Evaluating intervention options to achieve environmental benefits in the residential sector

Abstract

Government intervention schemes in the form of policy instruments and financial incentives or rebates can have a major influence on the adoption of technologies by residential consumers to reduce their natural resource consumption and greenhouse gas emissions. However, the pattern and rate of consumer uptake of voluntary schemes are not always well-understood or easily taken into account in future scenarios analyses. This paper presents an innovative extension of the Bass diffusion model that has been integrated with multi-criteria analysis to enable explicit consideration and balancing of the impacts of technology cost, financial benefits, demographic suitability and household income on the likelihood of adoption. This ‘Intervention Options’ model is formulated into a constrained integer programming problem to allow optimisation of the size and timing of government rebates to maximise adoption rate and, ultimately, environmental benefits. The model’s capability is demonstrated using an Australian case study of 25,000 households, and historical information on the uptake of solar hot water and solar photovoltaic panels in Brisbane, Queensland. Case study results reveal new insights and important context-relevant trends that could assist policy makers to substantially improve the effectiveness of intervention schemes to achieve environmental goals within desired budgets.

Appendix: Mathematical model and solution algorithm

The main decision variable is based on the cost of the technology: $x_{d,j}^{t}$ where j = 3 for upfront cost. For the optimisation, we let

$$ x_{d,3}^{t} = xb_{d}^{t} + c_{d}^{t} $$

where $xb_{d}^{t}$ is the base cost without rebate, and $c_{d}^{t}$ is the size (in AUD) of the incentive (rebate). For the optimisation, $c_{d}^{t}$ is the decision variable. In most practical circumstances, it would be undesirable for $c_{d}^{t}$ to be different for every demographic by location by building type $d \in D$. The decision variable $c_{d}^{t}$ may differ for building types (e.g. apartment versus separate house) and may need to differ under some means testing (e.g. $c_{d}^{t}$ = 0 for high income categories of $d \in D$). To accommodate this situation, let $DD^{k} \subset D$ be a subset of D where

$$ c_{d}^{t} = c_{d'}^{t} \, \forall d,d^{'} \in DD^{k} $$

(9)

That is, all incentives $c_{d}^{t}$ within category set $DD^{k}$ are of the same value.

$$ \cup DD^{k} \, \forall k = D,\;DD^{k'} \cap DD^{k} \, = 0 $$

That is the combination of all $DD^{k}$ category sets = DD.

The size of the incentive will have upper and lower bounds:

$$ lb_{d}^{t} \le c_{d}^{t} \le ub_{d}^{t} $$

(10)

where $lb_{d}^{t} ,ub_{d}^{t}$ are the lower and upper bounds of the incentive as defined by the user.

The budget constraint is:

$$ \sum\limits_{t \in T} {\sum\limits_{d \in D} {c_{d}^{t} \cdot A_{d}^{\prime } (t) \le B} } $$

(11)

where B is the total budget.

Constraints on maximum number of incentive changes across the planning horizon:

$$ h_{d}^{t} = \left\{ {\begin{array}{*{20}l} 1 & {{\text{if }}c_{d}^{t} - c_{d}^{t - 1} \ne 0} \\ 0 & {\text{otherwise }} \\ \end{array} } \right. $$

where:

$h_{d}^{t} \in \left\{ {0,1} \right\}$, an integer variable with value 0 or 1. $h_{d}^{o,t} = 1$ if incentive $c_{d}^{t}$ changes from time t − 1 to time t.

Then

$$ \sum\limits_{t \in T} {h_{d}^{t} \le \beta } \quad \forall o \in O,d \in D $$

(12)

where $\beta$ is an input parameter for the maximum number of changes in the incentives over the planning horizon.

Objective functions

Two objective functions are considered.

Objective 1

The first objective is to maximise the total number of adopters at a key time period t = T subject to the budget:

$$ {\text{Max}}\;Z = \sum\limits_{d \in D} {A_{d} (T)} $$

(13)

for the case of summing over all options.

Equation 13 is subject to the constraints given in Eqs. 1–7 and 9–12.

Objective 2

The second objective is to minimise the cost to achieve a desired level of adoption at key time period t = T

$$ {\text{Min}}\;Z = \sum\limits_{t \in T} {\sum\limits_{d \in D} {c_{d}^{t} \cdot A_{d}^{\prime } (t)} } $$

(14)

subject to

$$ \sum\limits_{d \in D} {A_{d}^{o} (T) > {\text{AD}}} $$

(15)

and the constraints in Eqs. 1–7 and 9–12, where AD is the minimal level of adoption.

Greedy search heuristic for optimising rebates

There is a wide range of suitable meta-heuristics for solving the sub-problems for $c_{d}^{t}$, including simulated annealing and tabu search, genetic algorithms and hybrid heuristics. Any of these methods can be applied and a comparison between methods is beyond the scope of this paper. In this study, we used a simple multi-start greedy search method.

The greedy search heuristic is based on the establishment of moves so as to transform a current solution to one of the neighbouring solutions. The greedy search escapes local optimal solutions by allowing up-hill (non-improving) moves to be performed when no down-hill (improving) moves are available. At each iteration, the neighbourhood (or a sample of it) is searched for which the move found in the search is applied, regardless of whether it is an uphill move. Instead of searching the entire neighbourhood, which can be very large, our algorithm will randomly select a subset N1, N2 for neighbourhoods 1 and 2 respectively. This sampling feature also helps escape local optimal solutions. Two neighbourhoods are applied:

• Neighbourhood 1: Modify the incentive (rebate) for categories $d \in DD^{k}$ from time period t′ through to time period t″. That is, set $c_{d}^{t}$ = v for all $d \in DD^{k}$ and $t^{'} \le t \le t^{''}$.

• Neighbourhood 2: Perform neighbourhood 1 twice simultaneously, where the incentives for one are modified upwards and the other are modified downwards.

The two neighbourhoods for the decision variables, $c_{d}^{t}$, complement one another during the heuristic routine. Neighbourhood 1 is applied more frequently when it is difficult to improve the solution due to constraints (10, 11, 12). After $\phi^{'}$ continuous iterations where the best solution is not improved, the search is intensified by replacing the current solution with the best so far. The greedy search used is described by Algorithm 1, based on objective 1 in Eq. 13.

Algorithm

Set $\phi$ = 0

Set $\phi^{'}$ = 25 which is the number of iterations without solution improvement.

C is the matrix solution of values $c_{d}^{t}$

Z′ represents the best found objective function overall.

Z″ represents the objective function of the working solution within the algorithm.

C′ represents the solution with the best found objective function overall.

C″ represent the working solution internally within the minor iteration of the algorithm.

N1 is the sample size for neighbourhood 1.

N2 is the sample size of neighbourhood 1 + 2.

Obtain an initial solution by running the model with no incentives (C = 0) and let Z be the objective function value.

Let Z′ = Z, Z″ = Z, C′ = C, C″ = C

REPEAT ! This is the major iteration

 REPEAT ! This is the minor iteration

  Randomly generate a sample of size N2 of solutions in the neighbourhood of C, whilst satisfying constraints 10–12.

  Let $Z_{i}$ be the solution from the N2 solutions with the largest objective function

  IF $Z_{i}$ > Z″, then Z″ = $Z_{i}$, C″ = $C_{i}$

  IF $Z_{i}$ > Z′, then Z′ = Z″, C′ = C″, $\phi$ = 0

  ADD $C_{i}$ to the tabu list of the TL most recent solutions, and remove the oldest solution

  ADD 1 to $\phi$

UNTIL $\phi = \phi^{'}$

Z″ = Z′, C″ = C′

UNTIL convergence criteria is achieved.

For this case study, convergence was achieved after 30 min of CPU time on a standard dual core 2.8 GHz PC, as improvement in the solution was negligible after that time.