# Freeway Traffic Congestion Reduction and Environment Regulation via Model Predictive Control
## 5 Problem Formulation  
This section mainly explains the optimization objectives in MPC_CPDMO-NSGA-II. This paper adopts TTS, TTD, TE and TF to evaluate the performance of the freeway, which are also used as the optimization objectives in dynamic multi-objective optimization algorithm CPDMO-NSGA-II. The prediction models for estimating these indicators are the integration of METANET and VT-micro model. (1) Total Time Spend (TTS)  
The TTS consists of two parts: (1) Total travel time (TTT); (2) Total waiting time (TWT). \[\min\qquad J_{1}(k_{c})=\sum\limits_{k=Zk_{c}}^{Z(k_{c}+N_{p})-1}T_{s}\cdot \left[\sum\limits_{m}\sum\limits_{i}\rho_{m,i}(k)\cdot L_{m,i}\cdot\lambda_{m,i }+\sum\limits_{o\in Ramp}\omega_{o}(k)\right]\enspace, \tag{26}\]  
where \(k\) is steps of sampling; \(k_{c}\) is steps of control; \(N_{p}\) is prediction horizon; and \(Z\) is the ratio of control period and sampling period. (2) Total Travel Distance (TTD)  
\[\max\qquad J_{2}(k_{c})=\sum\limits_{k=Zk_{c}}^{Z(k_{c}+N_{p})-1}\sum\limits_{ m}q_{m,i}\cdot T_{s}\enspace. \tag{27}\]  
In optimization, maximized objectives will be transformed into minimized problems with a minus. (3) Total Emissions (TE) and Total Fuel Consumption (TF)  
According to the VT-micro model, following equation is used to calculate the TE for various gases and the TF:  
\[J_{y}(k_{c})=\sum\limits_{k=Zk_{c}}^{Z(k_{c}+N_{p})-1}[\sum\limits_{i=1}^{3} \left(J_{y,m,i}^{t}(k)+J_{y,m,i}^{s}(k)\right)+J_{y,m,o}(k)], \tag{28}\]  
where \(y\in\{CO,HC,NO_{x}\}\). It should be noted that TE and TF are two different indicators, so they cannot be added together directly. In this paper, the problem is solved by standardization using the following:  
\[\min\qquad J_{3}(k_{c})=\frac{I_{CO}(k_{c})+I_{HC}(k_{c})+I_{NO_{x}}(k_{c})}{ I_{sl,emission}(k_{c})}\enspace+\frac{J_{FC}(k_{c})}{J_{sl,FC}(k_{c})}, \tag{29}\]  
where \(J_{sl,emission}(k_{c})\) is TE of the freeway with fixed speed limit strategy, and \(J_{sl,FC}(k_{c})\) is TF. They can be obtained through the follow-up simulation experiment. To ensure the safety of the environment, the control solutions should not have too much fluctuation. Besides, the flows of the freeway should not exceed capacity. Therefore, following constraints are considered in the optimization. (1) In the same control horizon, the difference of VSLs between adjacent segments is no more than 5 km/h:  
\[\left|v_{ctrl,i}(k_{c})-v_{ctrl,j}(k_{c})\right|\leq 5 \tag{30}\]  
(2) At the same segment, the difference of VSLs between adjacent control horizons is no than 10 km/h:  
\[\left|v_{ctrl,i}(k_{c})-v_{ctrl,i}(k_{c}+1)\right|\leq 10 \tag{31}\]  
(3) Between adjacent control horizons, the difference of RM rates is no more than 0.3:  
\[\left|r_{o}(k_{c})-r_{o}(k_{c}+1)\right|\leq 0.3 \tag{32}\](4) The range of RM rate is shown as follows:  
\[r_{o,\min}\leq r_{o}(k_{c})\leq 1 \tag{33}\]  
(5) The flows of the mainline and the on-ramp should not exceed the traffic capacity:  
\[q_{m,i}(k)-Q_{main,\max,i}\leq 0 \tag{34}\]  
\[q_{o}(k)-Q_{o,\max}\leq 0, \tag{35}\]  
where \(v_{ctrl,i}(k_{c})\) is the value of VSL applied to segment \(i\) at control horizon \(k_{c}\); \(Q_{main,\max,i}\) is the capacity of segment \(i\); \(Q_{o,\max}\) is the RM rate applied to the on-ramp \(o\) at control horizon \(k_{c}\); and \(Q_{o,\max}\) is the capacity of the on-ramp \(o\). Some parameters are haven been described in Section 3. Above constraints are referred to paper [26; 27]. Due to a shorter control period in this paper, some of them are modified in order to alleviate risk of slow response from drivers to changed control solutions. This paper employs static penalty function to dispose these constraints, so that the CPDMO-NSGA-II algorithm can search for non-dominated solutions in feasible domains. These penalties are added to existed objectives:  
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