Pontryagin's Minimum Principle based model predictive control of energy management for a plug-in hybrid electric bus

To improve computational efficiency of energy management strategies for plug-in hybrid electric vehicles (PHEVs), this paper proposes a stochastic model predictive controller (MPC) based on Pontryagin’s Minimum Principle (PMP), which differs from widely used dynamic programming (DP)-based predictive methods. First, short-time speed forecasting is achieved using a Markov chain model, based on real-world driving cycles. The PMP- and DP-based MPCs are compared under four preview horizons (5 s, 10 s, 15 s and 20 s), and the results show that the computational time of the DP-MPC is almost four times of that in the PMP-MPC. Moreover, the influence of predication horizon length on computational time and energy consumption is examined. Given a preview horizon of 5 s, the PMP-MPC holds a total energy consumption cost of 7.80 USD and computational time per second of 0.0130 s. When the preview horizon increases to 20 s, the total cost is 7.77 USD with the computational time per second increasing to 0.0502 s. Finally, DP, PMP, and rule-based strategies are contrasted to the PMP-MPC method, further demonstrating the promising performance and computational efficiency of the proposed methodology

Model predictive energy management for plug-in hybrid electric vehicles considering optimal battery depth of discharge

When developing an energy management strategy (EMS) including a battery aging model for plug-in hybrid electric vehicles, the trade-off between the energy consumption cost (ECC) and the equivalent battery life loss cost (EBLLC) should be considered to minimize the total cost of both and improve the life cycle value. Unlike EMSs with a lower State of Charge (SOC) boundary value given in advance, this paper proposes a model predictive control of EMS based on an optimal battery depth of discharge (DOD) for a minimum sum of ECC and EBLLC. First, the optimal DOD is identified using Pontryagin's Minimum Principle and shooting method. Then a reference SOC is constructed with the optimal DOD, and a model predictive controller (MPC) in which the conflict between the ECC and EBLC is optimized in a moving horizon is implemented. The proposed EMS is examined by real-world driving cycles under different preview horizons, and the results indicate that MPCs with a battery aging model lower the total cost by 1.65%, 1.29% and 1.38%, respectively, for three preview horizons (5, 10 and 15 s) under a city bus route of about 70 km, compared to those unaware of battery aging. Meanwhile, global optimization algorithms like the dynamic programming and Pontryagin's Minimum Principle, as well as a rule-based method, are compared with the predictive controller, in terms of computational expense and accuracy

A Pontryagin Minimum Principle-Based Adaptive Equivalent Consumption Minimum Strategy for a Plug-in Hybrid Electric Bus on a Fixed Route

When developing a real-time energy management strategy for a plug-in hybrid electric vehicle, it is still a challenge for the Equivalent Consumption Minimum Strategy to achieve near-optimal energy consumption, because the optimal equivalence factor is not readily available without the trip information. With the help of realistic speeding profiles sampled from a plug-in hybrid electric bus running on a fixed commuting line, this paper proposes a convenient and effective approach of determining the equivalence factor for an adaptive Equivalent Consumption Minimum Strategy. Firstly, with the adaptive law based on the feedback of battery SOC, the equivalence factor is described as a combination of the major component and tuning component. In particular, the major part defined as a constant is applied to the inherent consistency of regular speeding profiles, while the second part including a proportional and integral term can slightly tune the equivalence factor to satisfy the disparity of daily running cycles. Moreover, Pontryagin’s Minimum Principle is employed and solved by using the shooting method to capture the co-state dynamics, in which the Secant method is introduced to adjust the initial co-state value. And then the initial co-state value in last shooting is taken as the optimal stable constant of equivalence factor. Finally, altogether ten successive driving profiles are selected with different initial SOC levels to evaluate the proposed method, and the results demonstrate the excellent fuel economy compared with the dynamic programming and PMP method

Influence analysis of complex crack geometric parameters on mechanical properties of soft rock

Abstract Soft rocks, such as coal, are affected by sedimentary effects, and the surrounding rock mass of underground coal mines is generally soft and rich in joints and cracks. A clear and deep understanding of the relationship between crack geometric parameters and rock mechanics properties in cracked rock is greatly important to the design of engineering rock mass structures. In this study, computed tomography (CT) scanning was used to extract the internal crack network of coal specimens. Based on the crack size and dominant crack number, the parameters of crack area, volume, length, width, and angle were statistically analyzed by different sampling thresholds. In addition, the Pearson correlation coefficients between the crack parameters and uniaxial compression rock mechanics properties (uniaxial compressive strength UCS, elasticity modulus E) were calculated to quantitatively analyze the impact of each parameter. Furthermore, a method based on Pearson coefficients was used to grade the correlation between crack geometric parameters and rock mechanical properties to determine threshold values. The results indicated that the UCS and E of the specimens changed with the varied internal crack structures of the specimens, the crack parameters of area, volume, length and width all showed negative correlations with UCS and E, and the dominant crack played an important role both in weakening strength and stiffness. The crack parameters of the angle are all positively correlated with the UCS and E. More crack statistics can significantly improve the correlation between the parameters of the crack angle and the rock mechanics properties, and the statistics of the geometric parameters of at least 16 cracks or the area larger than 5 mm2 are suggested for the analysis of complex cracked rock masses or physical reproduction using 3D printing. The results are validated and further analyzed with triaxial tests. The findings of this study have important reference value for future research regarding the accurate and efficient selection of a few cracks with a significant influence on the rock mechanical properties of surrounding rock mass structures in coal engineering