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1.0Introduction
Traffic light control emanated in 1868 in the British Houses of Parliament in London, since inception it has been the major means of traffic control. The major purpose is to control and regulate the flow of vehicles at intersections using time as a reference to pass the various sections of the road. Although traffic control began in England, it has reached almost all countries in the world because of the opportunities that cause people to move to the urban areas also lead to an increase in traffic congestion.

Urban traffic congestion has been a challenge in both developed and developing countries. For example, according to the traffic congestion and Reliability Report (FHWA,2005) congestion has grown greatly throughout the developed and developing countries over the past twenty years, regardless of city size. A report estimates that In 2007, congestion caused urban American to travel 4.2 billion hours more, resulting in an extra 2.8 billion of fuel for a cost of $7.2billion,an increase of more than 50 percent over the previous decade”(Schrank and Lomax, 2009). Here in Nigeria, such data is not available, but with the naked eye, it is clear to see the effect of migration to the urban cities has on the traffic situation.

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Modelling of traffic signal controls is essential to the smooth running of traffic lights, various techniques have been used to do this, and the models depend on the laws and preference of the country making use of them. Optimization techniques have been used to reduce queuing delays and improve departure times at intersections by considering the arrival time, queuing length and departure time. The right of way and timing of the Green Amber and Red phases are also taken into consideration to save as much time as possible.

Most of the existing traffic signals work with pre-timed cycles. These cycles are determined by the historical data of the traffic demand of that particular area; this is then used to program the splitting for the green-amber-red phases. However in recent years due to an exponential increase in vehicular traffic these systems fail to fulfil the requirements, improper traffic signal controls at the intersection is, therefore, a major cause of traffic congestion.

1.1Aim and objectives
This research work aims to design an optimal traffic signal control for road networks. The objectives of this proposed work are:
To identify the traffic conflicts in major & minor streams in a particular intersection/junction.

To estimate the basic traffic stream parameters for the selected road.

To obtain optimal control signal sequence, for the intersection of interest.

To design and provide the signal timings for the intersection of interest.

1.2Problem Statement
The geometric increase in population has many effects on the community today and one of such areas is traffic. With the increase in population in Nigeria, about 60% of which are youths, more licensed drivers will emerge leading to the acquisition of more vehicles for transportation purposes; this leads to congestion and traffic jams.

The cost of traffic congestion is not limited to the numerous hours wasted in traffic jams; it also extends to the environment and health problems as well. Regarding health the consequences may include, physical discomfort, stress and even mental discomfort, all these lead to a shorter lifespan and lower quality of life. Environmentally traffic congestion leads to pollution and increase in fuel usage. This affects transportation of goods and services which leads to delays and also higher prices for consumers.

Several causes of traffic congestion exist from bad roads to reckless driving, weather, road accidents bad road designs and traffic signals. Traffic lights were developed to help to reduce these problems to a large degree, but it must be stated that this can only be effective if the individual drivers comply with the traffic light commands or else it will not be effective. A possible solution to traffic congestion problems would be the construction of more motorable roads, but the cost of construction and availability of land in most cities hampers this idea. A better option will be to maximise the use of currently available road networks by improving or optimising already working traffic lights. A traffic system is defined as the passage of vehicles through a road infrastructure. Roadways, controls (e.g., traffic signals, stop signs), drivers and vehicles are the four principal elements of these systems. Traffic control is the process by which the passage of vehicles through a road infrastructure is governed. It is quite evident that the efficiency of traffic control directly depends on the efficiency and relevance of the control methodologies. Poor traffic control can lead to traffic congestion, whereas well-designed traffic control plans, such as efficient traffic signal timings, can significantly reduce traffic congestion. In fact, in traffic systems that contain traffic signals, control of traffic light signal timings is one of the least expensive and most effective means of reducing vehicular congestion in metropolitan road networks. This is especially true in times of peak traffic flow, such as during morning and evening rush hours. Traffic signal management is one of the fastest methods to achieve traffic congestion improvements. While building new roads can take years, a new traffic signal plan can be implemented in a matter of weeks.

Traffic signals are controlled by a plan that controls when and how to change phases. The plans used to guide these changes, known as traffic signal management (control), vary in complexity. The four control parameters for traffic signal management are:
Offset: time between cycle starts of different signals;
Stage specification: outlines what options (go straight, turn left, turn right) a vehicle has at any given time at each intersection
Cycle time: the time it takes a signal to progress through all stages;
Split: a portion of a cycle that the signal is green for each direction.

1.3Justification
Effective traffic management is essential to the long-term success of any country’s development; since it promotes economic development and prevents most countries from going bankrupt. If not properly managed, it may lead to the collapsing of the economic market and most companies. It is, therefore, justifiable because it helps traffic engineers to verify whether traffic properties and characteristics such as speed (velocity), Density and flow among others determines the effectiveness of traffic flow. Hence a model can be used by engineers of the road network to plan toward preventing excessive traffic jam.

2.0Literature Review
Various optimization techniques have been used to optimize traffic flow and control example Haddad et al. (2010) developed optimal steady state-state control for isolated traffic intersections, using a discrete-event max plus model for green light switching. Emphasis was placed on shorting queuing length by evaluating arrival and departure times at the intersection. The optimal switching sequence can be computed for the steady-state problem with constant cycle length by solving a LP problem analytically. Similarly Aihara et al. (2013) developed two strategies aimed toward real time optimal control for traffic flow. Optimal “on-off” laws based on bilinear control problem with binary constraints. First of all lyapunov function method is used and gives out a feedback law for setting on-off signal laws at the traffic intersection. The second strategy uses optimization method for binary optimization problems where it is a real variable optimization in which the binary constraint is treated by the exact penalty method. The method is tested to solve the traffic control problem but it should be widely applicable to other models formulated as 0–1 integer optimal control problems. Both methods are tested and compared, and the tests demonstrate that the both methods provide very effective and efficient traffic control laws.

Using another approach Holden and Risbero (2011) introduced a model that describes heavy traffic on a network of unidirectional roads. The model consists of a system of initial boundary value problems for nonlinear conservative laws. Firstly the Riemann problem was formulated and solved for such a system and based on this, the existence of a solution to the Cauchy problem is then shown. A few examples were shown to explain how to solve the traffic flow problem. Shifting focus to queue length Schutter and Moore (1998) developed a model that showed the evolution of the queue lengths in each lane as function of t. it is shown that for a class of objective function an optimal light switching scheme can be obtained efficiently. After creating cthe model simulations with examples were done to show the merits of the model. The major advantage of this model is that the green-amber-red cycle is allowed to vary from one cycle to another. Going further Schutter (2002) derived methods to optimize performance measures such as average or worst case waiting times and queue lengths for a switched system with linear dynamics subject to saturation, quantitative properties of the system were considered and it was advised that other techniques of interest in qualitative properties such as, safety should be used. Also suggestion was given for future research if the extension of the results obtained to networks of dependent queues, i.e., a situation where the outputs of some queues will be connected the inputs of some other queues. If a moving horizon strategy in combination with a decentralized control solution, it can apply still the approach given and use measurements from one queue to predict the arrival rates at the other queues provided that the routing rates are known (i.e., the amount of cars, fluid,. . . that will be routed from the output of queue to the input of queue ) and the traveling times from one queue to another.

The phase sequence order was assumed to be pre-fixed. The switching time optimization presented could be used as an inner loop in a discrete optimization outer loop that also optimizes the phase order.
A totally different technique used by Huang and Chu (2008) explain the modelling, analysis and implementation of an urban traffic lights system using timed coloured petri nets (TCPN) models for an intersection. A basic traffic light system model is developed, where the structural analysis of TCPN model is clearly shown, which can be improved upon and help in designing the extended models. Also emphasis is placed on operational flow of the traffic lights systems, from the derived TCPN model by looking into the schedule of the signal timing plan of the traffic systems. The traffic systems with signal timing plan for a day are successful in converting TCPN models. This is helpful because it is used to obtain a TCPN model for a complex urban traffic lights system The main advantage of this proposed approach as opposed to others is the clear presentation of the system behaviour and readiness for implementation. Similarly, Ganiyu et al. (2011) utilized the Timed Coloured Petri Net (TCPN) formalism to model and simulate a multiphase traffic light controlled junction with an associated fixed timing plan using a T-type junction located in Federal Capital Territory, Abuja, Nigeria, as a case study. The development of the Timed Coloured Petri Net model is limited to fixed time control strategy as traffic is currently ruled in the considered intersection following the signal timing plan. Extra features such as presence of pedestrians and motorcycles at the junction are not considered. However, the developed Timed Coloured Petri Net (TCPN) model of the T-type junction if explored further could help in studying and improving traffic flow for vehicles based on the knowledge acquired from the simulation. The result of the work could serve as a guide in setting real-time actuated signal controller for the modelled scenarios.
Göttlich et al. (2015) presented a mathematical model for traffic light settings within a macroscopic continuous traffic flow, by introducing a time dependent mixed integer traffic flow network problem and studied a traffic light control problem. A general strategy to reformulate the model is presented based on partial differential into a large-scale mixed-integer problem and to apply this approach to the nonlinear traffic flow model. To obtain the computational results, the existing black-box Branch-and-Bound algorithm is improved by branching priorities and bounding heuristics using information of the under lying dynamics governed by the partial differential equations. However, the optimization algorithm in its present form is not suitable for online-optimization. Ideas are given for future developments where Future work might include further research on computational methods to reduce the computation time. Also Nguyen (2011) developed a flexible traffic model based on the recent development of traffic theory and applied the developed model to simulate traffic and signal control in order to identify the optimal signal control policy in several traffic scenarios. The paper starts with a review of the cell transmission model, the widely used model in traffic theory. Several impracticalities of the theory are pointed out and necessary modifications are employed in order to build a flexible computer based traffic model successfully. Finally, the developed model is used for simulation of different traffic scenarios with three common traffic signal control policies to identify their strengths and weaknesses, which is essential for optimizing traffic control and enables us to answer if the most widely used signal control policy (fixed time with offset) is the most effective one.

Hamacher and Tjandra (2001) reviewed models and algorithms for evacuation planning. The review covered macroscopic models quite extensively and sketched microscopic models. Both approaches are able to mirror the flows of evacuations over time. The former has its strength in its possibility to optimize the system (while neglecting individuals’ behaviour), while the latter is able to capture and utilize properties of each of the evacuees.

Under the macroscopic approach, minimum turnstile cost dynamic network flow models can be applied to estimate the average evacuation time per evacuee. Maximum dynamic flows and universal maximum flows can be used to estimate the maximum number of evacuees which can reach safety during any given time horizon for the evacuation. Quickest flow models allow the estimation of the minimum time required to bring a given number of evacuees to safety. Considering the source and propagation of hazards, availability of emergency service units and better organization, the evacuation region can be divided into some regions with different priority levels. Therefore, multiple objective models are presented to cope with this problem. Constant travel time is mostly assumed in the literature. This time can be obtained by taking the travel time of the average flow or travel time of a specific queuing level. In order to reflect the congestion phenomenon, it was shown how the constant travel time assumption can be strengthened by considering density dependent travel time. This approach will, however, significantly increase the complexity of the model.

Surisetty and Sekhar (2016) used Webster’s methods in combination with HCM 2000 delay equation to compute the traffic signal design of Kothavalasa T-Intersection in India to reduce problems like road accidents, conflicts and congestions at that intersection. Some design parameters taken into consideration include the number of lanes provided on each approach and for each movement. Traffic data which comprises of the volume, density, flow, saturation and PCU vales are taken and evaluated using the equations to decide the optimal green time for each lane at the intersection also how long the pedestrian walk signal should be was taken into consideration.
3.0Methodology
The proposed work looks to develop a mathematical model that will help solve traffic issues. The approach involves using equations for an intersection that will factor in arrival time and departure time, which will shorten the queuing length for vehicles at the traffic lights and also running simulations to show how the model will work in a real life scenario. The signal design procedure involves these steps. They include the
(1) Phase design
(2) Determination of amber time and clearance time
(3) Determination of cycle length
(4) Apportioning of green time
(5) The performance evaluation of the above design.

A four (4) phase intersection will be taken as a case study an example is shown in Figure 3.1 and the order of phase movements are shown in Figure 3.2
9817776584Figure 3.1: A four phase intersection

Figure 3.2: The four phases at the intersection
Cycle length will be considered for the various phases of the traffic light at various times of the day. A simple cycle is shown in Figure 3.3

Figure 3.3: Traffic light cycle
The amber or clearance time shall be at two (2) seconds to give ample time for a driver to come to a stop or begin to move.

The cycle length can be calculated using

C0=1.5L+5(1.0-i=1nY) (1)
Where;
Co = the cycle length
L = the total loss time
Y = the sum of critical phase flow ratios
But
Y=Yi (2)Or Y1=q1s1+Y2=q2s2+Y3=q3s3+Y4=q4s4 (3)
Where;
q = the normal flow for each approach
s = the saturation flow per unit time

The normal flow value ‘q’ can be calculated using a Poisson distribution,
q=Px=?-x ×e-?x! (4)Where;
µ = flow rate per hour
x = the number of vehicles in unit time
The length of green time can be calculated using the formula
G1=Y1YC0-L (5)Where;
G = the length of green time
Y = the sum of critical phase flow ratios
Co = the cycle length
L = the total loss time
The set-up and the model of the system
A four phase intersection will be considered as shown in Figure 1. There are four lanes L1, L2, L3 and L4, and on each corner of the intersection there is a traffic light (T1, T2, T3 and T4). For each independent traffic light there are three subsequent phases: green, amber, and red. We assume that the duration of the amber phase is fixed and equal to ya. Let t0, t1, t2 . . . be the main switching time instants, i.e., the time instants at which the traffic lights switch from one main phase to another (as shown in Table 1).
Period T1 T2 T3 T4
t0-t1 Red Green Red Green
t1-t2 Green Red Green Red
t2-t3 Red Green Red Green
t3-t4 Green Red Green Red
Table 1: The traffic light switching scheme.Define yk = tk+1 ?tk. Let li(t) be the queue length (i.e., the number of cars waiting) in lane Li at time instant t. In reality li(t) will be an integer valued function and the arrival and departure rates will vary as a function of time. Defining the arrival rate at lane L1 at time t as a(t) and the departure rate as d(t) and continues to y(t) (where y(t) is the amber phase). The following can be logically deduced;
The relationship between the queue lengths at the main switching time instants when considering lane L1, the traffic light T1 will be red, meaning there are arrivals at lane L1 and no departures.

dl1(t)dt= a1 (6)for t ? (t2k, t2k+1) and l1(t2k+1) = l1(t2k)+ a2y2k . Furthermore for values of k = 0, 1, 2, when the traffic light T1 is green, there are arrivals and departures at lane L1. Since the net arrival rate is a1-d1 and since the queue length l1(t) cannot be negative, we have
dl1(t)dt=a1-d1 if l1t;00 if l1t=0 (7)as the queue length of lane L1.

4.0Expected Results
At the end of this project work, a good mathematical model that optimizes traffic signal control and reduces queuing length as well as increases the rate of departure is expected.

References
Aihara K., Ito K., ; Nakagawa J.,(2013) Optimal control laws for traffic flow. Applied Mathematics Letters 26: 617-623.Ganiyu R.A.,Olabiyisi S.O, Omidiora E.O.,Okedira O.O. ; Alo O.O., (2011). Modelling and simulation of a multi-phase traffic light controlled T-type junction using timed coloured petri nets. American Journal of Scientific and Industrial Research 2(3): 428-437. ISSN: 2153-649X doi:10.5251/ajsir.2011.2.3.428.437
Ganiyu, R. A., Omodiora, E. O., Olabiyisi, S. O., Arulogun, O. T. and Okediran, O. O. (2011): “The underlying concepts of Coloured Petri Net (CPN) and Timed Coloured Petri Net (TCPN) models through illustrative example”, Accepted Manuscript for Publication in International Journal of Physical Science, Paper No: ASCN/2011/012, African University Journal Series, Accra, Ghana.

Gottlich S., Hertry M., ; Zieglar U.,(2015) Modelling and optimizing traffic light settings in road networks. Computers and operations Research 55: 36-51.

Haddad J., Schutter B.D.,Mahalel D., Ioslovich I.,; Gutman P. O. (2010). Optimal steady state control for isolated traffic intersection. IEEE transactions on Automatic Control Volume 55, Number 1.

Holden H., ; Risbero N.H. (1998). A mathematical model of traffic flow on a network of unidirectional roads

Huang Y.S., ; Chung T.H., (2008). Modelling and analysis of Urban traffic light control systems using timed coloured petri nets. Journal of Information Science and engineering 24: 875-890.

Nguyen.T., (2011). A flexible model for traffic simulation and traffic signal control optimization.Comp 3740 European Journal of Control 4: 260-276.

Ramesh Surisetty ; Soma N Sekhar Designing of a Traffic Signaling System at T-Intersection International Journal of Engineering Research and Application www.ijera.com ISSN : 2248-9622, Vol. 7, Issue 4, ( Part -3) April 2017, pp.82-86
Reddy B.Sudharshan ; Reddy N.Venkata Hussain (2016) Signal design for t-intersection by using Webster’s method in Nandyal town, Kurnool district of Andhra Pradesh. International research Journal of engineering technology Volume: 3, issue:4 page 1124-1131
Schutter B.D., & Moor B.D. (1998). Optimal traffic control for a single intersection. European Journal of Control 4: 260-276.

Schutter B.D., (2002). Optimizing acyclic traffic signal switching sequences through an extended linear compatibility problem formulation. European Journal of Operations Research 139: 400-415.

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