e r a t i v e L o ss l e ss L o ss y I t e r a t i v e L o ss y TransportLossyTransportLosslessIterativeLosslessLossyIterativeLossy 1.0 0.99 0.94 0.95 0.81 0.880.99 1.0 0.94 0.96 0.83 0.90.94 0.94 1.0 0.99 0.91 0.950.95 0.96 0.99 1.0 0.91 0.960.81 0.83 0.91 0.91 1.0 0.970.88 0.9 0.95 0.96 0.97 1.0
Onshore Wind T r a n s p o r t L o ss y T r a n s p o r t L o ss l e ss I t e r a t i v e L o ss l e ss L o ss y I t e r a t i v e L o ss y TransportLossyTransportLosslessIterativeLosslessLossyIterativeLossy 1.0 0.98 0.94 0.92 0.82 0.860.98 1.0 0.95 0.92 0.85 0.90.94 0.95 1.0 0.98 0.88 0.910.92 0.92 0.98 1.0 0.89 0.910.82 0.85 0.88 0.89 1.0 0.970.86 0.9 0.91 0.91 0.97 1.0
Solar T r a n s p o r t L o ss y T r a n s p o r t L o ss l e ss I t e r a t i v e L o ss l e ss L o ss y I t e r a t i v e L o ss y TransportLossyTransportLosslessIterativeLosslessLossyIterativeLossy 1.0 0.96 0.89 0.9 0.65 0.780.96 1.0 0.9 0.93 0.73 0.860.89 0.9 1.0 0.95 0.8 0.860.9 0.93 0.95 1.0 0.77 0.890.65 0.73 0.8 0.77 1.0 0.940.78 0.86 0.86 0.89 0.94 1.0
Hydrogen Storage T r a n s p o r t L o ss y T r a n s p o r t L o ss l e ss I t e r a t i v e L o ss l e ss L o ss y I t e r a t i v e L o ss y TransportLossyTransportLosslessIterativeLosslessLossyIterativeLossy 1.0 0.92 0.86 0.85 0.76 0.830.92 1.0 0.81 0.81 0.7 0.790.86 0.81 1.0 0.93 0.83 0.870.85 0.81 0.93 1.0 0.82 0.880.76 0.7 0.83 0.82 1.0 0.960.83 0.79 0.87 0.88 0.96 1.0
Battery Storage T r a n s p o r t L o ss y T r a n s p o r t L o ss l e ss I t e r a t i v e L o ss l e ss L o ss y I t e r a t i v e L o ss y TransportLossyTransportLosslessIterativeLosslessLossyIterativeLossy 1.0 0.99 0.93 0.93 0.91 0.920.99 1.0 0.94 0.93 0.92 0.920.93 0.94 1.0 0.98 0.96 0.960.93 0.93 0.98 1.0 0.95 0.970.91 0.92 0.96 0.95 1.0 0.980.92 0.92 0.96 0.97 0.98 1.0
Line T r a n s p o r t L o ss y T r a n s p o r t L o ss l e ss I t e r a t i v e L o ss l e ss L o ss y I t e r a t i v e L o ss y TransportLossyTransportLosslessIterativeLosslessLossyIterativeLossy 1.0 0.98 0.92 0.91 0.74 0.790.98 1.0 0.92 0.92 0.77 0.820.92 0.92 1.0 0.98 0.87 0.90.91 0.92 0.98 1.0 0.89 0.920.74 0.77 0.87 0.89 1.0 0.980.79 0.82 0.9 0.92 0.98 1.0
Link
Figure 4: Capacity correlation of optimised nodal investments among covered flow models distinguished by technology. N o d a l p r i c e [ E U R / M W h ] Share of Snapshots and Nodes [%]
TransportLossyTransport IterativeLosslessIterativeLossy
Figure 5: Nodal price duration curves (snapshots and nodes) for selected flow models. In the omitted section, pricesrise steadily and similarly for all models. Some models allow for negative nodal prices with occurence below 0.2%.
HVAC Line Capacity2 GW5 GW10 GW HVDC Link Capacity2 GW5 GW10 GW TechnologyOnshore WindOffshore Wind (AC)Offshore Wind (DC)SolarPumped Hydro StorageReservoir & Dam Run of riverHydrogen StorageBattery StorageHVAC LineHVDC Link Generation50 GW10 GW1 GWGeneration50 GW10 GW1 GW HVAC Line Capacity2 GW5 GW10 GW HVDC Link Capacity2 GW5 GW10 GW TechnologyOnshore WindOffshore Wind (AC)Offshore Wind (DC)SolarPumped Hydro StorageReservoir & Dam Run of riverHydrogen StorageBattery StorageHVAC LineHVDC Link Generation50 GW10 GW1 GWGeneration50 GW10 GW1 GW
Lossless Iterative Lossless
HVAC Line Capacity2 GW5 GW10 GW HVDC Link Capacity2 GW5 GW10 GW TechnologyOnshore WindOffshore Wind (AC)Offshore Wind (DC)SolarPumped Hydro StorageReservoir & Dam Run of riverHydrogen StorageBattery StorageHVAC LineHVDC Link Generation50 GW10 GW1 GWGeneration50 GW10 GW1 GW HVAC Line Capacity2 GW5 GW10 GW HVDC Link Capacity2 GW5 GW10 GW TechnologyOnshore WindOffshore Wind (AC)Offshore Wind (DC)SolarPumped Hydro StorageReservoir & Dam Run of riverHydrogen StorageBattery StorageHVAC LineHVDC Link Generation50 GW10 GW1 GWGeneration50 GW10 GW1 GW
Lossy Iterative Lossy
HVAC Line Capacity2 GW5 GW10 GW HVDC Link Capacity2 GW5 GW10 GW TechnologyOnshore WindOffshore Wind (AC)Offshore Wind (DC)SolarPumped Hydro StorageReservoir & Dam Run of riverHydrogen StorageBattery StorageHVAC LineHVDC Link Generation50 GW10 GW1 GWGeneration50 GW10 GW1 GW HVAC Line Capacity2 GW5 GW10 GW HVDC Link Capacity2 GW5 GW10 GW TechnologyOnshore WindOffshore Wind (AC)Offshore Wind (DC)SolarPumped Hydro StorageReservoir & Dam Run of riverHydrogen StorageBattery StorageHVAC LineHVDC Link Generation50 GW10 GW1 GWGeneration50 GW10 GW1 GW
HVAC Line Capacity5 GW10 GW20 GW HVDC Link Capacity5 GW10 GW20 GW Generation50 GW10 GWGeneration50 GW10 GW TechnologyOnshore WindOffshore Wind (AC)Offshore Wind (DC)SolarPumped Hydro Storage Reservoir & DamRun of riverHydrogen StorageBattery StorageTechnologyOnshore WindOffshore Wind (AC)Offshore Wind (DC)SolarPumped Hydro Storage Reservoir & DamRun of riverHydrogen StorageBattery Storage
Figure 6: Maps of cost-optimal capacity expansion results for the covered lossless flow models.
18s further underlined by the high correlations of optimised capacities shown in Figure 4. Po-tentially due to some placement degeneracies, lowest correlations are found for battery andhydrogen storage. Further notable differences concern grid reinforcement. The lossless andlossy transport models feature many new transmission lines in France and Scandinavia, whichdisappear as more accurate flow models are applied. The difference adds up to 20% less networkreinforcement. Likewise, the energy transmitted decreases as more constraints are imposed onpower transmission. In order to avoid grid losses, models that consider transmission losses andKVL transmitted up to 66% less energy than the transport model. The reduced spatial transportof power is then compensated by a shift towards hydrogen storage and controllable HVDClinks (e.g. in the West of Germany). Despite the involved conversion losses, balancing renew-ables in time through storage becomes more attractive. Additionally, to offset the energy lostby transmission but also the reduced amount of power transmission, lossy models feature morewind and solar generation capacity. This includes both more localised generation (e.g. moresolar panels in Southern Germany and more onshore wind turbines in Eastern Europe) wherepreviously there were few production sites, and more concentrated generation in the North Searegion to pair with the appended storage units. The added capacities raise the system cost. Intotal, the annual system costs increase by approximately 5.7% compared to iterative linearisedpower flow, or 8% relative to the transport model.Besides investments, we also compare electricity prices in an idealised nodal market byusing the dual variables of the nodal balance constraints. The price duration curves depicted inFigure 5 show that nodal prices are more evenly distributed in the lossless linear power flowcompared compared to the transport model. The even distribution of prices was also found inGunkel et al. [17]. The transport model and lossy transport model do not have the propertiesthat would allow negative prices. Negative nodal prices are a consequence of KVL and occurwhen increasing demand at a bus relieves a transmission line, allowing power to be exportedfrom somewhere cheap to somewhere expensive. This lowers the system cost and consequentlyresults in a negative price at that bus. Other constraints that can generally entail negativeprices are unit commitment constraints, but these are not considered in this contribution. Wefind that even for models with KVL and loss approximations, negative prices are rare ( ≤ 0.2% ).The major differences regarding nodal prices can be observed in the 10% of highest prices. Thetransport model features step-like price profiles, whereas the profiles of the other models aresmoother. The iterative lossy linearised power flow model possesses the highest yet smoothestprice duration curve.
Figure 7 relates optimised line flows 𝑝 𝓁,𝑡 to optimised losses 𝜓 𝓁,𝑡 for the lossy transportmodel and the iterative lossy linearised power flow model. The lossy transport model underes-timates losses under high loading conditions depending on the assumed constant loss factor andfails to reflect the quadratic relationship between losses and flow. On the contrary, the resultsalso confirm that approximating losses in linearised optimal power flow with a convex envelopedoes 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 fromthe parabola to above only occur when there is no cost (or even a benefit in the case of negativenodal prices) associated with higher losses; e.g. when energy is being curtailed, or when there issome 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 loss19ossy Transport Iterative Lossy Relative Line Flows (opt.) R e l a t i v e L o ss e s ( o p t . ) C o un t Relative Line Flows (opt.) R e l a t i v e L o ss e s ( o p t . ) C o un t Figure 7: Examination of convex envelope relaxation around loss formula 𝜓 = 𝑟𝑝 given in equation (38) for lossytransport model and the iterative lossy linearised power flow model in a two-dimensional histogram. The line flowsare normalised by their nominal capacity including the 𝑁 −1 security margin ( 𝑝 𝓁,𝑡 / 𝑝 𝓁 𝑃 𝓁 ) and maximum losses according tosecurity-constrained line capacity respectively, such that lines with different electrical parameters can be mapped ontothe same chart. The count refers to a tuple (𝓁, 𝑡) of line and snapshot. The black line depicts the normalised quadraticloss formula (38). Lossy Iterative Lossy
Relative Losses (opt.) R e l a t i v e L o ss e s ( s i m . ) C o un t Relative Losses (opt.) R e l a t i v e L o ss e s ( s i m . ) C o un t 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 𝜓 𝓁,𝑡 / 𝜓 max𝓁 according tosecurity-constrained line capacity 𝑝 𝓁 𝑃 𝓁 . The count refers to a tuple (𝓁, 𝑡) of line and snapshot. The black line indicatesperfect alignment of simulated and optimised losses. ossy Iterative IterativeIndicator Unit Transport Transport Lossless Lossless Lossy LossyRoot Mean Squared (RMSE) MW 1468 1059 790 679 298 60Mean Absolute (MAE) MW 775 707 269 207 194 35Pearson Correlation (R) – 0.91 0.94 0.97 0.98 0.99 0.998Coef. of Determination (R ) – 0.83 0.89 0.94 0.95 0.98 0.996Table 3: Flow errors compared accross covered flow models. overestimation are rare ( ≤ 0.2% ). These circumstances cause the generous convex relaxationto function well. Underestimating losses is also possible, albeit to a much smaller extent, as asmall fraction of the feasible space lies between the loss parabola and the tangents that form theconvex envelope. Recall that the loss parabola (38) is already an approximation of the cosineterms in equation (35).Figure 8 compares transmission losses retrieved from the optimisation problem to the sim-ulated losses from AC power flow for the iterating and non-iterating loss approximation. Likein Figure 7, we note that the iterative lossy formulation manages to sufficiently represent lossesobserved in the respective AC power flow simulation. However, when the iteration is skippedand hence line impedances are not updated according to their optimised capacities, losses areoverestimated. Figure 9 compares line flows from optimisation to simulated line flows from AC power flowfor each of the flow models in a two-dimensional histogram. Figure 10 displays the same infor-mation from a different perspective as duration curves of relative line loading for both simulatedand optimised flows (figure inspired by Brown et al. [19]). Table 3 quantifies the alignment ofoptimised and simulated flows with some standard absolute and relative measures of error thatare frequently encountered in the literature (cf. [29]): root mean squared error (RMSE), meanaverage error (MAE), Pearson correlation coefficient ( 𝑅 ), coefficient of determination ( 𝑅 ).First and foremost, the results reveal that the iterative lossy model matches simulated flowsalmost perfectly. Other formulations show deficiencies particularly under high loading con-ditions, but generally get the direction of flow right. The errors become significantly lesspronounced and produce less undesired line overloading, the more physical characteristics ofpower 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 isalready achieved with the transport model, indicating that despite its simplicity the model cancapture 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 highdeviations from the simulated flows. This can be explained with the aid of Figure 11. Thereare 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 modelthere is no penalty for moving power between the two nodes, such that the optimisation yieldsa random value. However, for the lossy transport there is an incentive to set the flow to zeroto avoid the losses, which is why exactly this phenomenon frequently occurs when there is noprice difference. The physical flow constraints enforced by KVL make it complicated to realisezero flow on a line. This is the reason why we do not observe many lines with zero flow for21 ransport Lossy Transport . . . . . . . . . . . . . Relative Line Flows (sim.) R e l a t i v e L i n e F l o w s ( o p t . ) C o un t . . . . . . . . . . . . . Relative Line Flows (sim.) R e l a t i v e L i n e F l o w s ( o p t . ) C o un t Lossless Iterative Lossless . . . . . . . . . . . . . Relative Line Flows (sim.) R e l a t i v e L i n e F l o w s ( o p t . ) C o un t . . . . . . . . . . . . . Relative Line Flows (sim.) R e l a t i v e L i n e F l o w s ( o p t . ) C o un t Lossy Iterative Lossy . . . . . . . . . . . . . Relative Line Flows (sim.) R e l a t i v e L i n e F l o w s ( o p t . ) C o un t . . . . . . . . . . . . . Relative Line Flows (sim.) R e l a t i v e L i n e F l o w s ( o p t . ) C o un t Figure 9: Two-dimensional histograms comparing simulated flows (AC power flow) and optimised flows of the indicatedflow models. Relative line flows are shown as 𝑝 𝓁,𝑡 /𝑃 𝓁 . The count refers to a tuple (𝓁, 𝑡) of line and snapshot. The blackline indicates perfect alignment of simulated and optimised flows. ransport Lossy Transport Lossless Share of Snapshots [%] R e l a t i v e L i n e L o a d i n g [ - ] approximated flow modelAC power flow 0 20 40 60 80 100 Share of Snapshots [%] R e l a t i v e L i n e L o a d i n g [ - ] approximated flow modelAC power flow 0 20 40 60 80 100 Share of Snapshots [%] R e l a t i v e L i n e L o a d i n g [ - ] approximated flow modelAC power flow Iterative Lossless Lossy Iterative Lossy
Share of Snapshots [%] R e l a t i v e L i n e L o a d i n g [ - ] approximated flow modelAC power flow 0 20 40 60 80 100 Share of Snapshots [%] R e l a t i v e L i n e L o a d i n g [ - ] approximated flow modelAC power flow 0 20 40 60 80 100 Share of Snapshots [%] R e l a t i v e L i n e L o a d i n g [ - ] approximated flow modelAC power flow 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 𝓁,𝑡 /𝑃 𝓁 . The count refers to a tuple (𝓁, 𝑡) of line and snapshot. Share of Snapshots and Selected Lines [%] N o d a l p r i c e d i ff e r e n c e [ E U R / M W h ] All flow variablesMean price difference: 2.05 EUR/MWhCongested flow variablesMean price difference: 8.30 EUR/MWhZero flow variablesMean price difference: 0.52 EUR/MWh
Figure 11: Duration curves (lines and snapshots) of nodal price differences for lines experiencing no flow, congestedlines, and all lines. T r a n s p o r t L o ss y T r a n s p o r t L o ss l e ss I t e r a t i v e L o ss l e ss L o ss y I t e r a t i v e L o ss y S n a p s h o t s n o t c o n v e r g e d [ % ] Synchronous ZoneContinentalNordicBalticIrelandGreat Britain
Figure 12: Share of snapshots where the Newton-Raphson algorithm for solving the AC power flow equations did notconverge distinguished by colour-coded synchronous zone. eak Memory Consumption Solving Time T r a n s p o r t L o ss y T r a n s p o r t L o ss l e ss I t e r a t i v e L o ss l e ss L o ss y I t e r a t i v e L o ss y P e a k M e m o r y [ G B ] T r a n s p o r t L o ss y T r a n s p o r t L o ss l e ss I t e r a t i v e L o ss l e ss L o ss y I t e r a t i v e L o ss y S o l v i n g T i m e [ h ] Figure 13: Comparison of computational performance in terms of peak memory consumption and solving time. models that enforce KVL. Conversely, Figure 11 also shows that congested lines cause highnodal price differences.In some cases the Newton-Raphson algorithm does not converge. Typical causes can be highvoltage angle differences, voltage drops, and reactive power flows. The power flow simulationis run separately for each snapshot and each synchronous zone, so we can check individuallywhat prevalent network characteristics, in combination with the underlying flow models, causethe failure to converge. The resulting share of snapshots not converged for each synchronouszone is presented in Figure 12. Almost exclusively, difficulties are observed in the Nordic syn-chronous zone which possesses many long (aggregated) lines, which lead to high voltage angledifferences. With regard to the whole European system, the number of snapshots where no con-vergence is reached is low. We observe better convergence rates for more detailed flow modelsand the issue is found to become less problematic as the spatial resolution of the transmissionnetwork is increased.Given that high voltage angle differences diminish the accuracy of the linear power flowapproximation, a maximum of up to ±30 ◦ is commonly tolerated in the literature [28, 56, 57].This domain links to the range beyond which the relative error of the small-angle approxima-tion of the sine exceeds 5%. Since the cosine approximation is a second order Taylor seriesexpansion, unlike the first order sine approximation, it does not reduce the acceptable rangeof angle differences further (cf. Table C.6). We observe that across all flow models a majorityof voltage angle differences lies within an uncritically low range where the sine approximationis quite precise (cf. Figure C.14). The share of voltage angle differences outside ±30 ◦ reducesconsiderably with more physically accurate grid modelling (5% for transport model versus 1.5%for lossy model, cf. Table 2). The computational performance of the different flow models, both in terms of memory andcomputation time, is shown in Figure 13. More variables and constraints leads to higher peakmemory consumption. The spectrum ranges from 70 GB to 130 GB (around factor 2). Partic-ularly the loss approximation raises memory requirements significantly in relation to addedKVL constraints or constant efficiencies, also depending on the number of tangents used forthe convex envelope. Solving times range between 5 hours and 50 hours (factor 10). Lossy andlossless transport model are solved the fastest by far. The lossless linearised power flow model24equires almost twice the time. Iteration has the biggest impact on solving times, multiplyingwith the number of iterations. Finally, we notice that the lossy formulations are more prone tonumerical issues, which could be circumvented by increasing the numeric accuracy parameterof the solver at the cost of computational speed.
The disregard of voltages and reactive power flows during optimisation ranks among theseverest shortcomings of the presented flow models. The cost and required capacities for re-active power control are not assessed. The confinement to linear formulations may also beconsidered 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 assessthe performance of flow models in low-voltage distribution grids. This is especially relevant inview of further closing the granularity gap. Furthermore, losses on the distribution level are notdirectly modelled but taken into account only through the electricity demand. Typically, thescale of losses is higher than at the transmission level, as for instance the German Federal Net-work Agency reports [55]. In 2019, losses at the transmission level amounted to 1.2%, whereaslosses at the distribution level were as high as 3%. Moreover, the relations between ambienttemperature, dynamic line rating and losses are not addressed. Higher ambient temperaturesreduce the amount of power a transmission line can transmit safely but simultaneously increasethe resistance, affecting the losses.Although the clustered transmission system is of course also simplified due to computa-tional constraints, we could observe consistent results for spatial aggregation to 100, 200 and250 nodes. However, the extent of network clustering also affects the length of modelled linesand we note that for very long lines with voltage angle differences beyond ±30 ◦ in highly ag-gregated grid models the standard equivalent Π model may not be suitable [53].
7. Conclusion
In the present contribution we discuss best practices for incorporating two inherently non-linear phenomena, power flow and transmission losses, into linear capacity expansion problemsthat 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 selectionof a suitable representation of power flows depends on the application, the user’s availabilityof computational resources, and the level of spatial detail considered. A highly aggregated net-work will not benefit from detailed power flow modelling, whereas modelling losses is criticalin the presence of continent-spanning power transmission at sub-national detail. The presentpaper provides a detailed comparison to facilitate this choice.We find that already as little as three tangents are sufficient to accurately approximate thequadratic losses, which in turn are an approximation of the trigonometric losses. We do notobserve excessive fictitious losses despite the broad convex relaxation. However, we concludethat accounting for changing impedances as lines are expanded is essential. Otherwise, losseswill be overestimated.The literature predominantly employs the lossless transport model in design studies, whichcan already capture the main features of a cost-efficient system, but is too inexact for subsequentnonlinear power flow calculations. However, a representation of power flows that considers25ransmission losses as well as both Kirchhoff laws allows us to bridge between techno-economicmodels and more detailed electrotechnical models.
Acknowledgement
F.N. and T.B. gratefully acknowledge funding from the Helmholtz Association under grantno. VH-NG-1352. The responsibility for the contents lies with the authors. This work is licensedunder a Creative Commons “Attribution 4.0 International” license. cb CRediT Author StatementFabian Neumann:
Conceptualization, Methodology, Investigation, Software, Validation,Formal analysis, Visualization, Writing – Original Draft, Writing – Review & Editing
VeitHagenmeyer:
Writing – Review & Editing, Project administration, Funding acquisition
TomBrown:
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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Milano, PSAT documentation: version 2.0.0, Tech. rep. (feb 2008).URL ∼ ee521/Material/20120927/psat-20080214.pdf [53] P. Kundur, Power System Stability and Control, McGraw-Hill, 1994.[54] J. Machowski, J. Bialek, J. Bumby, Power System Dynamics: Stability and Control, 2nd Edition, Wiley-Blackwell,2008.[55] German Federal Network Agency (Bundesnetzagentur), Monitoring report 2019, Tech. rep. (nov 2019).URL [56] K. Dvijotham, D. K. Molzahn, Error bounds on the DC power flow approximation: A convex relaxation approach,in: 2016 IEEE 55th Conference on Decision and Control (CDC), 2016, pp. 2411–2418. doi:10/d64w .[57] K. Purchala, L. Meeus, D. Van Dommelen, R. Belmans, Usefulness of DC power flow for active power flow analysis,in: IEEE Power Engineering Society General Meeting, 2005, 2005, pp. 454–459. doi:10/d4ww62 . ppendix A. AC Power Flow Problem Solved with Newton-Raphson Appendix A.1. Without Distributed Slack
Given nodal power imbalances 𝑆 𝑛 at any given snapshot for each bus 𝑛 the AC power flowequations are given by 𝑆 𝑛 = 𝑃 𝑛 + 𝑖𝑄 𝑛 = 𝑉 𝑛 𝐼 ∗𝑛 = 𝑉 𝑛 (∑ 𝑚 𝑌 𝑛𝑚 𝑉 𝑚 ) ∗ , (A.1)where 𝑉 𝑛 = |𝑉 𝑛 | 𝑒 𝑖𝜃 𝑛 is the complex voltage, whose rotating angle is taken relative to the slackbus and 𝑌 𝑛𝑚 is the bus admittance matrix, based on the branch impedances and shunt admit-tances (including those attached to buses).For the slack bus 𝑛 = 0 it is assumed |𝑉 | is given and that 𝜃 = 0 ; 𝑃 and 𝑄 are to be found.For the PV buses, 𝑃 and |𝑉 | are given; 𝑄 and 𝜃 are to be found. For the PQ buses, 𝑃 and 𝑄 aregiven; |𝑉 | and 𝜃 are to be found.Considering PV and PQ as sets of buses, then there are | PV | + 2 | PQ | real-valued equationsto solve: Re [𝑉 𝑛 (∑ 𝑚 𝑌 𝑛𝑚 𝑉 𝑚 ) ∗ ] − 𝑃 𝑛 = 0 ∀ PV ∪ PQ (A.2)Im [𝑉 𝑛 (∑ 𝑚 𝑌 𝑛𝑚 𝑉 𝑚 ) ∗ ] − 𝑄 𝑛 = 0 ∀ PQ (A.3)We need to find 𝜃 𝑛 for all PV and PQ buses and |𝑉 𝑛 | for all PQ buses.These equations 𝑓 (𝑥) = 0 are solved using the Newton-Raphson method, with the Jacobian 𝜕𝑓𝜕𝑥 = ( 𝜕𝑃𝜕𝜃 𝜕𝑃𝜕|𝑉 |𝜕𝑄𝜕𝜃 𝜕𝑄𝜕|𝑉 | ) (A.4)and the initial guesses 𝜃 𝑛 = 0 and |𝑉 𝑛 | = 1 for unknown quantities. For more details see forexample Grainer and Stevenson [51]. The total active slack power, which balances remainingmismatches of power generation and demand resulting from the AC power flow equations, isfully allocated to the slack bus. This can be a crude assumption, particularly for large networkswith a high penetration of renewables. Appendix A.2. With Distributed Slack
A better alternative is to distribute the total active slack power across all generators inproportion to their capacities (or another distribution scheme) [52]. The active power flowequations are altered toRe [𝑉 𝑛 (∑ 𝑚 𝑌 𝑛𝑚 𝑉 𝑚 ) ∗ ] − 𝑃 𝑛 − 𝑃 slack 𝛾 𝑛 = 0 ∀ PV ∪ PQ ∪ slack (A.5)where 𝑃 slack is the total slack power and 𝛾 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 theslack bus since it is now part of the distribution scheme.29e distributed slack approach extends the Jacobian by an additional row for the derivativesof the slack bus active power balance and by an additional column for the partial derivativeswith respect to 𝛾 𝜕𝑓𝜕𝑥 = ⎛⎜⎜⎜⎝ 𝜕𝑃 𝜕𝜃 𝜕𝑃 𝜕|𝑉 | 𝜕𝑃 𝜕𝛾𝜕𝑃𝜕𝜃 𝜕𝑃𝜕|𝑉 | 𝜕𝑃𝜕𝛾𝜕𝑄𝜕𝜃 𝜕𝑄𝜕|𝑉 | 𝜕𝑄𝜕𝛾 ⎞⎟⎟⎟⎠ . (A.6)If 𝛾 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 𝓁 = 𝑟 𝓁 + 𝑖𝑥 𝓁 is composed of ohmicresistance 𝑟 𝓁 and reactance 𝑥 𝓁 Likewise, the admittance 𝑦 𝓁 = 𝑔 𝓁 +𝑖𝑏 𝓁 is composed of conductance 𝑔 𝓁 and susceptance 𝑏 𝓁 Impedance and admittance are reciprocals ( 𝑦 𝓁 = 𝑧 −1𝓁 ), hence we obtainthe relations 𝑔 𝓁 + 𝑖𝑏 𝓁 = 1𝑟 𝓁 + 𝑖𝑥 𝓁 , (B.1) 𝑔 𝓁 + 𝑖𝑏 𝓁 = 𝑟 𝓁 − 𝑖𝑥 𝓁 (𝑟 𝓁 + 𝑖𝑥 𝓁 )(𝑟 𝓁 − 𝑖𝑥 𝓁 ) , (B.2) 𝑔 𝓁 + 𝑖𝑏 𝓁 = 𝑟 𝓁 − 𝑖𝑥 𝓁 𝑟 + 𝑥 . (B.3)By splitting real and imaginary parts we can express conductance and susceptance in terms ofimpedance and reactance: 𝑔 𝓁 = Re [ 𝑟 𝓁 − 𝑖𝑥 𝓁 𝑟 + 𝑥 ] = 𝑟 𝓁 𝑟 + 𝑥 , (B.4) 𝑏 𝓁 = Im [ 𝑟 𝓁 − 𝑖𝑥 𝓁 𝑟 + 𝑥 ] = 𝑥 𝓁 𝑟 + 𝑥 . (B.5) Appendix C. Additional Figures and Tables
Labels for used mathematical symbols are included in Table Appendix C. An overview ofapplied assumptions in selected equations of the present paper is given in Table C.5. Histogramsof voltage angle differences for the evaluated flow models are plotted in Figure C.14. Relativeerrors of first and second order small-angle approximations of trigonometric functions are listedin Table C.6. Information on the distribution of line lengths in the considered network aredisplayed in Figure C.15. Optimised capacities and energy generation by carrier are summarisedin Table C.7 and Table C.8. 30 ymbol Description set of buses set of lines set of cycles in cycle basis 𝑔 𝓁 conductance 𝑏 𝓁 susceptance 𝑟 𝓁 resistance 𝑥 𝓁 reactance 𝑧 𝓁 impedance 𝑦 𝓁 admittance |𝑉 𝑖 | voltage magnitude 𝜃 𝑖 voltage angle 𝑝 𝓁(𝑖,𝑗) = 𝑝 𝓁 active power flow 𝑞 𝓁(𝑖,𝑗) = 𝑞 𝓁 reactive power flow 𝜓 𝓁 power loss 𝐾 𝑖𝓁 incidence matrix 𝐶 𝓁𝑐 cycle incidence matrix 𝑝 𝑖 nodal power injection 𝑃 𝓁 line capacity 𝑃 𝓁 maximum line capacity 𝑝 𝓁 maximum per-unit flow 𝑚 𝑘 slope of loss tangent for interval 𝑘𝑎 𝑘 offset of loss tangent for interval 𝑘𝑛 number of intervals for loss tangents Table C.4: Nomenclature
Assumptions |𝑉 𝑖 | ≈ 1 𝑏 𝓁 ≫ 𝑔 𝓁 𝑥 𝓁 ≫ 𝑟 𝓁 sin(𝜃 𝑖 − 𝜃 𝑗 ) cos(𝜃 𝑖 − 𝜃 𝑗 ) 𝑞 𝓁 = 0 Formulas ≈ 𝜃 𝑖 − 𝜃 𝑗 ≈ 1 − (𝜃 𝑖 −𝜃 𝑗 ) 𝑔 𝓁 ≈ 𝑟 𝓁 𝑥 −2𝓁 (21) x 𝑏 𝓁 ≈ 𝑥 −1𝓁 (22) x 𝑝 𝓁 = 𝑏 𝓁 (𝜃 𝑖 − 𝜃 𝑗 ) (27) x x x x ∑ 𝓁 𝐶 𝓁𝑐 𝑝 𝓁 𝑏 −1𝓁 = 0 x x x x 𝑝 𝓁 = 𝑥 −1𝓁 (𝜃 𝑖 − 𝜃 𝑗 ) (28) x x x x x ∑ 𝓁 𝐶 𝓁𝑐 𝑝 𝓁 𝑥 𝓁 = 0 (31) x x x x x 𝜓 𝓁 = 2𝑔 𝓁 [1 − cos(𝜃 𝑖 − 𝜃 𝑗 )] (34) x x 𝜓 𝓁 = 2𝑔 𝓁 [1 − cos(𝑝 𝓁 𝑏 −1𝓁 )] x x x x 𝜓 𝓁 = 2𝑟 𝓁 𝑥 −2𝓁 [1 − cos(𝑝 𝓁 𝑥 𝓁 )] (35) x x x x x 𝜓 𝓁 = 𝑟 𝓁 𝑝 (38) x x x x x x Table C.5: Overview of applied assumptions in respective equations. ransport Lossy Transport Lossless
90 80 70 60 50 40 30 20 10 0 10 20 30 40 50 60 70 80
Voltage Angle Difference [Degrees] F r e q u e n c y
90 80 70 60 50 40 30 20 10 0 10 20 30 40 50 60 70 80
Voltage Angle Difference [Degrees] F r e q u e n c y
90 80 70 60 50 40 30 20 10 0 10 20 30 40 50 60 70 80
Voltage Angle Difference [Degrees] F r e q u e n c y Iterative Lossless Lossy Iterative Lossy
90 80 70 60 50 40 30 20 10 0 10 20 30 40 50 60 70 80
Voltage Angle Difference [Degrees] F r e q u e n c y
90 80 70 60 50 40 30 20 10 0 10 20 30 40 50 60 70 80
Voltage Angle Difference [Degrees] F r e q u e n c y
90 80 70 60 50 40 30 20 10 0 10 20 30 40 50 60 70 80
Voltage Angle Difference [Degrees] F r e q u e n c y Figure C.14: Distribution of voltage angle differences for the indicated flow models. 𝜃 𝑖 − 𝜃 𝑗 sin(𝜃 𝑖 − 𝜃 𝑗 ) ≈ 𝜃 𝑖 − 𝜃 𝑗 cos(𝜃 𝑖 − 𝜃 𝑗 ) ≈ 1 − (𝜃 𝑖 −𝜃 𝑗 ) ±10 ◦ ±20 ◦ ±30 ◦ ±40 ◦ Length [km] F r e q u e n c y short linesmedium-length lineslong linesdistribution of line lengths Figure C.15: Distribution of line lengths in 250 node networks by line classification according to Kundur [53]. Of alllines, 15% fall into the category of long lines, while 6% are classified as short lines. ossy Iterative IterativeCarrier Unit Transport Transport Lossless Lossless Lossy LossyOffshore Wind (AC) GW 139 137 139 137 141 135Offshore Wind (DC) GW 209 215 207 208 217 212Onshore Wind GW 328 346 343 347 401 393Run of River GW 34 34 34 34 34 34Solar GW 431 461 440 456 535 517Pumped Hydro GW 55 55 55 55 55 55Hydro Dam GW 100 100 100 100 100 100Hydrogen Storage GW 121 128 131 127 150 146Battery Storage GW 47 45 44 46 42 43HVDC Links TWkm 48 45 60 63 69 67HVAC Lines TWkm 167 160 146 143 91 103Table C.7: Optimised capacities by technology for different flow models.Lossy Iterative IterativeCarrier Unit Transport Transport Lossless Lossless Lossy LossyOffshore Wind (AC) TWh 525 515 526 515 509 504Offshore Wind (DC) TWh 935 967 922 927 983 953Onshore Wind TWh 875 908 907 901 988 980Run of River TWh 139 140 139 138 140 140Solar TWh 510 539 508 521 586 576Hydro Inflow TWh 387 387 387 387 387 387Table C.8: Energy by carrier for different flow models.ossy Iterative IterativeCarrier Unit Transport Transport Lossless Lossless Lossy LossyOffshore Wind (AC) GW 139 137 139 137 141 135Offshore Wind (DC) GW 209 215 207 208 217 212Onshore Wind GW 328 346 343 347 401 393Run of River GW 34 34 34 34 34 34Solar GW 431 461 440 456 535 517Pumped Hydro GW 55 55 55 55 55 55Hydro Dam GW 100 100 100 100 100 100Hydrogen Storage GW 121 128 131 127 150 146Battery Storage GW 47 45 44 46 42 43HVDC Links TWkm 48 45 60 63 69 67HVAC Lines TWkm 167 160 146 143 91 103Table C.7: Optimised capacities by technology for different flow models.Lossy Iterative IterativeCarrier Unit Transport Transport Lossless Lossless Lossy LossyOffshore Wind (AC) TWh 525 515 526 515 509 504Offshore Wind (DC) TWh 935 967 922 927 983 953Onshore Wind TWh 875 908 907 901 988 980Run of River TWh 139 140 139 138 140 140Solar TWh 510 539 508 521 586 576Hydro Inflow TWh 387 387 387 387 387 387Table C.8: Energy by carrier for different flow models.