## 3 Optimization-Based Strategies
### Instantaneous Optimization
#### 3.1.2 Model Predictive Control (MPC)
To improve prediction accuracy, intelligent transportation systems (ITSs) with some vehicular telemetry technologies including onboard GPSs, geographical information systems (GISs), and advanced traffic flow modeling techniques [49; 62] can be used to access information of the traffic or road conditions. Based on the traffic data from telematics, the battery SOC profile can be scheduled systematically by considering the effect of road conditions, battery and real-time implementation ability, which can efficiently improve fuel economy based on the MPC strategy [63]. In addition, a multi-step Markov prediction method can be used to predict the driving conditions, and the DP method can be used to solve the optimization problem within the prediction horizon [64]. As for the optimization of the short-term horizon, the methods used could be the PMP algorithm, the DP algorithm, or an intelligent algorithm, such as the genetic algorithm (GA), particle swarm optimization (PSO), simulated annealing (SA), quadratic programming (QP), and so forth. As described in [65], a predictive energy management strategy is proposed with an online correction algorithm in the optimal energy management strategy of PHEVs, which is optimized with the dynamic neighborhood particle swarm optimization (PSO) algorithm. In [66], several quadratic equations were employed to determine the engine fuel rate with respect to battery power, using the QP and SA methods together to find the optimal battery power commands and engine-on power. The Markov chain is a series of transition probabilities from one of the limited states at instant \(k\) to another state of all possible states at instant \((k+1)\), which is mainly used to model and solve dynamic decision-making problems. The key of the Markov chain is to obtain the transition probability matrix \(T\), whose element can be defined as:  
\[T(i,j)=Pr\Big{[}x(k+1)=x_{i}\big{]}x(k)=x_{j}\Big{]}(i,j\in 1,2,\ldots,N) \tag{7}\]where the matrix \(T\) represents the transition probability from the state \(x_{j}\) at the moment \(k\) to the state \(x_{i}\) at the moment \(k+1\). In the MPC-based strategy, the future torque/power demand of powertrain can be modeled by Markov chain; therefore, this problem can be regarded as a kind of stochastic constrained optimization problem for nonlinear systems. Based on the above theory, a novel stochastic model predictive control (SMPC) [67; 68] strategy is developed to contribute the practical application of the MPC-based strategy. To improve real-time performance, an SMPC-based energy management strategy for PHEB uses state reconstitution method to guarantee the continuity of practical application, and employs time-varying predictive steps by an online accuracy estimation method and a corresponding threshold to maintain the prediction accuracy [24]. In [25], the SMPC is modified with the equivalent consumption minimization strategy for PHEB, which considers the reference SOC trajectory in the finite predictive horizons to eliminate undesirable working points. Based on SMPC with learning (SMPCL), a driver-aware vehicle control was developed in [69], using Markov chain to adapt to changes in driver behavior, and the QP method was used for optimization to handle larger state dimension models. The simulation results showed that the performance of the proposed approach was close to that of MPC with full knowledge of future driver power requests in standard and real-world driving cycles. Due to the fact that the road conditions significantly influence the battery charging and discharging processes of HEVs, it is worth considering that the previewed road grade can be used to predict the future power demand to improve the performance of energy management without the route being determined in advance. Ref. [70] proposed an SMPC-based energy management strategy to maintain battery SOC within its boundaries and achieve good energy consumption performance.