A participatory multi-criteria approach for power generation and transmission planning

Abstract

The energy sector continues to undergo substantial structural changes. Currently, the expansion of renewable energy sources and the decentralisation of energy supply lead to new players entering the market who pursue different objectives and have different preferences. Thus, multiple and usually conflicting targets need to be considered. Moreover, recent public reactions towards infrastructure projects highlight the importance of considering public acceptance as a key dimension of decision making in the energy sector. As a result, decision processes grow more complex at all levels from political to strategic, tactical and operational decisions in companies. We therefore present an approach combining power systems analysis considering grid constraints and multi-criteria decision analysis. The approach focusses on multi-dimensional sensitivity analyses allowing for simultaneous variations of the different preference parameters determined within the decision analysis aimed at facilitating preference elicitation and consensus building in group decisions. The focus of the paper is the demonstration of the presented approach for a power generation and transmission planning case study in the context of the energy transition in Germany.

Appendices

Appendix 1: Mathematical description of the grid expansion approach within PERSEUS-NET

This appendix contains the mathematical details of the grid expansion approach implemented within PERSEUS-NET as indicated in Sect. 3.1. Below, we focus on the main modifications of the load flow equations carried out to consider grid expansion on the basis of Slednev et al. (2014).

As described in Sect. 3.1, the endogenous grid expansion is modelled as a mixed integer problem, since the line susceptance as well as the thermal limit depend on the expansion stage of the transmission grid. The introduced binary decision variables take into account that at each expansion stage, the line susceptance and the thermal limit may only be increased by discrete levels. Table 5 provides an overview of the nomenclature.

Table 5 Nomenclature PERSEUS-NET

Starting with the line flow constraint, the original equation is extended by an additional dimension $i$, see Eq. (10). The equation now needs to be solved for the existing transmission line realisation $i_0$ as well as for each expansion level ($i_1, i_2, \ldots$), where each level $i$ consists in a realisation of certain discrete investment alternatives. Equation (10) expresses the active power flow $f^*_{n,n',i,t,s}$ over each possible realisation of the expansion stage $i$ between two grid nodes $n$ and $n'$ as the summed product of the phase angle difference $\theta _{n'',t,s}$ at grid node $n''$ and the elements $h_{n,n',n'',i,t}$ of the so-called transfer admittance matrix. While the columns of the transfer admittance matrix correspond to the nodes, the rows refer to the lines of the grid. The transfer admittance matrix is determined as the product of the transposed vector of susceptances¹ and the transposed incidence matrix.²

$$\begin{aligned} f^{*}_{n,n',i,t,s}&= \sum \nolimits _{n'' \in N} h_{n,n',n'',i,t} \cdot \theta _{n'', t, s} \nonumber \\&\forall n,n' \in N, \forall t \in T, \forall s \in S, \forall i \in I \end{aligned}$$

(10)

In addition, further auxiliary conditions are included in order to restrict the actual power flow according to the finally realised expansion stage. Therefore, Eq. (11) splits the previously calculated potential line flow $f^*_{n,n',i,t,s}$ into the two components $f_{n,n',t,s}$ and $r'_{n,n',i,t,s}$.

$$\begin{aligned} f^{*}_{n,n',i,t,s}&= f_{n,n',t,s} + r'_{n,n', i, t, s} \nonumber \\&\forall n,n' \in N, \forall t \in T, \forall s \in S, \forall i \in I \end{aligned}$$

(11)

The above splitting enables setting the auxiliary variable $r'_{n,n',i,t,s}$ to zero in a second step in case that a certain expansion stage $i$ is realised. In this case, the value of the corresponding potential line flow $f^*_{n,n',i,t,s}$ is assigned to the resulting line flow variable $f_{n,n',t,s}$ describing the flows on existing lines. For a sufficiently high scalar $M$, which should be higher than the absolute highest possible line flow, Eq. (12) realises this second step.

$$\begin{aligned}&M \ge | r'_{n,n', i, t, s} | + x_{n,n',i,t} \cdot M \nonumber \\&\forall n,n' \in N, \forall t \in T, \forall s \in S, \forall i \in I \end{aligned}$$

(12)

The binary variable $x_{n,n',i,t}$ in (12), defining whether a certain expansion stage is realised, restricts the auxiliary variable $r'_{n,n',i,t,s}$ to zero in case that $x_{n,n',i,t}$ is set to 1. In addition, inequality (13) restricts the active power flow over a line according to the respective expansion stage.

$$\begin{aligned}&|f_{n,n',t,s}| \le \left( 1-x_{n,n',i,t}\right) \cdot M + tl_{n,n',i,t} \nonumber \\&\forall n,n' \in N, \forall t \in T, \forall s \in S, \forall i \in I \end{aligned}$$

(13)

Equations (10–13) increase the problem complexity on linear scale. In contrast, the application of the power flow constraints on the nodal balance results in a significant complexity increase since the nodal balance needs to be guaranteed for every possible combination of expansion stages of adjacent transmission lines. Analogously to Eq. (10), the dimension of the nodal balance power flow constraint is extended by adding the index k. This index denotes all possible combinations of expansion stages of adjacent transmission lines at one grid node and enables considering the varying network characteristics within the nodal balance. Equation (14) therefore states that the net injection into a bus $g^*_{n,k,t,s}$, in case of a certain network characteristic ($k \in K$), corresponds to the balance of power flows over adjacent lines.

$$\begin{aligned} g^{*}_{n,k,t,s}&= \sum \nolimits _{n' \in N} b_{n,n',k,t} \cdot \theta _{n', t, s} \nonumber \\&\forall n \in N, \forall t \in T, \forall s \in S, \forall k \in K \end{aligned}$$

(14)

The actual net injection into a bus $g_{n,t,s}$ is, again, determined by a binary variable $y_{n,k,t}$ defining the finally realised combination of expansion stages of adjacent transmission lines at a grid node. Equations (15) and (16) ensure that only power flows over realised transmission lines are considered within the nodal balance.

$$\begin{aligned}&g^{*}_{n,k,t,s} = g_{n,t,s} + r''_{n,k,t,s} \nonumber \\&\forall n \in N, \forall t \in T, \forall s \in S, \forall k \in K \end{aligned}$$

(15)

$$\begin{aligned}&M \ge | r''_{n, k, t, s} | + y_{n,k,t} \cdot M \nonumber \\&\forall n \in N, \forall t \in T, \forall s \in S, \forall k \in K \end{aligned}$$

(16)

In order to ensure the consistency within the line flow and nodal balance equations, (17) restricts the realised amount of transmission line expansion stages and (18) their combination at a grid node to 1 respectively. Note that, up to this point, the line flow and nodal balance equations are solved independently. This approach requires an additional solution of power flow equations only for nodes and lines with investment options instead of solving the line flow (10) and nodal balance (14) equation of each node and line for every possible variation of the network characteristic. The advantage is a significant reduction of the problem complexity.

$$\begin{aligned}&\sum \nolimits _{i \in I} x_{n,n',i,t} = 1 \nonumber \\&\forall n, n' \in N, \forall t \in T \end{aligned}$$

(17)

$$\begin{aligned}&\sum \nolimits _{k \in K} y_{n,k,t} = 1 \nonumber \\&\forall n, \in N, \forall t \in T \end{aligned}$$

(18)

Finally, in order to guarantee an overall valid solution, the consistency between the investment decision resulting from the line flow equation (10) and the nodal balance equation (14) has to be guaranteed. First, the relation between the binary variable $x_{n,n',i,t}$, which defines the possible grid expansion stages between two nodes, and the binary variable $y_{n,k,t}$, taking into account the investment decision on adjacent lines, has to be defined. Equation (19) couples both variables by means of an incidence matrix, whose elements are equal to 1 if a certain expansion stage $i$ is realised in relation to investment options on adjacent lines $k$.

$$\begin{aligned}&\displaystyle x_{n,n',i,t} = \sum \nolimits _{k \in K} y_{n,k,t} \cdot ls_{n,n',n,k,i,t} \nonumber \\&\displaystyle \forall n, n' \in N, \forall t \in T, \forall s \in S, \forall i \in I \end{aligned}$$

(19)

By additionally relating the variable $x_{n,n',i,t}$ to the investment decision $y_{n',k,t}$ at node $n'$ the consistency between the two binary variables is extended to the complete transmission grid.

$$\begin{aligned}&\displaystyle x_{n,n',i,t} = \sum \nolimits _{k \in K} y_{n',k,t} \cdot ls_{n,n',n',k,i,t} \nonumber \\&\displaystyle \forall n, n' \in N, \forall t \in T, \forall s \in S, \forall i \in I \end{aligned}$$

(20)

Appendix 2: Empirical weight distributions elicited within the online survey

As described in Sect. 4.2, the inter-criteria preference elicitation was carried out within the online survey by asking the participants for pairwise comparisons of the considered criteria on a nine-point-scale as in the analytic hierarchy process (AHP). Since the pairwise comparison procedure may lead to inconsistent preference statements, these statements were checked for consistency and inconsistent statements were removed. Moreover, Sect. 4.2 points out that the weights derived from the pairwise comparisons vary significantly between the participants. In addition to the weight intervals, determined as supersets of the individually derived weights, as described in Sect. 4.2, the empirically determined weight distributions, which result from the survey, are shown in Fig. 8. While the weights of security of supply and environmental sustainability are more or less evenly distributed (though with variations and peaks in the lower as well as the upper parts of the respective weight intervals), Fig. 8 clearly indicates that the empirical weight distributions of economic competitiveness and public acceptance are right-skewed.

Fig. 8

Empirical weight distributions resulting from the pairwise comparisons