Designing Efficient Renewable Energy Portfolios for Optimal Coverage of European Power Demand under Transmission Constraints

Abstract

In this paper, we consider spatially and technologically diversified portfolios of renewable energy resources (RES) on a European scale. These portfolios efficiently allocate wind and solar production capacity with a view on matching load and restricting conventional generation capacity in each country. We investigate the composition of optimal portfolios taking into account the existing cross-border transmission capacity as well as future expansion scenarios. For the purposes of our analysis, we develop several multi-criteria optimization models and apply them to a rich dataset of daily electricity load, RES capacity factors, and transmission capacity constraints for 32 European countries during the period 2010–2015. By exploring different scenarios for the composition of efficient portfolios and cross-border network connectivity, we quantify the effectiveness of the resource aggregation strategy in terms of reducing the energy production deficit or surplus in each country. Empirical evidence shows that the portfolio perspective provides a solid basis for strategic planning on the future expansion of the European grid.

1. Introduction

According to the “European Energy Roadmap 2050” [1], the European Commission requires the decarbonization of the European Union (EU) energy system with the objective of reducing by 2050 carbon dioxide emissions by 80–95% compared to 1990 levels. This strategic roadmap aims to ensure reliable and economically competitive electricity produced from renewable energy sources (RES) such as wind, solar, and hydro power, accounting for at least 55% of total energy consumption, as well as from low-emission sources, such as nuclear power plants and thermal generating units equipped with carbon capture and storage technologies. The overriding challenge arising from the expected high penetration of renewables in the energy mix is dealing with the variability of generation due to the stochastic nature of wind speed and solar irradiation.

In parallel to decarbonization actions, the European Commission [2] obliges Member States to ensure a high level of security in electricity supply. This is defined as the ability of an energy system to reliably supply consumers with electricity. In light of the lack of cost-effective energy storage systems and abundant transmission capacity, much of the electricity produced must be used instantly. Hence, there is a need to efficiently balance electricity production with consumption. This is a non-trivial exercise taking into account the richness of the statistical laws governing electrical load (seasonality, short memory, persistence, etc.), which manifest themselves in different time scales.

Before the broad uptake of variable generation technologies, load matching was a standard procedure for transmission system operators (TSO), as the generation of conventional (thermal and hydro) plants could be scheduled accordingly to meet demand fluctuations. In power systems with high penetration of variable resources, load matching is severely hindered by the fact that generation largely depends on unstable and unpredictable weather conditions. This increases the risk of experiencing load mismatches and the need for using costly conventional generation capacity to close production gaps. Developing strategies for managing the RES generation risk becomes of paramount importance to the future viability of clean energy systems.

One strategy to control production intermittencies is to distribute generating capacity across distant sites or different generation technologies. This is called spatial or technological risk diversification, respectively. A stream of research studies offers empirical evidence that by interconnecting variable wind or solar generation units located in remote sites it is possible to smooth out the volatility of energy production in each area [3,4]. This is explained by the fact that in a well-diversified portfolio the inability to meet load due to the low productivity of one region (or technology) can be overcome by energy produced in another area or by a different-type generator possibly installed in the same region. In order to make optimal use of the spatial variety of clean energy resources, it is essential to consider interconnecting sites over long distances, even across country borders.

The ability to transfer energy across space is a critical factor for the creation of diversified RES portfolios. Traditionally, transmission system operators optimize the electricity supply at the national level using mostly domestic energy sources. However, for the implementation of future decarbonization plans, a crucial role is to be assigned to the European Network of Transmission System Operators for Electricity (ENTSO-E), which is expected to upgrade the interconnection of European countries in order to achieve the EU’s energy and economic policy goals [5].

With a view to achieving efficient balancing of the energy supply and demand while taking network constraints into consideration, in this paper we present a multi-objective optimization model for efficient deployment of RES generating units. We design portfolios of wind and solar power plants that service the load optimally, in the sense of jointly minimizing renewable energy unavailability and spillover in each country. In parallel to this objective, we investigate the sufficiency of the European transmission system capacity to meet the needs of the overall generation mix. For an empirical assessment of the efficiency of our methodology, we use a rich dataset of daily load measurements, wind/solar capacity factors, and transmission capacity data for 32 European countries [Austria (AT), Belgium (BE), Bulgaria (BG), Bosnia and Herzegovina (BA), Switzerland (CH), the Czech Republic (CZ), Germany (DE), Denmark (DK), Estonia (EE), Greece (EL), Spain (ES), Finland (FI), France (FR), Croatia (HR), Hungary (HU), Ireland (IE), Italy (IT), Lithuania (LT), Luxembourg (LU), Latvia (LV), Montenegro (ME), North Macedonia (MK), the Netherlands (NL), Norway (NO), Poland (PL), Portugal (PT), Romania (RO), Serbia (RS), Sweden (SE), Slovenia (SI), Slovakia (SK), and the United Kingdom (UK)] during the period 2010–2015. These countries were selected based on data availability and their level of connectivity with neighboring countries.

The rest of this paper is organized as follows. Section 2 provides a literature review of studies related to the risk management of renewable energy resources, and Section 3 discusses the contributions of this paper beyond the state-of-the-art. Section 4 develops the multi-criteria linear optimization framework proposed for the allocation of RES portfolio capacity shares, and Section 5 presents empirical results. Finally, Section 6 concludes the paper and indicates directions for further research.

2. Literature Review

The problem of managing the risk of electricity generation from variable energy sources is receiving increasing attention from researchers in various scientific areas. A significant body of literature recommends the geographical dispersion of RES power plants as a means of diversifying away the production uncertainty of a particular site (termed volumetric risk). Other studies have considered geographical diversification through the distribution of capacity among different generation technologies (e.g., wind and solar) in order to mitigate resource-specific variability.

Thomaidis et al. [4] applied Markowitz’s [6] portfolio theory to select optimal RES aggregation plans in the southern Iberian Peninsula (Spain). Based on simulated wind and solar generation data, they computed the efficient frontier of optimal harvesting plans that maximize energy supply for a certain level of generation risk, as measured by the daily variability of renewable energy production. Their empirical results indicate the potential of pooling resources to stabilize the aggregate energy supply while reducing the intensity of the harvesting plan. Out of 4474 total candidate sites, efficient portfolios only committed 34 grid nodes for solar and 42 nodes for wind power plant development. The key element in diversifying volumetric risk is the complementarity of wind and solar energy fields, which is the topic of the present study. Using meteorological simulations for the period 1980–2015, Santos-Alamillos et al. [7] applied mean variance analysis to explore different scenarios for the re-allocation of wind and solar capacity in Europe. Their empirical study identified three main geographical areas of key importance for achieving the portfolio objectives. These regions are the Iberian Peninsula (due to its strong wind and solar potential), the United Kingdom and Scandinavia (due to their rich wind resources), and the Northern Mediterranean region (characterized by good wind and solar potential). They concluded that a “better” re-distribution of RES capacity could lead to an increase of approximate of $30\%$ in the energy yield or a reduction of $37.5\%$ in production variability, as compared to the 2014 allocation of renewable generating units.

Based on measurement data from ten weather stations on the Corsican coast, Cassola et al. [8] concluded that the division of the island into three zones and the interconnection of carefully selected generation sites could lead to a reduction in the variability of the wind energy supply along with an improvement in the overall energy yield. The benefits of geographical diversification of volumetric risk were found to be substantial, despite the relatively small area of the island. In the study of Roques et al. [9], wind capacity data for five European countries (Austria, Denmark, France, Germany and Spain) were examined in a mean variance analysis framework. Their study took into account constraints on the transmission of energy across borders. The empirical results show that insufficient grid connectivity of European countries significantly reduces the efficiency of wind energy harvesting plans. Based on this finding, the authors stress the importance of improving the cross-border interconnections to ensure the future decarbonization of European power systems.

As mentioned earlier, the reliability of power systems depends on real-time balancing of energy production and demand. As both renewable energy generation and load are volatile, researchers have attempted to investigate the feasibility and cost-effectiveness of developing power systems composed exclusively of renewable energy sources. Based on wind and solar meteorological data, Rodriguez et al. [10] generated technically and economically optimal scenarios for a simplified pan-European power system composed of RES and backup conventional capacity. An optimization model was developed to identify renewable generation mixes that minimize the amount of reserve power and of installed and transmission capacity. Their cost analysis prescribed 50% of the total energy supply to be covered by RES, with the share of wind power being as high as 97%. Under the considered levels of electricity demand, their optimal plan assumed approximately 600 GW of installed wind capacity, 60 GW of solar power, 320 GW of conventional capacity, and a five-fold increase in existing transmission capacity. Jacobson et al. [11] explored the feasibility of creating generation mixes based exclusively on wind, solar, and hydro energy with the ability to match demand despite their intermittent output. Empirical evidence suggests that there is an abundance of different generation mixes that could attain energy equilibrium in 139 countries by 2050 in a cost-effective way. Energy storage in the form of heat or hydrogen and demand-side management strategies to control peak load levels are particularly important in achieving this objective. Heide et al. [12] investigated the seasonal behavior of wind and solar energy production in Europe with the goal of turning the mirror seasonality of these energy sources into an optimal generation mix with the ability to meet load efficiently, which is one of the objectives of the present study as well. Their analysis indicated that in the ideal scenario of a 100% clean power system, 55% of the total energy supply should be provided by wind farms and the rest by solar power generating units, while for lower levels of RES penetration the share of wind power should be increased.

3. Contribution to the State-of-the-Art

In this paper, we attempt to enhance the portfolio selection strategy by seeking to minimize on a daily basis the mismatch between (wind and solar) energy production and the demand for electricity. The optimization process explicitly takes into account various scenarios about the cross-border transmission capacity. By introducing load tracking as a portfolio selection target, we derive capacity allocation plans with optimal control over generation variability, especially large deviations from the target demand. Using a bi-objective mathematical programming framework, we derive the Pareto set of optimal RES generation mixes for various interconnection scenarios and levels of decision-maker aversion to energy deficits or surpluses. The relative efficiency of these sets is quantified by means of the hypervolume indicator. As a final step, we carry out a sensitivity analysis to determine the contribution of each country to the achievement of the production goals. Our empirical study is based on a panel of 32 countries and 2191 operational days. Even an experimental setup of such a moderate size results in to very large multi-objective optimization problems, including roughly 2.5 million decision variables and 1.5 million constraints. To address the computational challenges associated with the scale of the problem, we resort to advanced linear programming solvers such as MOSEK ([13]).

Our paper extends the existing literature in many directions. First of all, optimal portfolios are not exclusively selected based on the aggregate productivity profile (as in recent literature [4,7,8,9,12]) but also with reference to the load patterns observed in each country. Gearing renewable energy generation towards load often has a dramatic effect on the composition of the RES mix. Further inclusion of cross-border transmission capacity in the optimization model places additional constraints on the energy harvesting plan, pushing it towards more intense capacity allocations. For instance, empirical results show that the optimal solution in cases of limited network connectivity and poor domestic RES energy supply entails up-scaling of the installed wind and solar capacity to meet demand.

The experimental setup adopted in this paper does not explicitly take into account energy production costs, as is traditional in similar works (see, e.g., [10]). Instead, we restrict our attention to the potential of clean energy resources to maintain the stability of the European power system. In other words, our cost function measures deviations from the load level of each country. We thereby avoid cases in which the optimization process indicates efficient capacity allocation plans with an increased share of conventional (mostly thermal) power plants due simply to their lower investment and maintenance costs. This design choice allows us to fully investigate the extent to which wind and solar resources could gradually replace fossil fuels under existing and future European grid configurations. Even in the current setup, production, and investment costs are indirectly taken into account by the two objectives of the optimization program, as detailed in Section 4. In particular, the first objective function (1a) tries to minimize the need for conventional (mostly) thermal capacity to cover energy deficits created in each operational hour by inadequate renewable energy supply. The second optimization criterion (1b) attempts to restrict excessive renewable installed capacity by controlling the amount of wind and solar generation that exceeds the load and consequently has to be curtailed from the grid.

4. Methodology

4.1. Model Presentation

The bi-objective optimization model presented in this paper aims to investigate the optimal allocation of installed wind and solar power capacity in order to ensure the minimization of mismatches between produced and demanded energy. In this way, it explicitly takes transmission capacity constraints into account. The countries participating in the power system are represented as single nodes $i\in \mathcal{I}=\{1,2,\cdots ,N\}$ in a graph, while their interconnection is expressed by the graph vertices. The assumption being made is that energy deficits will be balanced by conventional power plants (Balancing Energy, BE) while surpluses will be curtailed from the grid (Curtailed Energy, CE). In particular, the objective functions are formulated as follows (see the Nomenclature at the end of this paper for a detailed account of mathematical symbols used in the optimization program):

$$\begin{array}{ccc}\hfill \underset{C_i^w,C_i^s,F_{tij},C{E}_{ti},B{E}_{ti}\in \mathbb{R}}{minimize}& \hfill & \hfill \sum _{i\in \mathcal{I}}\sum _{t\in \mathcal{T}}B{E}_{ti}\end{array}$$

(1a)

$$\begin{array}{ccc}\hfill \underset{C_i^w,C_i^s,F_{tij},C{E}_{ti},B{E}_{ti}\in \mathbb{R}}{minimize}& \hfill & \hfill \sum _{i\in \mathcal{I}}\sum _{t\in \mathcal{T}}C{E}_{ti}\end{array}$$

(1b)

The decision variables are the installed wind and solar power capacity level in each country $\left(C_i^w,C_i^s\right)$, the cross-border energy transmission $\left(F_{tij}\right)$, the energy deficit covered by conventional capacity $\left(B{E}_{ti}\right)$, and the energy surplus cut off from the grid $\left(C{E}_{ti}\right)$, at each time period $t\in \mathcal{T}=\{1,2,\dots ,T\}$. The convention we adopt regarding the cross-border transmission variables is $F_{tij}<0$ for a net import to node i from node j at time t and $F_{tij}>0$ for a net export from node i to node j at time t, where the indices $i.j$ take values in the set $\mathcal{I}=\{1,2,\dots ,N\}$ and $j>i$. The optimization constraints are presented in Equations (1c) & (1g)–(1i).

To begin with, Equation (1c) expresses the energy equilibrium that should be met at each node i and time period t. This equation states that for each country i and at each time period t, the energy production from wind and solar generating units $(W{G}_{ti}+S{G}_{ti})$ and the balancing energy $\left(B{E}_{ti}\right)$ should be equal to the load $\left(L_{ti}\right)$, the cross-border energy transfer $\left(E{T}_{ti}\right)$, and the energy cut off from the grid $\left(C{E}_{ti}\right)$. The energy produced by RES is expressed as the product of the installed capacity multiplied by the capacity factor of the corresponding energy source $(C{F}_{ti}^w,C{F}_{ti}^s)$ at a given time index t; see Equations (1d) & (1e).

$$\begin{array}{c}\hfill W{G}_{ti}+S{G}_{ti}+B{E}_{ti}=L_{ti}+E{T}_{ti}+C{E}_{ti},\forall i\in \mathcal{I},\forall t\in \mathcal{T}\end{array}$$

(1c)

where

$$\begin{array}{c}\hfill W{G}_{ti}=C_i^{w}C{F}_{ti}^w,\forall i\in \mathcal{I},\forall t\in \mathcal{T}\end{array}$$

(1d)

$$\begin{array}{c}\hfill S{G}_{ti}=C_i^{s}C{F}_{ti}^s,\forall i\in \mathcal{I},\forall t\in \mathcal{T}\end{array}$$

(1e)

$$\begin{array}{c}\hfill E{T}_{ti}=\sum _{j\in \mathcal{I};j>i}{F}_{tij}-\sum _{j\in \mathcal{I};j<i}{F}_{tji},\forall i\in \mathcal{I},t\in \mathcal{T}\end{array}$$

(1f)

Our optimization problem includes additional constraints associated with energy transmission. Equation (1g) imposes energy conservation over the entire grid, i.e., at any time t, total energy exports should be equal to total imports. Constraint (1h) represents the bounds due to limitations in the capacity of the transmission network that energy flows should satisfy in each direction.

$$\begin{array}{c}\hfill \sum _{i,j\in \mathcal{I};j>i}{F}_{tij}=0,\forall t\in \mathcal{T}\end{array}$$

(1g)

$$\begin{array}{c}\hfill -NT{C}_{ij}^{-}\le F_{tij}\le NT{C}_{ij}^{+},\forall i,j\in \mathcal{I};j>i,\forall t\in \mathcal{T}\end{array}$$

(1h)

Finally, constraint (1i) imposes non-negativity of the installed RES capacity, the energy deficit, and the energy surplus.

$$\begin{array}{c}\hfill C_i^w,C_i^s,B{E}_{ti},C{E}_{ti}\ge 0,\forall i\in \mathcal{I},\forall t\in \mathcal{T}\end{array}$$

(1i)

4.2. Practical Considerations

The optimization model was programmed in the CVX ([14]) and MATLAB R2021a environments and equipped with a MOSEK ([13]) solver. Experiments were performed on the High Performance Computing Infrastructure and Resources grid at Aristotle University of Thessaloniki (AUTh). Pareto-optimal sets were derived by solving an equivalent single objective formulation in which each objective function was weighted by a coefficient $w_i$, $i=1,2$ (assuming $w_1,w_2\ge 0$ and $w_1+w_2=1$). Due to their complementarity, the weights $w_1,w_2$ express the relative preference (or aversion) of the policymaker with respect to the occurrence of energy deficits or surpluses, e.g., a high value of $w_1$ implies that the decision-maker prioritizes minimizing energy deficits, assigning lesser weight to the existence of redundancies. To generate the Pareto front, we set $w_1=w$ and $w_2=1-w$ for a new variable $0\le w\le 1$ and solved a sequence of single objective optimization problems for selected values of w in an equidistant partition of the $[0,1]$ real interval (101 points in total). Starting from the extreme scenario of absolute aversion to energy deficits ($w_1=1,w_2=0$) and moving to the case of full aversion to energy surpluses ($w_1=0,w_2=1$) with a step of $0.01$, we were able to to obtain a fine representation of the Pareto-optimal set.

In order to investigate the benefits of horizontal (geographical) and vertical (technological) diversification of RES-generating portfolios, we assumed three interconnection scenarios:

1.

No cross-border connection (No Interconnection);

2.

Cross-border connection with constrained transmission capacity

(Constrained Interconnection);

3.

Cross-border connection with unconstrained transmission capacity

(Unconstrained Interconnection).

4.3. Sensitivity Analysis

The purpose of the sensitivity analysis was to determine the contribution of each country to the efficiency of the generation portfolio. Among the many methods proposed in the literature for measuring“efficiency”, we picked one that is better suited to the concept of a generation mix. Intuitively, if the wind or solar resources of a country $i=1,2,\dots ,N$ do not contribute very much to the fulfillment of the portfolio targets, the Pareto-optimal set resulting from a reduced size portfolio (excluding country i from the generation mix) should not be significantly inferior to the full Pareto set (including all countries). For the quantification of the dominance of a set of Pareto-optimal solutions, we used the hypervolume indicator [15], which is a measure of the hypervolume of the region of the objective functions space which includes all feasible solutions dominated by the elements of the Pareto set with respect to a reference point. Figure 1a illustrates the calculation of the hypervolume indicator in the case of a bi-objective optimization problem, i.e., where the goal is to minimize both objectives. Because the space of feasible solutions $\mathcal{S}$ is a subset of ${\mathbb{R}}^2$, the hypervolume indicator measures the area of the shaded region included between the Pareto set and the upper and right edge of the rectangle. The position and spread of the rectangle is determined by the choice of the reference point, which by definition is the feasible solution dominated by all points of the Pareto set. As shown in Figure 1a, the coordinates of the reference point are made up of the worst value of each objective function, hence the term nadir point. This seems to be a reasonable metric for comparing the efficiency of all Pareto-optimal solutions, as picking a reference point closer to the front would result in the calculation of a hyperbolic-shaped index omitting the contribution of elements of the Pareto set. The hypervolume indicator can be easily extended to higher-dimensional spaces resulting from the combination of multiple criteria in the formulation of the optimization problem.

To compare the efficiency of two fronts (which is the goal of sensitivity analysis), it is important to choose a common reference point. As shown in Figure 1b, in all experiments we placed the reference point according to the worst value attained by any of the objective functions in the fronts under comparison. In this way, we make sure that all segments of the fronts are taken into account in the calculation of the hypervolume indicator; when a Pareto set clearly dominates another (by being located to the left), its hypervolume indicator is always higher. The latter property can be seen in Figure 1b, where the area marked by the hypervolume metric in the Pareto set ‘pf2’ is smaller than that of ‘pf1’.

To measure the relative efficiency loss when excluding one country from the generation mix, we calculated the percentage change in the hypervolume indicator, where we used the value of the index corresponding to the full generation mix as a starting point (including all countries). When excluding one country from the generation mix, the Pareto front can be expected to move to the right. As seen from the example of Figure 1b, this results in decreased efficiency and a negative sign for the percentage change of the hypervolume indicator.

5. Empirical Study

5.1. Sample Data

5.1.1. Load Data

Load data were derived from the ENTSO-E’s website [16], and correspond to the total hourly electricity demand (in GWh) in each of the reference countries. For the purpose of reducing computational requirements, hourly load observations were consolidated into daily ones, forming a data panel with a cross-sectional dimension that includes the 32 countries of the grid and the temporal dimension the 2191 operational days in the years 2010–2015. In Figure 2, we provide an overview of the statistical features of this dataset; the maps presented in this paper were created using data from the World Food Programme database [17]. It is evident that the average daily load for the majority of countries is below 500 GWh. Nonetheless, there exist countries, such as Germany, France, Italy, Spain and the United Kingdom, with considerably higher electricity consumption. This can be attributed to the increased population and economic activity in these countries. Furthermore, it can be observed that the variability of daily consumption in these countries is more pronounced. Panel (d) of Figure 2 illustrates the seasonal signature of the aggregate load in the examined pool of countries. The electricity demand in summer months is significantly lower than in the winter.

5.1.2. RES Capacity Factor Data

Wind and solar energy capacity factors were derived from the database of the EMHIRES (European Meteorological derived HIgh-resolution Renewable Energy Sources generation time series ([18,19]) project. EMHIRES provides estimates of the wind and solar energy production attained by each country on average over designated hourly time intervals per GW of installed capacity. Capacity factors are estimated based on high-quality NASA atmospheric reanalysis data. Reanalysis calibrates short-run forecasts generated by physics-based meteorological models against the latest weather observations to consistently reproduce atmospheric conditions. In this way, it provides a much more rich description of wind and solar energy fields across space and time than the thin historical production records from sparsely distributed power plants.. The capacity factors range between 0 and 1, the latter meaning that weather conditions are favorable enough that the power plant production is near its nominal rating. For the purpose of our analysis, hourly data were consolidated into daily time series, the main statistical features of which are analyzed in Figure 3 and Figure 4.

Beginning with wind data, Figure 3 shows a “no free lunch” feature of aeolic energy production. Countries such as Spain, Austria, Ireland, and the United Kingdom are very productive, although the high expected energy yield (as measured by the average capacity factor) is typically associated with high generation risk (proxied by the standard deviation of daily capacity factors). The generation risk profile in the case of solar energy technologies is somewhat different, as shown in Figure 4. There exist countries, such as Belgium and the Netherlands, with unfavorable risk–yield trade-offs (i.e., daily variability is disproportionately high compared to expected production), while other countries, such as Spain, the average energy delivery largely counterbalances the variability of the resource.

Comparing Figure 3 and Figure 4, many differences can be seen in the production profiles of the two resources. Both energy yield and generation risk are considerably higher in the case of wind. In addition, seasonality seems to manifest itself in mirror patterns. As Panel (d) of Figure 3 and Figure 4 suggests, the aggregate wind portfolio is more productive in winter months, albeit with higher levels of production uncertainty. Solar energy production reaches its peak in the summer season, and the range of daily variability is relatively constant across calendar months. The boxplots showcase a complementarity property between the two resources, which has previously been documented elsewhere in studies assuming various temporal and spatial scales ([4,12,20,21,22,23,24]). This is revisited in Section 5.3.2, where we discuss optimal energy harvesting plans combining both resources.

5.1.3. Cross-Border Net Transmission Capacity Data

Data on cross-border electricity transmission capacity were obtained from the ENTSO-E transparency platform ([25,26]). ENTSO-E tables report peak cross-border electricity flows that occurred between the summer season of 2010 and the winter season of 2010–2011. By averaging the observed flows during the summer and winter season, we derived an estimate of the maximum allowed amount of electrical power that can be transmitted across borders per country pair and direction of flow. The results of this analysis are depicted in Figure 5. Panel (a) maps the calculated transmission capacity for each country pair, while Panel (b) provides the exact data. Each cell $(i,j)$ of the table corresponds to the maximum amount of energy that is allowed to flow from country i to j, where i is the row index.

5.2. Pareto Optimal Sets

For all interconnection scenarios, we depict the optimal wind and solar portfolios in a space of three performance metrics, as shown in Figure 6a. Apart from the portfolio scores with respect to the two criteria used in the formulation of the optimization programs (balancing and curtailed energy), we show the total installed capacity of optimal RES generation plans. We experimented with the inclusion of the total installed RES capacity as a third objective in the optimization problem on top of the minimization of the need for balancing energy and power curtailments. Experimental results (not presented here) show that the third criterion does not contradict the rest, as is apparent from Figure 6a. Due to this property, we chose to omit total capacity and restrict ourselves to the bi-objective optimization framework.

From an inspection of the Pareto fronts of Figure 6c, it is apparent that technological and geographical diversification of the RES portfolio both lead to an increase in the efficiency of the generation mix. For each interconnection scenario, the Pareto set of the generation plans combining wind and solar resources dominates the frontier of wind-only portfolios. An interconnected array of wind and solar power plants attains a reduced level of energy deficits or surpluses compared to energy harvesting plans focusing on wind resources only. This property holds for all connectivity scenarios, although the efficiency gains of the mixed portfolio are more apparent with an increase in transmission capacity.

Figure 6b shows that if policymakers are willing to accept moderate levels of energy deficits in all countries, they can attain the portfolio goals by linearly increasing the installed capacity of the generation plan. When the degree of aversion towards the occurrence of deficit events is high, the need for expanding RES generation capacity increases exponentially, the effect being more pronounced in scenarios of poor cross-border connectivity. This is explained by the fact that when the ability to transfer energy across borders is limited, each country has to rest on its own resources and upgrade the RES installed capacity to meet the energy deficit goal. Pooling wind and solar resources reduces the need for capacity expansion. Because the Pareto front of the composite portfolios in the case of unlimited interconnection dominates the rest, we conclude that efficient exploitation of clean energy resources requires investing in additional transmission capacity. Similar conclusions can be drawn for the relationship between total installed capacity and the amount of energy spillover. The better the connectivity of a country, the easier it is for the system operator to transfer excess energy production across border, reducing the need for curtailments. As expected, the total RES capacity is positively related to the amount of energy that is cut off from the grid. This implies, among other things, that the total capacity should be reduced if policymakers seek to minimize power curtailments.

5.3. Composition of Optimal Portfolios

In this section, we present the composition of optimal power portfolios in terms of installed RES capacity and the energy flows they entail. The presentation focuses on a set of reference portfolios emerging from the solution of the bi-objective optimization problem with equal weights placed on both criteria ($w_1=w_2=0.5$). These portfolios are of particular interest, as they balance the need for conventional capacity (to close energy deficit gaps) and the amount of RES production that should be curtailed from the grid.

5.3.1. The Case of Wind Energy

Figure 7 illustrates the optimal wind portfolio for three network connectivity scenarios. Figure 7a covers the extreme case of no interconnection, which, albeit fictitious, is useful for quantifying the benefits of expanding the grid; Figure 7b refers to the current state of the European grid, and Figure 7c shows the full connectivity scenario, which points to the future. The capacity allocated to each country (measured in GWs) can be read from the colorbars. A supplementary vector field illustrates the direction and the amount of energy that is exchanged across borders during the study period. The width of each arrow is set according to the percentage of sample days at which country i exports energy to country j.

In the first two cross-border connection scenarios, shown in Figure 7a,b, the installed capacity generally follows the average electricity demand in each country. The implications of this are direct. Countries with relatively high average loads, such as Germany, France, and Italy, should expand renewable capacity to make up for instances in which the transmission network is congested and no adequate energy can be imported from countries with an abundance of RES generation at that time. All countries productively contribute to these energy mixes; even in the case of constrained interconnection they are required to cover a large part of their load, as the potential for importing energy is limited. This feature of the capacity allocation plan radically changes in the case of unrestricted transmission capacity, as covered by Figure 7c. In this scenario, only a fraction of countries participate in the generation mix and the rest benefit from imported energy. “Active” countries are selected based mainly on two criteria: low level of RES generation variability (as in the case of Norway and Portugal), and high average capacity factor (as in the case of Ireland). Greece is an important constituent of the generation plan in this “utopic” connectivity scenario.

It is interesting to look into the optimal energy flows in the second interconnection scenario, which corresponds to the current state of the transmission network. Figure 7b shows that countries with relatively high loads (such as Germany and France) tend to be net importers, while countries with lower demand for electricity and rich wind resources, such as Portugal and Ireland, tend to be net exporters of wind energy.

Another aspect of the optimal solution is the values of the energy balance variables for each country in the mix. The area plots of Figure 8 show the time evolution of the three sources of energy in each country: wind, balancing (offered by conventional power plants), and imports. The amount of energy that country i imports at day t is calculated as $IM{P}_{ti}=-{\sum }_{j\in \mathcal{I};j>i}min(F_{tij},0)+{\sum }_{j\in \mathcal{I};j<i}max(F_{tji},0)$; similarly, the energy exported from country i is equal to $EX{P}_{ti}={\sum }_{j\in \mathcal{I};j>i}max(F_{tij},0)-{\sum }_{j\in \mathcal{I};j<i}min(F_{tji},0)$. To facilitate comparison, we scale in each time instance the depicted indices by the total available energy, which is the sum of the total domestic generation and the imports. Figure 9 shows the energy usage, depicting the time series of electrical load, curtailed energy, and exports for each country resulting from application of the optimal generation plan.

Figure 8 shows that all countries except for Luxembourg and Latvia are able to largely cover a part of their electricity needs from domestic wind resources. In the majority of countries, the need for conventional capacity to close energy gaps has a strong seasonal signature. In summer months, when wind energy productivity tends to be lower, energy deficit episodes are more frequent and intense, and the need for reserve conventional capacity is increased. In the Balkan countries, imports make up a large share of the domestic generation mix due to the fact that electricity consumption in these countries is relatively low and existing transmission capacity is sufficient to serve a significant part of the load. In contrast, in Germany and France imports are not a significant energy source, as the average load in these states is so high that imported energy is only capable of covering a small portion of the demand.

Two distinctive features of the optimal generation plan involving wind resources are worth mentioning. As the area plots of Figure 9 show, the wind energy portfolio is efficient overall in terms of load tracking, as only a small percentage of the total energy production in each country is curtailed. Austria, Switzerland, Denmark, the Czech Republic, and the Netherlands are exporting countries, mainly transferring energy to Germany to cover the increased demand for electricity. Estonia, Croatia, and Montenegro are heavy exporters as well.

5.3.2. Pooling of Resources

Figure 10 illustrates the structure of the optimal portfolios and the resulting energy flows for the case in which both resources (wind and solar) are utilized. The top row of displays provides spatial information on the wind leg of the portfolio, whereas the bottom row analyses the solar energy share. As can be seen, the spatial allocation of wind capacity bears great resemblance to the portfolios examined in Section 5.3.1. However, the total share of wind capacity in the generation mix is low, as the optimization algorithm had a more effective option for matching load through the pooling of resources. The general recommendation of the optimal solution in the composite case is that countries should invest more into solar generating capacity in search of greater reliability and smoothness of the production profile.

Combining wind and solar resources results in a more effective generation plan, as clearly illustrated by Figure 11. Compared to pure wind portfolios, the balancing energy needs are significantly reduced in all countries. This is a consequence of the complementarity that characterizes the production profiles of wind and solar power plants. In fact, solar energy mainly services the load in the summer season when winds are weaker. This results in a simultaneous reduction in energy deficits and the total installed capacity, increasing the efficiency of the composite generation plan.

5.4. Sensitivity Analysis

This section presents the results of our sensitivity analysis of the optimal generating plans. Among the many aspects that a sensitivity analysis can address, we are particularly interested here in quantifying the contribution of each member country to the fulfillment of the portfolio objectives. To address this issue, we calculated 32 new Pareto-optimal sets by sequentially removing each country from the panel. The derived Pareto sets are compared to the global solution and efficiency losses are measured with respect to the rate of change in the corresponding hypervolume indicators (see Section 4.3). In order for the comparison to be meaningful, for each state excluded from the panel we removed the corresponding coefficients from the objective functions. In other words, we investigated whether the situation in the remaining countries changes dramatically when the reference state does not participate in the generation mix. Obviously, such an exclusion leads to a reduction in the value of the hypervolume indicator, which is consistent with the overall deterioration in the efficiency of the capacity allocation plan. In the following discussion, we restrict our attention to the composite portfolio alternative, which was shown to be superior in the case of restricted network connectivity.

Figure 12 shows the percentage change in the efficiency of the Pareto set with the removal of each country from the production mix. Overall, countries with exporting capabilities, such as France, Switzerland, Austria, and Sweden, cause a larger percentage drop (in absolute terms) in the efficiency of the generation network. This means that their wind and solar resources are valuable for keeping the energy balance in countries of close (network) proximity. In contrast, states with high load levels, such as Germany, Spain, and Italy, cause a relatively smaller decrease in the overall generation efficiency. Their resources are mainly used for smoothing out domestic imbalances, and when these assets are removed from the mix they can be substituted by wind and solar energy provided by other nodes of the grid.

6. Conclusions and Future Research

The purpose of this paper was to explore the extent to which clean energy resources (wind and solar) can serve the need for electricity on a continental scale and become a viable alternative to fossil fuels. Using a panel of 32 European countries, we derived optimal allocation plans for RES capacity that take into account the statistical features of domestic load as well as the possibility of transmitting electricity across borders. The bi-objective optimization model developed in this paper highlights the importance of creating spatially and technologically diversified portfolios to address increasing load under the current configuration of the European power grid. Specifically, the results of our empirical study suggest that investments in RES capacity must unavoidably be proportional to the level of domestic electricity demand due to limitations in transmission capacity, regardless of the richness of resources in each country. However, it is clear that expanding transmission capacity and mixing wind and solar resources could help system operators to improve the overall efficiency of the generation plan. This strategy has tangible benefits in terms of obviating unnecessarily large installation of RES generating units, reducing the spillover of clean energy, and minimizing the need for conventional (mostly thermal) capacity. Above all, our study stresses the fact that it is essential in order to meet the European Commission’s objectives regarding the reduction of pollutant emissions [1] to both upscale wind and solar generating capacity in each country and to invest strategically in a pan-European cooperative for the sharing of clean energy resources.

This study can be extended in many possible directions. One challenge is to consider other forms of renewable energy, such as hydro or biomass, as part of the generation mix. Although in principle these generation technologies are dispatchable and could be used for peak shaving, they are not a viable alternative for each country. Furthermore, the incorporation of hydropower stations in the optimization framework is not a trivial task, as in each time iteration it is necessary to take into account the available water resources based on the limited reservoir capacity of each stations. It might be interesting to consider energy storage methods (batteries, pumped storage power plants) that could shift possible surpluses created by extensive RES production in time and space during periods of low domestic demand and network congestion. Nonetheless, the adoption of these technologies on a grid scale is challenging from a both technical and an economic point of view.

The results of this and similar studies on renewable energy harvesting are unavoidably sensitive to the models and techniques used to quantify wind and solar resources in each country. An assumption made in this paper is that past capacity factor realizations are reflective of the future profile of wind and solar energy generation. In this regard, the outcomes of climatology research could be quite insightful in terms of providing typical patterns of the wind or solar resources in each country as well as in identifying possible “structural breaks” that might be infused in the underlying probability laws due to global climate change.