Textbox 1: MFMA clause on infrastructure projects
5.3 Decision recognition and diagnosis
The framework developed combines two multi-objective optimisation techniques to generate energy portfolios of action for a community. These two techniques are the GP and the MAVT technique (Refer to Section 3.3.2 and Section 3.3.5 for definitions). This combination allows the extension of the GP technique to a MAVT technique. The two procedures are discussed
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89 in the next section. The first section focuses on GP technique and the second section addresses the MAVT technique.
5.3.1 Selecting portfolios of action: goal programming
In this first iteration, portfolios are developed using the GP technique. As discussed in Section 3.3.2, the GP technique has been applied to the evaluation of energy access problems of a combinatorial nature (Ramanathan and Ganesh, 1994; 1995). Two elements must exist for GP evaluation, namely goals and constraints. The energy access problem qualifies for GP, given different stakeholders' different perspectives and goals that can be translated to objectives and constraints. A goal can be expressed as a constraint on the goal value expression plus or minus a deviation variable set which is equal to the desired goal level.
One of the key goals of the 17 wide-ranging SDGs, SDG seven, is “ensuring access to affordable, reliable and sustainable and modern energy”. Financial constraints, however, hamper investments in new and clean technologies to achieve considerable reductions in carbon emissions. Given the financial constraints, standard GP decision variables in the energy access problem include lifecycle costs, carbon emissions, thermal efficiency, time and capacity factors. Other constraints include limitations related to the time available for maintaining systems. Information related to the decision variables is often found in several national plans and published case studies.
GP allows decision analysts and decision-makers to investigate the role that different energy access strategies can play in national efforts to reduce carbon emissions at the same time meeting the desired outcomes. The identification of feasible alternatives can only be achieved if the pool from which they are drawn is adequately diversified. The objectives are derived from the expert panel outlined in Section 5.1.
The framework developed employed the augmented Chebychev approach (Wierzbicki, 1980, 1999). Under the model developed, standard linear programming was extended into the augmented Chebychev GP form for the energy planning problem. While the Chebyshev formulation is nondifferentiable and the formulation used is differentiable, the model was optimized using an LP framework that does not use derivatives. The LP framework is useful in solving the problem.
In the energy planning problem, the minimisation of !! objectives is desired. The target, therefore, is to minimize the undesirable #!" deviational variables with #!# playing no
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90 substantial role. We therefore first supressed the superscripts on the #!"deviational variables and just wrote #! with deviational constraints as:
!! − #!" ≤ &! Equation 3
for ' = 1, … ,.
! represents the objective, ' criteria/objectives which may be scenario related, - = number of technologies considered. The augmented Chebychev GP formulation designed for the study, thus, sought to minimize ./0!$%& 1!#!+ 3 ∑&!$%1!#! for 3 > 0. As discussed above, for the study, a new 7 variable was introduced to deal with non-linearity with an additional constraint now formulated as:
7 ≥ 1!#! for ' = 1, … ,. Equation 4
a) Decision variables
In the model developed, different energy supply options (labelled 9) are to be considered for a representative community and to be deployed to form an energy mix in GJ (labelled : = 1, … , n). The decision variables for the model are thus defined as:
9' : provided (installed) capacity in GJ for technology : (: = 1, … , n)
=('!: quantity of installed technology : allocated to serving energy needs of six different uses > (> = 1, … , n) under conditions of three different scenarios ' (' = 1, … , n).
The model developed considers climate change scenarios and these scenarios should be obtained from the strategic dialogues. The scenarios are based on a year in future labelled
@$a with @ representing time and a representing the selected year.
b) Constraints
The model developed for this study incorporated hard constraints based on the estimated household energy requirements for satisfying the basic needs discussed above. Two classes of hard constraints are thus defined as:
Capacity constraint: ∑ =( ('! ≤ 9' for all : and for each scenario '.
Demand constraint: ∑ =' ('!≥ demand for use > under scenario ' for all > and '. =('! denotes GJ of usable power for use > under scenario ' delivered by an allocation of 1 GJ of installed capacity of technology : to this usage. In some cases, this parameter had a value 0 (not
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91 appropriate), in many cases it may be 1, but represent efficiency in other cases (e.g., wind and solar power under certain conditions).
c) Objectives
Using the goal programming formulation, the analysts obtain the decision-makers desirable levels of performance, known as goals (g)) for each objective, k (k = 1, … , K). In obtaining any feasible solution, non-negative deviational variables are introduced (#!" /-# #!#). At least one of the deviational variables should be zero, such that z) +– #!"= g) with z representing objectives k representing criteria/objectives which may be scenario related. The minus and the plus superscripts indicate whether a goal is to be minimised, over, or underachievement of the goal priority
The final objective function for the linear programming procedure to implement the augmented Chebychev GP metric for the energy access problem with the new 7 can thus be formulated as:
!%= ∑&'$%9',' Equation 5
!*! = ∑ ∑ =( ' ('!F('! Equation 6
!+! = ∑ ∑ =( ' ('!7('! Equation 7
!,! = ∑ ∑ =( ' ('!G('! Equation 8
Minimise 7 + H ∑&!$%1!#! Equation 9 for ' = 1, … ,.
Subject to the following constraints:
Capacity constraint: ∑ =( ('! ≤ 9' for all : and for each scenario '.
Demand constraint: ∑ =' ('!≥ demand for use > under scenario ' for all > and '.
The epsilon variant, 3, is a moderately small positive number greater than 0 (between 0.02 or 0.10). The assumed ∈, ought to be small enough not to substantially affect the optimal value of the !!, but large enough to ensure a Pareto optimality solution. As discussed in Section 3.3.2, the GP technique is popular in MCDA because of its simplicity. Finding optimal solutions
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92 using the GP technique is straightforward to implement and has been universally and successfully implemented elsewhere.