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This paper addresses a maintenance plan optimization problem for building energy retrofitting with a novel differential evolution-based algorithm. The problem manifests substantial complexity due to the parallel optimization of the maintenance time scale, maintenance instants, and maintenance rates, with multiple objectives and time scales. The problem is formulated based on an impulsive and switched phenomenon, with objectives of maximizing the energy performances and economic indicators for the retrofitting project over a finite horizon. A multiscale differential evolution algorithm is accordingly proposed to be the numerical solver. The proposed algorithm is able to optimize the decision variables simultaneously, especially the maintenance time scale that can influence the number of other decision variables. Simulation results illustrate the effectiveness of the proposed approach.

While buildings result in a significant amount of energy consumptions [

The building energy retrofitting refers to activities in existing buildings that apply energy conservation interventions to improve efficiency or conserve energy or water, or manage demand [

The maintenance plan optimization problem introduced in [

Wang et al. [

Obviously, the aforementioned investigation is limited by the constraint of fixed time scheduling. In fact, the time scheduling is a common decision variable in the reliability engineering area [

In this paper, the preceding control system modelling is extended, where an impulsive and switched approach is employed and solved. The maintenance actions are considered to be a sequence of impulsive effects at variable instants that switch the retrofitted items to different working states. The numbers of items under the different working states, i.e., the system states, jump with such impulsive effects, whose impacts are indicated by the maintenance rates of PM and CM. The time scheduling describes the sequence of the impulsive effects, where the number of time instants is also variable. In this way, a hybrid system with impulsive effects and switching phenomena [

The remainder of the paper consists of four sections. Section

The maintenance plan optimization problem is established based on the retrofitting problem. Given a retrofitting project, the performances over time are estimated on a life-cycle cost analysis basis; i.e., the varying performances, operational costs, and maintenance costs are all taken into account. More details about such a retrofitting planning problem can be referred to [

The maintenance plan optimization problem is modelled on a discrete time system basis. Given a series of time instants

As aforementioned, the performances reveal a dynamic nature under the joint impacts of failures (malfunctioning) and maintenance. As a result, an idea of the control system approaches is employed for the maintenance plan optimization. Generally, we employ

The mathematical description of the system dynamics is hereby explained. Consider the following discrete dynamical system over

Equation (

Similarly,

According to (

The estimation of the aggregate performances requires two parts of information, the population and individual item performances. The population is estimated by the preceding system dynamics modelling, and the item performances are characterized by a series of performance indicators that are explained as the following. Firstly, an assumption is made such that the retrofitted items can be categorized into several groups; items from the same group are considered as homogeneous ones with the same energy and reliability performances. Let

Let

For a dynamical system consisting of

In the above equations,

A general form of the multiscale problem is employed as the following:

To solve such a multiscale minimization problem, we propose a multiscale-based approach, namely, multiple-scale DE (MSDE). The MSDE is actually an improvement of the conventional DE algorithm. Therefore, the concepts of basic DE will be introduced firstly and thereafter the implementation of MSDE.

The differential evolution (DE) is a popular category of the evolutionary algorithm that is suitable to solve the continuous optimization problem. The DE algorithm can hopefully, although not guaranteed, discover a satisfactory solution after sufficient iterations for most types of optimization problems. In the DE algorithm, the term “individuals” are adopted to refer to the candidate solutions that represent the possible values of the decision variables. These individuals are moved around in the search-space via a series of mathematical operations including

In order to implement the DE algorithm, a population of NP individuals is generated at the initial state to cover as more search space as possible. Therefore, a uniform randomization is usually adopted at the initial stage. As fixed-scale problems can be regarded as a special case of multiscale problems, the following minimization problem

At each generation, an individual

More mutation operators can be found in [

After the mutation phase, DE performs a binomial crossover operation to

The offspring are selected from the target vectors and the trail vectors, which is called selection operation. DE employs a greedy “one-to-one” replacement scheme. For each pair of

In MSDE, several subpopulations constitute the whole population of candidate solutions. Each subpopulation corresponds with a specific explicit variable

The procedure of MSDE can be illustrated in the following steps:

A population of

At each iteration, within subpopulation

After every

The pseudocode of MSDE is given in Algorithm

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3: Initialize

4: Evaluate the fitness for each individual;

5: Set

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8: Randomly choose (

9: Find the local best

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11: Randomly choose

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34: Find the best subpopulation

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36: Mutate

37: Initialize subpopulations

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40: Find global best solution

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The flowchart of MSDE.

The better fitness of the local best reveals a trend to find a better solution subject to the corresponding scale. Therefore, such an explicit variable is kept after the competition, and the other explicit variables have to be modified. Furthermore, a growth of the subpopulation with best explicit variable is encouraged; as a result, the worst subpopulation has to adopt a same explicit variable of the best subpopulation.

The major differences between the basic DE and MSDE are as the following. The MSDE has several subpopulations and accordingly multiple scales over its iterations, while basic DE has only one population and one fixed scale. During iterations of MSDE, the proper scale is identified simultaneously upon searching the optimum of other decision variables, while original DE lacks a mechanism to optimize the problem scale as well as the implicit vectors. Therefore, computation burden of the MSDE base solution is less than original DE to a multiscale problem.

In order to verify the effectiveness of the proposed approach, a case study is established and a simulation is performed based on the case study data. The case study is a part of a practical office building retrofitting project. Two categories of retrofitted items are involved. The first category is the compact fluorescent lamp (CFL). A type of 15 W CFLs is applied to replace the old inefficient lights. The CFLs are considered to be unrepairable items, with only one working state. The population decay model of the CFLs is given by

Parameters for population decay and state transition models.

Retrofits | ||||
---|---|---|---|---|

15 W retrofit CFL | 0.0947 | 0.775 | N/A | N/A |

Air conditioner |
N/A | N/A | 0.095 | 0.029 |

Air conditioner |
N/A | N/A | 0.04 | 0.033 |

Air conditioner |
N/A | N/A | N/A | 0.05 |

Table

Characteristics of retrofitted items.

Retrofits | Quantities | Unit price ($) | Unit energy saving (kWh) | Unit cost saving ($) | Preventive cost ($) | Corrective cost ($) |
---|---|---|---|---|---|---|

15 W retrofit CFL | 338 | 14 | 105.6 | 11.9 | N/A | 14 |

Air conditioner |
42 | 380 | 4320 | 486.65 | N/A | 175 |

Air conditioner |
0 | 380 | 3542.3 | 397.95 | 52 | N/A |

Air conditioner |
0 | 380 | 2651.75 | 278.35 | 70 | N/A |

Some configurations of the simulation are given as the following. For this case study, the sustainability period is set to be 10 years. The sampling interval is one month; i.e., there are 120 time instants over the sustainability period. The baseline energy performance is 8,685,311.7 kWh, which covers the pre-retrofit consumptions of lights and air conditioners. The baseline performance is extracted from a three-year energy bill of the inspected building. A 12% targeted energy saving is to be achieved by the retrofitting; as a result, the minimal energy saving is 1,042,237.4 kWh. Furthermore, for the economical performance calculation, the payback period time limit is set to be 24 months, which is a common acceptable payback period limit in practice. For the maintenance cost calculation, a constant cost at each maintenance instant is set to be $200. Several maintenance budgets are provided to establish different maintenance scenarios: $20,000 indicates a scenario with limited budget, $40,000 indicates a relatively insufficient one, and $65,000 indicates a sufficient one. As the comparison, a set of fixed maintenance time schedules are employed as the following:

Some simulation results are illustrated as the following. Table

Performances of maintenance plans with different budget limits.

Cases | Budget limit ($) | Energy savings (kWh) | Percentage saved over target | IRR | Payback period (months) | NPV ($) | Maintenance cost ($) | Total investment ($) |
---|---|---|---|---|---|---|---|---|

20,000 | 1,167,215 | 11.99% | 72.15% | 18.72 | 51427.78 | 19,986 | 40,678 | |

20,000 | 1,190,464 | 14.22% | 76.23% | 16.92 | 52949.3 | 19,998 | 40,690 | |

40,000 | 1,670,773 | 60.30% | 79.20% | 18.12 | 68393.86 | 39,987 | 60,679 | |

40,000 | 1,702,041 | 63.31% | 79.27% | 17.64 | 70076.16 | 39,986 | 60,678 | |

65,000 | 1,829,741 | 75.56% | 78.81% | 18.64 | 70608.34 | 52,382 | 73,074 | |

65,000 | 2,001,191 | 92.01% | 80.44% | 17.28 | 77670.84 | 58,046 | 78,738 |

Optimal maintenance time schedules.

Time schedule | Budget limit ($) | Maintenance number | Maintenance instants |
---|---|---|---|

20,000 | 6 | {14,26,38,50,62,74,86} | |

20,000 | 5 | {14,26,38,50,62,74} | |

40,000 | 12 | {8,18,28,…,98} | |

40,000 | 12 | {8,18,28,…,98} | |

65,000 | 34 | {5,8,11,…,104} | |

65,000 | 34 | {8,11,14,…,107} |

The maintenance rates over time with different maintenance budgets.

Energy saving trajectories of the optimal time schedule and maintenance plan with different maintenance budgets.

From the simulation, the most important difference between the MSDE and the comparative results are the maintenance scale. A proper selection of the maintenance scale can result in significant improvement in the performances. A preliminary investigation reveals that, with a fixed time maintenance time scale, the population size can be reduced to half to achieve a similar performance. However, identifying a proper scale takes quite a number of trail-and-error processes. The conventional method can spend a lot more time than the MSDE approach. In general, the effectiveness of employing MSDE to solve maintenance plan optimization problems with multiple categories of decision variables is verified by the simulation, and the MSDE is promising to solve impulsive and switched optimal control problems with similar formulation.

In this paper, a maintenance plan optimization problem for building energy retrofitting that simultaneously involves variable maintenance time scale, maintenance time scheduling, and maintenance rates, is investigated and addressed. An impulsive and switched optimal control formulation is employed to model the novel problem based on existing maintenance plan optimization studies. By adopting a weighted sum of two objectives, the aggregate energy savings and internal rate of return over a finite time horizon, an optimal control problem that aims at finding the optimal combination of the maintenance time scale, instants, and rates is formulated. Given that the maintenance time scale can vary upon searching optimum, the problem becomes a multiscale one. A multiscale differential evolution algorithm is proposed to find the numerical solution to the multiscale optimal control problem with acceptable computational burdens. We believe that this approach can be applied to a general category of minimization problems with variant scales and facilitates the solution of other impulsive and switched optimal control problems. The effectiveness of the proposed approach is verified by simulation results.

The current stage work calls for further studies from two major aspects: firstly, the maintenance plan optimization problem with variable time scheduling needs more investigation, such as the controller design and discussion on the robustness of the impulsive and switched modelling; secondly, the MSDE requires detailed studies on the mutation mechanism and shuffling period, such that the performance of MSDE can be further improved.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there is no conflict of interest regarding the publication of this paper.

This work was supported by the National Nature Science Foundation of China (grant numbers 61803162, 61873319, 61803054).