• Undesirable output

To capture the dependence between the generation of electricity based on fossil sources and the CO2 emissions that they incur, we discriminated between energy from clean generation sources and energy generated from fossil fuels. This strategy allowed us to capture the proportional variations between the non-separable desirable output—fossil-generation—and the CO2 emissions, known in the DEA literature as the assumption of weak disposability and introduced by Faere et al. [9].

As an undesirable output, we used the CO2 emissions generated by the electricity generation activity, measured in MTm. Due to the availability of information, we used observed data for 2016 and 2017, while we estimated the data for the period 2000–2015 from the CO2 emissions from electricity and heat production in each country in 2016 and the electricity generated from fossil sources in the same year. For Guyana, we created an estimate for the entire period using the regression from other countries because of the lack of information regarding CO2 emissions from electricity and heat production for this country. We think that this measure represented a good proxy for CO2 emissions generated by the electricity sector considering the high R-square of the regression of 0.9886.

• Inputs

We incorporated three inputs: the gross domestic product (GDP) per capita, the installed capacity of non-fossil generation sources and the installed capacity of fossil generation sources. These last two variables were also used to capture the inter-temporal dependence of electricity generation, entering the model as link variables.

The GDP of each country has been used in different studies within the DEA methodology as a desirable output [22,32–35]. We believe that, within the productive process presented by each DMU, one of the main inputs is the GDP per capita, in the sense that high-income countries can benefit from greater technological innovation and make greater efforts in R&D to improve energy efficiency [36]. This decision to use GDP per capita as an input was also based on studies that have evaluated the causality between electricity generation and economic growth, finding a unidirectional relationship for economic growth and electricity generation [37–39]. This indicator was measured in billions \$2015 PPP. In addition, Dyson et al. [25] recommended the use of type of variables to control the lack of homogeneity in the units tested.

The installed capacity has been used in different studies as an input for electricity generation. For example, Yunos and Hawdon [21] and Li et al. [24] used the installed capacity of fossil sources as an input without taking into account the different non-fossil sources of generation. In addition, Whiteman [20], Chen et al. [36] and Dogan and Tugcu [19] used the installed capacity of non-fossil sources in a disaggregated manner. This variable is measured in GW.

• Link variables

In this research, we considered that there is a dynamic component within the electricity generation sector that depends on the installed capacity for the different generation sources. The main reasons for this is as follows: (1) the level of installed capacity available for each country in year *t* is determined by the installed capacity level in the immediately preceding term, *t* − 1 [28]; (2) it can be assumed that the installed capacity in each country is a quasi-fixed input, and because of the large investment that this entails, it makes it difficult to adjust this to optimum levels every year [40]; and (3) the level of installed capacity in a year *t* has impacts on the generation in year *t* + 1, taking into account the fact that this input also functions as a warehouse, either for electricity, through batteries, or of potential generation—for example, the electricity power output that depends on the water flow in the penstock and the water accumulated in the reservoir [41].

#### *3.3. Model Approach*

To measure the efficiency of electricity generation in the 24 countries mentioned, we propose a dynamic slack-based DEA model and assume a constant return to scale (CRS). The model is based on the dynamic slack-based DEA model proposed by Tone and Tsutsui [30], which has been expanded to include undesirable outputs and to capture the assumption of weak disposability between the generation of electricity from fossil sources and emissions of CO2, presented in Tone [42].

The model structure is represented in Figure 1. We observe *n* countries over *T* terms. At each term *t*, each country uses its respective inputs (GDP, non-fossil-fuel installed capacity and fossil-fuel installed capacity) to produce the desirable output (non-fossil and fossil electricity generation). A variation in fossil generation implies a proportional variation of the undesirable output (CO2). The link variables connect consecutive terms (1, ... , *t* − 1, *t*, *t* + 1, ... , T); in our model, the level of installed capacity available for each country in term *t* determines the installed capacity in the immediately succeeding term, *t* + 1, and is determined by the installed capacity in the immediately preceding term, *t* − 1.

**Figure 1.** Model structure.

The dynamic DEA model defines a production possibility set for each term based on the observed output and input values of the DMUs in each term *t*.

Following Zhou and Liu [43], the maximization of the desirable output and minimization of the undesirable output can be reached with an additive DEA model with the next objective function:

$$\max \; SDO\\_NF\_{\ell t} + SDO\\_F\_{\ell t} + SILO\\_CO\_{2\ell t} \,\tag{3}$$

However, continuing to follow Zhou and Liu [43], this model cannot produce efficiency measures directly; thus, output-oriented efficiency must be measured for each year, and the overall efficiency measure for *DMUo* must be calculated while replacing the slacks in the following equations:

$$\tau\_{at}^{\*} = \frac{1}{1 + \frac{1}{3} \left( \frac{\text{SDO\\_NF}\_{it}}{\text{IDO\\_NF}\_{it}} + \frac{\text{SDO\\_F}\_{it}}{\text{IDO\\_F}\_{it}} + \frac{\text{SLO\\_CO}\_{i\text{t}}}{\text{IDO\\_CO}\_{i\text{t}}} \right)}; \ t = 2000, \ \dots, \ 2016 \tag{4}$$

$$
\tau\_o^\* = \frac{1}{17} \sum\_{t=\overline{2000}}^{2016} \tau\_{ot}^\* \tag{5}
$$

where a country *o* will be globally efficient (τ∗ *<sup>o</sup>* = 1) if and only if *SDO*\_*NOFot* = *SDO*\_*Fot* = *SUO*\_*CO*2*ot* = 0; ∀ *t* = 2000, ... , 2016. In other words, the country will be efficient throughout the period if it is efficient in each year. It should be noted that the evaluation of efficiency for the last year is lost because temporary interdependence is introduced into the proposed model. We chose an *Energies* **2020**, *13*, 6624

output-oriented measure of efficiency as we aimed to evaluate, given the set of inputs, if there were deficiencies in the desirable outputs or excesses in the undesirable output.

The production possibility set for the *DMUo* (country *o*, with *o* = 1, ... , 24) under a CRS is defined by Equations (1)–(9).

• Equations (1)–(3) are associated with constraints on inputs:

$$GDP\_{ot} = \sum\_{j=1}^{24} GDP\_{jt} \lambda\_j^t + S\_- GDP\_{ot} \tag{6}$$

The *GDP* of country "*o*" must be less than or equal to the linear combination of the *GDP* of all countries in each term *t*. The difference is the slack variable of the *GDP* of country *o* in term *t* (*S\_GDP*).

$$I\mathcal{C}\\_NF\_{ot} = \sum\_{j=1}^{24} I\mathcal{C}\\_NF\_{jt}\lambda\_j^t + S\mathcal{I}\mathcal{C}\\_NF\_{ot} \tag{7}$$

The non-fossil installed capacity (*IC\_NF*) of country *o* must be less than or equal to the linear combination of the non-fossil installed capacity of all countries in each term *t*. The difference is the slack variable of the non-fossil installed capacity of country *o* in term *t* (*SIC\_NF*).

$$I\mathcal{C}\\_F\_{\text{ot}} = \sum\_{j=1}^{24} I\mathcal{C}\\_F\_{j\text{i}}\lambda^t\_j + S\mathcal{I}\mathcal{C}\\_F\_{\text{ot}} \tag{8}$$

The fossil installed capacity (*IC\_F*) of country *o* must be less than or equal to the linear combination of the fossil installed capacity of all countries in each term *t*. The difference is the slack variable of the fossil installed capacity of country *o* in term *t* (*SIC\_F*).

The equation associated with the constraint on the separable desirable output is as follows:

$$DO\\_NF\_{ot} = \sum\_{j=1}^{24} DO\\_NF\_{jt}\lambda\_j^t - SDO\\_NF\_{ot}.\tag{9}$$

The electricity generation from non-fossil sources (*DO\_NF*) of country *o* must be greater than or equal to the linear combination of electricity generation from the non-fossil capacity of all countries in each term *t*. The difference is the slack variable of the electricity generation from non-fossil capacity of country *o* in term *t* (*SDO\_NF*).

Equations (5) and (6) capture the assumption of weak disposability between the electricity generation from fossil sources and the emission of CO2. A variation of the non-separable desirable output is designated by α*tDO*\_*Fot* and is accompanied by the same proportional variation in the non-separable undesirable output designated by α*tUO*\_*CO*2*ot*. Equation (5) represents the constraint on the non-separable desirable output. Equation (6) is the constraint of the non-separable undesirable output:

$$
\alpha\_t D O\\_F\_{ot} = \sum\_{j=1}^{24} D O\\_F\_{jt} \lambda\_j^t - S D O\\_F\_{ot} \tag{10}
$$

The electricity generation from fossil sources (*DO\_F*) of country *o* must be greater than or equal to the linear combination of electricity generation from the fossil capacity of all countries in each term *t*. The difference is the slack variable of the electricity generation from the fossil capacity of country *o* in term *t* (*SDO\_F*).

$$
tau\\_CO\_{\\_CO\_{\\_2ot}} = \sum\_{j=1}^{24} \,\!\!\/ MO\_{CO\_{24}} \Lambda\_j^t + \,\!\!\/ SO\\_CO\_{2ot} \tag{11}
$$

*Energies* **2020**, *13*, 6624

The CO2 emissions caused by the electricity generation activity (*UO\_CO2*) of country *o* must be less than or equal to the linear combination of CO2 of all countries in each term *t*. The difference is the slack variable of the CO2 of country *o* in term *t* (*SUO\_CO2*).

Two carry-over equations that guarantee the continuity of the link flows between the terms *t* and *t* + 1 are as follows:

$$\sum\_{j=1}^{24} IC\\_N F\_{jt} \lambda\_j^t = \sum\_{j=1}^{24} IC\\_N F\_{jt} \lambda\_j^{t+1}; \ t = 2000, \dots, \ 2016 \tag{12}$$

$$\sum\_{j=1}^{n} IC\\_F\_{jt} \lambda\_j^t = \sum\_{j=1}^{n} IC\\_F\_{jt} \lambda\_j^{t+1}; \ t = 2000, \dots, \ 2016\tag{13}$$

The installed capacity in non-fossil and fossil sources in each term *t* is determined by the respective installed capacity in term *t* − 1.

The assumption of a CRS in the production possibility set is captured by the following condition:

$$\sum\_{j=1}^{24} \lambda\_j^t \ge 0 \tag{14}$$

Additionally, non-negativity conditions are as follows:

$$\text{S\\_GDP}\_{\text{l\\_}} \text{ SDO\\_NF}\_{\text{l\\_}} \text{ SDO\\_F}\_{\text{l\\_}} \text{SIC\\_NF}\_{\text{l\\_}} \text{SIC\\_F}\_{\text{l\\_}} \text{SIC\\_CO}\_{\text{l\\_}} \ge 0 \tag{15}$$

We test the CRS assumption using the following test introduced by Banker [44]:

$$F\_{\bar{j}} = \frac{\sum\_{j=1}^{N} \left(\Theta\_{\bar{j}}^{\text{CCR}} - 1\right)^2}{\sum\_{j=1}^{N} \left(\Theta\_{\bar{j}}^{\text{BCC}} - 1\right)^2} \tag{16}$$

where θˆ*CCR* is the calculated efficiency measure that assumes a CRS, as proposed by Charnes et al. [7], and θˆ*BCC* is the calculated efficiency measure that assumes a variable return to scale (VRS), as proposed by Banker et al. [45]. This calculated value is asymptotically F-distributed with (*N*, *N*) degrees of freedom. If not rejected, the CRS is accepted.
