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Edited by: Onuegbu Ugwu, Federal University, Ndufu-Alike, Nigeria

Reviewed by: Bao-Jie He, University of New South Wales, Australia; Ali Behnood, Purdue University, United States

This article was submitted to Sustainable Design and Construction, a section of the journal Frontiers in Built Environment

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

A key decision in the design of urban lighting is the location of the luminaries that are used to illuminate the specified region. The decision needs to account for coverage requirements identified in certain areas, based on safety considerations and nature of work activity, along with ensuring the cost effectiveness of the installation pattern adopted. In this work, a novel approach is presented via a multi-objective mathematical optimization model that results in a sustainable layout of light poles in urban region. A maximal coverage objective, with implicit demand cover, is formulated as a measure of the social requirement in urban lighting, which models security and safety associated with night-time lighting of the urban region. At the same time, the economical aspect of the layout is considered via minimizing the installation cost of the lighting layout. A realistic case example is then solved using the ϵ-constraint method. A Pareto optimal front for the case considered is constructed and analyzed.

Light pollution is a considerable issue that is facing a significant number of cities around the world (Meier et al.,

The use of public lighting in cities is crucial to enhance the quality of life for humans (Brandi and Geissmar,

Intelligent light systems or energy efficient lighting, such as the use of LED (Beatley,

On closer inspection, it becomes evident that in many urban areas, the use of night-time lighting is excessive, leading to wasteful lighting that contributes to the light pollution issues faced (Maithili et al.,

Prior to discussing the literature on optimizing the lighting layout, a couple of “light-specific” terms are defined. Luminance, which is measured in candela per square meter, describes the amount of lighting that is reflected off a given surface. Illuminance on the other hand describes the measurement of the amount of light that falls onto a given surface; it is measured in lux (Schreuder,

Even though several studies exist in the literature that examine the optimization of lighting operations, very few focus on a sustainable lighting layout in urban regions that strikes a balance between the economic, social and environmental objectives of sustainability, simultaneously. Genetic algorithms were deployed to optimize the design of office lighting in order to maximize energy efficiency while satisfying visual requirements for workers (Cassol et al.,

To fill in the gap, this paper presents a novel approach that is based on a mathematical optimization model for solving the urban light location problem (ULLP) proposed in this study. The novelty of this research lies in the integration of 2 important sustainability criteria which are highly relevant in the planning of the lighting requirements in an urban region, namely safety, through sufficient lighting coverage of the zone, and the economic cost involved, through minimizing installation costs. The importance of adequate lighting in terms of deterring crime and reducing crime rates in cities has been reported in Arvate et al. (

This paper is organized as follows: in the next section, the notion of multi-objective optimization is presented. The problem description is then provided, along with the framework proposed for solving the ULLP. A binary integer programming (BIP) model proposed for the ULLP is then formulated. The paper then describes the applications of the model on a realistic case example where the problem is solved via a mono and multi-objective approach. Insights and concluding remarks are presented at the end.

Applications of multi-objective optimization are wide and varied in the built environment (Wang et al.,

Assuming a minimization problem, a solution of a multi-objective optimization problem, ^{*} is Pareto optimal if there does not exist another feasible solution

The concept of Pareto optimality is adopted in this paper in order to solve the proposed multi-objective ULLP. A description of the ULLP is provided next.

The ULLP can be described as follows. Assume that the urban region being considered is a planar area, which can be divided into zones, Figure

Discretising the urban regions into zones,

Framework for generating sustainable urban lighting layout.

Figure

A number of assumptions are made: first, the lighting requirement of each zone is represented via a number positioned at the centroid of that zone. Second, in order to service a particular zone, the light pole installed needs to be placed within the grid division of that zone. Any light pole placed outside the zone does not contribute toward the coverage requirement of that respective zone. Third, the light poles are placed at the centroid of the girds. Fourth, for each grid,

A framework is outlined in order to describe the process that is followed when solving the ULLP. The first component of the framework, labeled, Space discretization, has been explained via Figure

The second component of the framework, labeled Illumination pre-processing, computes the requirements of the illuminance level at each grid point of the discretized zone, due to placing the light-pole at the centroid of the grids. The computation of the illuminance levels is conducted in line with the requirements enlisted by the International Commission on Illumination (CIE,

The supply of light due to a pole placed on a grid within the urban zone will largely depend on the characteristics of the lighting fixture to be placed (i.e., lumen output from all luminaires within the light fixture, maintenance factors of the luminaires and atmospheric loss factors). Each lighting fixture is modeled as a source with a specific coverage; the coverage area is calculated based on the distribution of light. In order to model the distribution of light from a single source, an inverse relationship between the brightness of the light and the distance from the light source is utilized (Simons and Bean,

The light supply is calculated based on the lumen method (Rea, _{ml} denote the illuminance in lx due to light pole

Figure

An example to demonstrate the calculation of the illuminance level based on the B-plane and the use of I-tables is given in Figure

Inverse square law for calculating illuminance.

Example of illuminance calculation based on B-Plane.

In this section, the set notation and formulations for the BIP mathematical optimization model proposed to solve the ULLP are described.

Sets:

_{i}: Set of permitted coverage at zone

_{ikl}: Set of grid points within urban region _{i}for light pole type

Parameters:

_{i}: Critical level of providing coverage for region

Variables:

_{ik} ∈ {0, 1}: Binary variable which equals 1 if urban area

_{ml} ∈ {0, 1}: Binary variable which equals 1 if grid cell

Two objective functions are formulated in the model. The first objective function, Equation (2), assesses the coverage at each zone. In particular, it is formulated as a proxy to the level of safety and security of the zone. Equation (2) is maximized since maximizing the coverage through increasing the illumination levels results in less security threats (robbery, assault etc.) as demonstrated in previous literature (Bromley et al.,

The second objective function seeks to minimize the total cost of installation associated with the placement of the light poles. Different zones will have different installation requirements and hence the cost function,_{ml} , across the zones of the urban region will differ.

A number of constraints are defined in the ULLP in order to delineate the feasible region of the problem. The first constraint formulated, Equation (4) requires each zone to be covered up to a certain coverage level. Since each zone will have different lighting requirements, the set _{i} is constructed and deployed in Equation (4) to ensure an appropriate selection of a level of coverage that suits the respective zone lighting requirement.

The second constraint, Equation (5), states that each grid cell within a defined zone can contain a maximum of 1 light pole.

The third constraint, Equation (6), specifies that if region _{ikl} that can achieve the level of coverage _{ikl} is determined from the pre-processing step of illuminance, through calculating the light level while changing the location of the pole across all girds of the zone, utilizing the concept of the B-Plane in Figure

The set of constraints, Equations (7, 8) define the domain of the integer variables

The ϵ-constraint approach is adopted to solve the multi-objective optimization problem. In its simplest form, the model reformulates the set of objective functions given so that one is optimized whilst the rest are executed as constraints (Chankong and Haimes,

The coverage objective function is represented by_{1}(_{2}(_{u}(_{2}(

In order to demonstrate the applicability of the proposed model, a realistic case example is considered. An urban area composed of 9 regions, and located 70 km south-west from the City of Wollongong in New South Wales, Australia, is proposed to be a commercial and touristic hub. The divisions of the 9 regions is displayed in Figure

Three levels of coverage are considered for the lighting namely 1 (60% coverage), 2 (80% coverage) and 3 (100% coverage). Three types of light poles are assumed. The number of each light pole type required to produce the specified level of coverage is given in Table

Number of light poles required.

1 | 1 | 1 |

2 | 1 | 2 |

3 | 1 | 3 |

1 | 2 | 2 |

2 | 2 | 3 |

3 | 2 | 4 |

1 | 3 | 3 |

2 | 3 | 4 |

3 | 3 | 5 |

Lighting requirement in each zone.

1 | 150 |

2 | 60 |

3 | 110 |

4 | 50 |

5 | 50 |

6 | 40 |

7 | 70 |

8 | 130 |

9 | 40 |

In order to construct the set _{i}, the importance of lighting for each zone is specifed as shown in Figure

The case example of Figure

Solving the problem where only coverage maximization is formulated as an objective function, with no limit on the number of light poles that can be used, and having the requirement of ensuring coverage to all areas of the urban region, results in a distribution as is displayed in Figure

Distribution of lighting around the urban region where maximal coverage is ensured, with black indicating 150 lx of lighting in the zones of the region.

In terms of security, the impact of the distribution shown in Figure

The problem is re-solved, where this time a limit of 9 light poles is placed through introducing the following constraint to the model, Equation (9):

The resulting distribution of lighting is shown in Figure

Distribution of illuminance around the urban region where a limit is placed on number of lighting.

The Pareto optimal front yielded through solving the problem via the ϵ-constraint method is shown in Figure

Pareto Optimal front. Point A is one where cost installation is less emphasized, compared to Point B.

One of the Pareto optimal solutions is presented in Figure

Distribution of illuminance based on maximized coverage levels around the zones, with little emphasis on light installation cost, Point A.

In Figure

Distribution of illuminance based on maximized coverage levels around the zones, with greater influence of light installation cost on solution produced, Point B.

Compared to Figure

Results of Figures

A number of insights can be yielded from the proposed optimization model. First, the fact that social measures of lighting distribution around an urban region (i.e., increased security and safety due to well-lit regions) is an important consideration to account for when deciding on an installation pattern for lighting in urban regions, can be achieved through formulating the problem as an implicit coverage one. The coverage of each zone of the urban region is assumed as accumulative, with multiple lighting poles providing the required level of illumination specified for the given zones. Second, as the optimization results of Figures

This paper presented a mathematical optimization model for the sustainable location of light poles in urban regions. The model is based on a multi-objective approach, where coverage of the region under consideration is maximized in an attempt to satisfy security requirements, while the cost associated with installation of the lighting poles is minimized. Demand of the region is assumed as a priori input, and the supply of lighting is based on the luminaries utilized, which is calculated via a pre-processing step through use of the inverse square law and isolux diagrams obtained from luminaire manufacturers. The urban space is discretized into zones, with zones further divided into girds. Integer variables are then introduced to highlight where the light poles will be positioned in the urban space. The ϵ-constraitnt approach was adopted to yield the Pareto front. A clear trade-off between coverage and installation cost was highlighted. In addition, the use of a limit on the number of poles leads to a limited number of lights being installed in locations where the requirement for lighting is assessed to be critical. A distinction between solutions obtained using single vs. multiple optimization objective was also observed, with the introduction of the installation cost objective yielding an economic saving of 79% on average.

The proposed method finds applicability in many urban design related fields, including, street lighting, retail lighting and park lighting. The limitation of this work lies in the implicit modeling of light pollution, through accounting for social and environmental measures via only a single objective function that acts as a proxy (i.e., coverage objective function). Even though this yields a sustainable solution, as was demonstrated in this work, a further extension of this work can involve the formulation of a multi-objective optimization problem with additional objective functions that explicitly measure energy wasted and light pollution at each respective urban zone modeled. This can then be contrasted with the multi-objective approach introduced in this paper.

AH performed the work involved in developing and implementing the optimization framework. AA contributed to problem definition and supervised the work conducted leading to the paper.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The reviewer, BH, declared a shared affiliation, with no collaboration, with one of the authors, AA, to the handling Editor.