## 3 Optimization-Based Strategies
### Instantaneous Optimization
#### 3.1.2 Model Predictive Control (MPC)  
Common approaches for online energy management reduce the computational time by implementing optimization for a short-time finite horizon of the entire trip and repeating it at every time step. The MPC [56; 57; 58] strategy, as one of the instantaneous optimization control strategies, and the rolling time domain control, have been extensively studied theoretically and applied in different fields, and is able to achieve an approximate fuel economy to DP and can be implemented online with limited computation and memory resources. The MPC strategy depends on a short-term prediction of driver-demanded power in the future at each moment, which is different from DP, which requires the whole driving cycle in advance. Model predictive control (MPC) is based on rolling optimization, which converts the optimization process into a limited prediction horizon to reduce calculations and has the potential for real-time control. The short-term prediction can be studied from the perspective of the time domain or the distance domain. In the time domain, each location has the same time of \(\Delta s\), and control inputs are updated with a fixed time step. Similarly, in the distance domain, each location has the same distance of \(\Delta l\), and control inputs are updated with a fixed distance step. The majority of studies have implemented the prediction from the perspective of the time domain. With respect to the distance domain, a distance-based ecological driving scheme with long-term speed optimization and short-term adaptation is proposed in [59]. The optimal speed profile for the entire route is optimized by road conditions, and the QP method is used for optimization to save computation time. In the short term, the speed at the next location is adapted by the spacing to the preceding vehicle. The main procedures of the MPC-based strategy are as follows, and the main research areas are summarized in Figure 5:  
Figure 5: Depiction of the main research areas of the MPC-based strategy. * Predict vehicle velocity in a prediction horizon, and obtain optimal control trajectory in this horizon based on minimizing objective function with multivariable constraints. If taking fuel consumption, the performance of driving, and battery SOC reference trajectory into consideration, the cost function can be written as: \[J=\int_{t_{0}}^{t_{h}}\!\!\Big{[}w_{m}\!\Big{(}\dot{m}_{f}(t)\Big{)}^{2}+w_{s}( SOC(t)-SOC_{r})^{2}+w_{v}(v(t+1)-v_{tar}(t+1))^{2}\Big{]}dt\] (6) where \(t_{0}\) is the current time, \(t_{h}\) is the terminal time during a prediction time horizon, \(w_{m},w_{s},w_{v}\) are penalty weights, \(SOC_{r}\) is the value of reference SOC, \(v\) is vehicle speed, \(v_{tar}\) is the target speed. * Implement the first element of the optimal control sequence in the corresponding vehicle model and send the feedback adjustment of the estimated SOC to the optimization section after the vehicle has responded;
* Move the entire prediction horizon one step forward, that is, refresh the optimization problem with the latest measured value at each moment;
* Repeat the steps 1 to 3. In general, the Markov Chain method, Monte Carlo Method [60] or artificial neural network algorithm can be utilized to predict vehicle speed. Markov Chain Monte Carlo Method (MCMC) can be adopted to predict vehicle velocity with post-processing algorithms including average filtering and quadratic fitting to moderate fluctuations of the prediction results, which can effectively improve the predictive accuracy and enhance the control performance of MPC-based strategy [24]. In [61], a velocity predictor based on current driving environment and vehicle information is developed to predict future driving conditions, which are established by the radial basis function neural network (RBF-NN). Meanwhile, the nonlinear model predictive control with forward dynamic programming is utilized to construct the master controller, which obtains optimal control variables online.