The Near-Optimal Feasible Space of a Renewable Power System Model

Models for long-term investment planning of the power system typically return a single optimal solution per set of cost assumptions. However, typically there are many near-optimal alternatives that stand out due to other attractive properties like social acceptance. Understanding features that persist across many cost-efficient alternatives enhances policy advice and acknowledges structural model uncertainties. We apply the modeling-to-generate-alternatives (MGA) methodology to systematically explore the near-optimal feasible space of a completely renewable European electricity system model. While accounting for complex spatio-temporal patterns, we allow simultaneous capacity expansion of generation, storage and transmission infrastructure subject to linearized multi-period optimal power flow. Many similarly costly, but technologically diverse solutions exist. Already a cost deviation of 0.5% offers a large range of possible investments. However, either offshore or onshore wind energy along with some hydrogen storage and transmission network reinforcement are essential to keep costs within 10% of the optimum.


I. INTRODUCTION
As governments across the world are planning to increase the share of renewables, energy system modeling has become a pivotal instrument for finding cost-efficient future energy system layouts.Energy system models formulate a cost minimization problem and typically return a single optimal solution per set of input parameters (e.g.cost assumptions).
However, feasible but sub-optimal solutions may be preferable for reasons that are not captured by model formulations because they are difficult to quantify [1].Public acceptance of large infrastructure projects, such as many onshore wind turbines or transmission network expansion, ease of implementation, land-use conflicts, and regional inequality in terms of power supply are prime examples of considerations which are exogenous to most energy system models.Bypassing such issues to enable a swift decarbonization of the energy system may justify a limited cost increase.Thus, providing just a singular optimal solution per scenario underplays the degree of freedom in designing cost-efficient future energy systems.Instead, presenting multiple alternative solutions and pointing out features that persist across many near-optimal solutions can remedy the lack of certainty in energy system models [2], [3].Communicating model results as a set of alternatives helps to identify must-haves (investment decisions common to all near-optimal solutions) and mustavoids (investment decisions not part of any near-optimal solution) [4].The resulting boundary conditions can then inform political debate and support consensus building.
A common technique for determining multiple near-optimal solutions is called Modeling to Generate Alternatives (MGA) which uses the optimal solution as an anchor point to explore the surrounding decision space for maximally different solutions [1].Other methods, such as scenario analysis, global sensitivity analysis, Monte Carlo analysis and stochastic programing, that address uncertainty in energy system modeling, concern parametric uncertainty, i.e. how investment choices change as cost assumptions are varied [3], [5].Conversely, MGA explores investment flexibility for a single set of input parameters, by which MGA becomes a complement rather than a substitute for methods sweeping across the parameter space.
Evidence from previous work suggests many technologically diverse solutions exist that result in similar total system costs for a sustainable European power system [13], [14].These two studies research the sensitivity of cost input parameters or the relevance of transmission network expansion for low-cost power system layouts with 30 regions.
Preceding studies that applied MGA to long-term energy system planning problems or retrospective analyses are reviewed in Table I.This work is the first to apply MGA to a European pan-continental electricity system model which includes an adequate number of regions and operating conditions to reflect the complex spatio-temporal patterns shaping cost-efficient investment strategies in a renewable system.Furthermore, the co-optimization of generation, storage and transmission infrastructure subject to linear optimal power flow (LOPF) constraints is unique for MGA applications.
The goal of this work is to systematically explore the wide array of similarly costly but diverse technology mixes for the European power system, and derive a set of rules that must be satisfied to keep costs within pre-defined ranges.Additionally, we investigate how the extent of investment flexibility changes as we apply more ambitious greenhouse gas (GHG) emission reduction targets up to a complete decarbonization and allow varying levels of relative cost increases.
The remainder of the paper is structured as follows: Section II guides through the problem formulation, the employed variant of MGA, sources of model input data, and the experimental setup.The results are presented and discussed from different perspectives in Section III and critically appraised in Section IV.The work is concluded in Section V.

II. METHODOLOGY A. Problem formulation for long-term power system planning
The objective of the long-term power system planning model is to minimize the total annual system costs, comprising annualised 1 capital costs c * for investments at locations i in generator capacity G i,s , storage capacity H i,s , and transmission line capacities F ij,s of technology s, as well as the variable operating costs o * for the generator dispatch g i,s,t : where representative snapshots t are weighted by w t such that their total duration adds up to one year; The objective function is subject to a set of linear constraints that define limits on (i) the capacities of generation, storage and transmission infrastructure from geographical and technical potentials, (ii) the availability of variable renewable energy sources for each location and point in time derived from re-analysis weather data, and (iii) the allowable levels of greenhouse-gas emissions in addition to (iv) linearized multiperiod optimal power flow (LOPF) and (v) storage consistency equations.The optimization problem is implemented in the 1 The annuity factor   open-source modeling framework PyPSA [15].The full set of equations for this convex linear program (LP) is documented in detail in Brown et al. [15].

B. Modeling to generate alternatives (MGA)
As shown in Fig. 1, following a run of the original model, the objective function is encoded as a constraint such that the original feasible space is limited by the optimal objective value f * plus some acceptable relative cost increase .
Other than preceding studies, this paper pursues a more structured approach to MGA to span the near-optimal feasible space.The search directions are not determined by the Hop-Skip-Jump (HSJ) algorithm that seeks to minimize the weighted sum of variables of previous solutions [5], but by predefined groups of investment variables.Consequently, the new objective function becomes to minimize and maximize sums of subsets of generation, storage and transmission capacity expansion variables given the -cost constraint.
The groups can be formed by region and by technology.Examples for thought-provoking search directions are to minimize the sum of onshore wind capacity in Germany or the total volume of transmission expansion (cf.Section II-D).This process yields boundaries within which all nearoptimal solutions are contained and can be interpreted as a set of rules that must be followed to become nearly cost-optimal.
Preprint submitted to 21st Power Systems Computation Conference (PSCC 2020) In fact, by arguments of convexity, it can be shown that nearoptimal solutions exist for all values of a group's total capacity between their corresponding minima and maxima.The original problem is convex as it classifies as a linear program.Neither adding the linear -cost constraint nor introducing an auxiliary variable z that represents the sum of the group of variables alter this characteristic.Hence, for any total z ∈ [z min , z max ] a near-optimal solution exists, however not for any combination of its composites leading to this total.

C. Model input data
The exploration of the near-optimal feasible space is executed for the open model dataset PyPSA-Eur of the European power system with a spatial resolution of 100 nodes and a temporal resolution of 2 hours for a full year [16].The chosen levels of geographical and temporal aggregation reflect, at the upper end, the computational limits to calculate a large ensemble of near-optimal solutions and, at the lower end, the minimal requirements to expose transmission bottlenecks and account for spatially and temporally varying renewable potentials with sufficient detail [17].
Following a greenfield approach (with the exception of the transmission grid and hydropower installations), we allow simultaneous capacity expansion of transmission lines, HVDC links and various types of storage units and generators: solar photovoltaics, onshore wind turbines, offshore wind turbines with AC or DC grid connections, battery storage, hydrogen storage and, ultimately, open-and combined cycle gas turbines (OCGT/CCGT) as sole fossil-fueled plants.Run-of-river and pumped-hydro capacities are not extendable due to assumed geographical constraints.The corridors for HVDC links, limited to 30 GW, are taken from the TYNDP 2018.Individual transmission line capacities may be expanded continuously up to four times their current capacity, but not reduced.The dependence of line capacity expansion on line impedance is addressed in a sequential linear programing approach [18].Full details on the data model workflow of PyPSA-Eur and the underlying datasets can be found in [16].Cost assumptions are listed in Table III.Fig. 2. Maps of transmission line expansion and regional generator and storage capacities for a 100% renewable power system for the (i) optimal solution and (ii) minimal transmission volume within a 10% cost increase.

D. Experimental setup
The MGA analysis is run within a parallelized workflow for different deviations ∈ {0.5%, 1%, 2%, 3%, 4%, 5%, 7.5%, 10%} from the cost-optimal solution and for system-wide greenhouse-gas emission reduction targets of 80%, 95% and 100% compared to emission levels in 1990.This allows to follow the propagation of investment flexibility for increasing optimality tolerances and more ambitious climate protection plans.The alternative objectives are to minimize and maximize the generation capacity of all (i) wind turbines, (ii) onshore wind turbines, (iii) offshore wind turbines, (iv) solar panels, and (v) gas turbines.Moreover, we search for the minimal and maximal deployment of (vi) hydrogen storage, (vii) battery storage, and (viii) power transmission infrastructure.This setup yields 384 near-optimal solutions.On average, each problem required 6.5 hours and 31 GB of memory to solve with Gurobi.
For slacks ∈ {1%, 5%, 10%} and a 95% emission reduction target a 3-hourly resolved model is run for country-wise minima and maxima of the investment groups above, resulting in additional 1584 near-optimal solutions.On average, each problem required 3.5 hours and 22 GB of memory to solve.

A. Optimal solutions
Before delving into near-optimal solutions, we first outline the characteristics of the optimal solutions for different emission reduction levels (cf.Table II).A system optimized for a 100% emission reduction is strongly dominated by wind energy.More than half of the electricity is supplied by offshore wind installations.Onshore wind turbines provide another quarter.In contrast, photovoltaics account for only 16% of electricity generation.Strikingly, a system targeting a 95% reduction in greenhouse gases uses significantly less onshore wind generators but more solar energy in comparison to a completely decarbonized system, while keeping the share of offshore wind generation constant.Thus, for the last mile from 95% to 100% more onshore wind generation is preferred to phase out the last remaining gas-fired power plants.The total system costs scale non-linearly with more tight emission caps.Achieving an emission reduction of 95% is roughly a quarter more expensive than a reduction by 80%, while a zeroemission system is almost 50% more expensive.
Also, the map in Fig. 2-(i) shows the optimal regional distribution of the capacities of power system components for a fully renewable European power system.Generation hubs tend to form along the coasts of North, Baltic and Mediterranean Sea, whereas inland regions produce little electricity.Expectedly, solar energy is the dominant carrier in the South, while wind energy prevails close to the coasts of the North Sea and the Baltic Sea.Most grid expansion can be found in Germany, France and the United Kingdom and individual HVDC links are built with capacities of up to 30 GW.The routes and capacities of HVDC links are well correlated with the placement of wind farms.

B. The near-optimal feasible space
In this section we extremise different groups of investments in generation, storage and transmission infrastructure.As an example, Fig. 2-(ii) depicts a system that seeks to deviate from the optimum by minimizing the volume of transmission network expansion up to a total cost increase of 10%, for instance as a concession to better social acceptance.With the results particularly the NordSued link connecting Northern and Southern Germany manifests as a no-regret investment decision up to a capacity of 15 GW in the context of full decarbonization.It is one of the few persistent expansion routes.All other transmission expansion corridors are (to a significant extent) not compulsory.Missing transmission capacities can be compensated by adding storage capacity and more regionally dispersed power generation.Nevertheless, some transmission network reinforcement is indispensable to remain within the given cost bounds.These results are also broadly aligned with findings in [13], [17].
Beyond this example, the MGA results offer insights about the structure of the near-optimal space.The intent is to portray a set of technology-specific rules that must be satisfied to keep costs within pre-defined ranges .Note, that the discontinuity created by restricts the accuracy of the solution space.Fig. 3 reveals that wind generation, either onshore or offshore, is essential to set up a cost-efficient European power system for all three evaluated emission reduction levels.Whilst already a small cost increase of 0.5% yields investment flexibilities in the range of ±100 GW, even a 10% more costly solution would still require more than 500 GW of wind generation capacity for a fully renewable system: two-thirds of the optimal system layout.However, even for a zero-emission system a cost increase of just 4% enables abstaining from onshore wind power, and a 7.5% more expensive alternative can function without offshore wind farms.
The investment flexibility develops non-linearly with increasing slack levels .Even a minor deviation from the cost optimum by 0.5% creates room for maneuver in the range of ±200 GW for onshore and ±150 GW offshore wind installations, which indicates a weak tradeoff between onshore and offshore wind capacities very close to the optimum.Nonetheless, dispensing with both is not viable.Furthermore, 10% of total system costs must be spent to rule out solar panels, but already a slack of 1% allows to reduce the solar capacity by a third.
Price et al. observed in their model that investment flexibility in generation infrastructure decreased as more tight caps on carbon-dioxide emissions were imposed [6].While it is true that more ambitions climate protection plans incur more must-haves (i.e.minimum requirements of capacity), for the case-study at hand the viable ranges of marginally inferior solutions increase as more total capacity is built.
As Fig. 4 exhibits, even for a complete decarbonization of the European power system building battery storage is not essential, although they are deployed in response to e.g.minimizing network reinforcement.Conversely, once weatherindependent dispatch flexibility from gas-fired power plants is unavailable, it becomes imperative to counter-balance with hydrogen storage.The cutback of hydrogen infrastructure under these circumstances goes along with building excess generation capacities and multiplied amounts of curtailment.
The reinforcement of the transmission network becomes more pivotal the more the power system is based on renewables.Aiming for an emission reduction by 80% a 2% more expensive variant can get by without any grid reinforcement.Reducing emissions by 100% still requires some additional power transmission capacity at a 10% cost deviation.However, within this range, the transmission volume can deviate from almost double of today's network capacities to merely a marginal reinforcement (cf.Fig. 2).
MGA iterations were also applied from a country-wise perspective.Remarkably, any one country could completely forego any one generation or storage technology and remain within 5% of the cost optimum when targeting a 95% reduction in greenhouse gas emissions.In this case, neighboring countries must offset the absence of this technology.

C. Correlations
So far, the study of the near-optimal feasible space did not capture the interdependence between different system components apart from the envelopes the analysis provided for each technology.Shifting to the extremes of one technology will diminish the investment flexibility of other carriers.Fig. 5 demonstrates how diversity in capacity mixes rises if more leeway is given in terms of system costs.The striking variety in capacity totals is largely attributable to the lower capacity factors of solar compared to wind energy.
Hennen et al. suggested to present the intertwining of technologies through Pearson's correlation coefficient across all near-optimal solutions [4].Fig. 6 confirms many of the previously noted connections.Hydrogen storage substitutes gas turbines and is positively correlated with onshore and offshore wind capacity, while battery deployment rather matches with solar installations.Likewise, transmission expansion occurs in unison with onshore and offshore wind deployment.Thereby, hydrogen storage and transmission become complements for the high renewable energy scenarios.As a note of caution, it should be mentioned that CCGT and OCGT as well as AC-and DC-connected offshore wind installations have high correlations because they are grouped in the MGA iterations.

D. Distributional equity
Surveys suggest that an equal distribution of power supply is preferred among the population and may increase their willingness to participate in the energy system transformation process [19].Sasse et al. and Drechsler et al. investigate the trade-offs between least-cost and regionally equitable solutions in Switzerland and Germany by using concepts of the Lorenz curves and the Gini coefficients as equity measures [9], [19].
In the context of power, the Lorenz curve can relate the cumulative share of electricity generation of regions to their cumulative share of electricity demand as shown in Fig. 7.For more ambitious emission reduction targets, less equitable solutions are favored from a cost-perspective, however, they may not be in line with the public attitude.The Gini coefficient is the corresponding scalar measure of uniformity and is determined by multiplying the area between the Lorenz curve and the identity line by two.A Gini coefficient of 1 represents the most unequal distribution, while 0 corresponds to the situation where every region produces (in total) as much electricity as they consume.
Preceding studies have, in general, noted that the focus on wind power tends to be detrimental to a regionally balanced distribution of electricity generation [19], whereas photovoltaic power supply is a key factor for an even distribution of the power supply [9].This observation is consistent with the results of the European power system at hand.But as we consider the transmission grid and energy storage options, we can further extend on these findings.In Fig. 8 we track the equity of solutions in relation to the transmission network volume as we approach the boundaries of either technology category.Substantially more regionally equitable solutions are attainable for a limited cost-increase when diverting attention away from transmission network expansion.Utilizing less energy storage rather discourages equitable generation patterns; in the opposite direction, this is not the case for a zeroemission system.Note that the equity measures are only an observed side-effect and not the objective of a particular search direction of the MGA method.There is no guarantee that no more equitable solutions exist within the near-optimal feasible space.
IV. CRITICAL APPRAISAL This paper covers the electricity sector only.Brown et al. suggested that with an increasing coupling of energy sectors the benefit of the transmission system decreases [20].It is not far-fetched, that the near-optimal feasible space, in general, might look very different with a tightened sectoral integration.
Within the computational constraints, it is moreover desirable to further enhance the spatial and temporal resolution to better reflect curtailment caused by transmission bottlenecks and factor in extreme weather events [17].
This work further neglects parametric uncertainty.Coupling this paper's variant of MGA with Monte Carlo simulation (cf.[8]) would allow to derive a more sophisticated version of Fig. 3 and 4 with fuzzy boundaries that represent the probability with which the respective capacities of system components are contained within the near-optimal feasible space.
V. SUMMARY AND CONCLUSIONS This work sheds light on the flatness of the near-optimal feasible decision space of a power system model with European scope for ambitious climate protection targets.
An understanding of alternatives beyond the least-cost solutions is indispensable to develop robust, credible and comprehensible policy guidelines.Therefore, we derived a set of technology-specific boundary conditions that must be satisfied to keep costs within pre-defined ranges using the modeling-to-generate-alternatives methodology.These rules permit well-informed discussions around social constraints to the exploitation of renewable resources or the extent to which the power transmission network can be reinforced in discussions.
Indeed, we observed high variance in the deployment of individual system components, even for a fully renewable system.Already a minor cost deviation of 0.5% offers a multitude of technologically diverse alternatives.It is possible to dispense with onshore wind for a cost increase of 4%, and to forego solar for 10%.Nevertheless, either offshore or onshore wind energy plus at least some hydrogen storage and grid reinforcement are essential to keep costs within 10% of the optimum.

SUPPLEMENTARY MATERIAL
For supplementary material and reading the reader is referred to the source code repository on Github (https://github.com/pypsa/pypsa-eur-mga)and the documentations of PyPSA (https://pypsa.readthedocs.io,[15]) and PyPSA-Eur (https://pypsa-eur.readthedocs.io,[16]).The storage round-trip efficiency consists of charging and discharging efficiencies η 1 • η 2 .d The installed facilities are not expanded in this model and are considered to be amortized.e For all technologies a discount rate of 4% is assumed.f This relates a storage unit's energy capacity to its power capacity; it is the maximum duration the storage unit can discharge at full power capacity.

F
.N. and T.B. gratefully acknowledge funding from the Helmholtz Association under grant no.VH-NG-1352.

1 −
(1+r) −n r converts the overnight investment of an asset to annual payments considering its lifetime n and cost of capital r.

Fig. 1 .
Fig.1.Illustration of the near-optimal feasible space and the modeling to generate alternatives (MGA) methodology for a two-dimensional problem for the search-directions relating to dimension x 1 .

Fig. 3 .
Fig. 3. Solution space of renewable generation infrastructure by technology for different levels of slack and emission reduction targets.

Fig. 4 .
Fig. 4. Solution space of storage and transmission infrastructure by technology for different levels of slack and emission reduction targets.

Fig. 5 .Fig. 6 .
Fig. 5. Composition of generation and storage capacities for various nearoptimal solutions with 100% renewables.Each subplot corresponds to a slack level and an optimization sense.The labels of the bar charts indicate which group of investment variables is included in the objective.

Fig. 7 .
Fig.7.Lorenz curves for the optimal solutions at different emission reduction levels relating the cumulative share of electricity generation to the cumulative share of demand in the 100 regions of the European power system model.

Fig. 8 .
Fig. 8. Development equitable power generation measured by the Gini coefficient in relation to the capacity of the power transmission network measured in TWkm.The four plotted search directions are minimization and maximization of each, total transmission volume and storage capacity.
Gas turbines have a CO 2 emission intensity of 0.19 t/MW th .b This includes fuel costs of 21.6 e/MWh th .
a c