Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 1857186, 22 pages

https://doi.org/10.1155/2017/1857186

## Energy Near-Optimal Control Strategies for Industrial and Traction Drives with a.c. Motors

^{1}Faculty of Electrical Engineering, University of Žilina, Žilina, Slovakia^{2}Faculty of Mechanical Engineering, University of Žilina, Žilina, Slovakia

Correspondence should be addressed to Ján Vittek; ks.azinu.lef@kettiv.naj

Received 11 October 2016; Revised 18 November 2016; Accepted 24 November 2016; Published 24 January 2017

Academic Editor: Mohammed Nouari

Copyright © 2017 Ján Vittek et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The main contribution of this paper is a new rest-to-rest position control system for use with electric drives employing a.c. motors that is near-optimal with respect to combined electrical and frictional energy waste minimization. The friction has constant, linear, and quadratic components with respect to the rotor speed. The closeness to optimality is assessed by simulation, comparing the energy loss of the new control system with that predicted by computed optimal controls. The application of the near-optimal control system is rendered straightforward by using a symmetrical trapezoidal speed-time profile. This is provided by an energy saving reference position generator whose output is faithfully followed by means of a feedback control law based on forced dynamics control yielding prescribed closed loop dynamics, together with a matched zero dynamic lag precompensator. For load torque consisting of constant, linear, and quadratic components also maneuver time is optimized if it can be chosen arbitrary. Two case studies, one applied to position control of rotational drive and second one applied to train movement, confirm the possibilities of achieving energy savings.

#### 1. Introduction

As a result of environmental concerns and minimization of energy consumption the calculation of energy optimal reference trajectories for electric drives is significant for possible energy saving [1]. The paper presents applications of energy optimal control theory based on Euler-Lagrange minimization to the rotational movement of industrial drives and to the traction motor of railway vehicle. Both applications assume load which consists of constant, linear, and quadratic components with respect to the rotor speed.

Two generally valid position control strategies respecting electrical losses of motor as well as mechanical losses minimization are developed and verified for comparison of their energy demands. The first strategy is based strictly on mathematical, Euler-Lagrange approach [2] while the second one is based on loss model exploiting symmetrical trapezoidal speed profile for electrical and mechanical losses minimization. The second approach is characterized with truly finite settling time and for prescribed movement allows precise prediction of all components of consumed energy. Total consumed energy of both control strategies is evaluated via time integral of the input power. For comparison of possible energy savings of both control strategies the investigations are completed with position control based on modified trapezoidal speed profile and control strategy based on triangular speed profile.

Principles of* vector control* and * forced dynamics control* are used for energy demands evaluation of all presented control strategies. Vector control of a.c. drives as a novel control strategy had opened possibility of controlling torque and field components of stator current independently and brought wide potential for considerable reduction of drive losses [3]. Significant energy savings also contributed to rapid spread of variable speed drives in industry and transportation. For simulations of energy saving position control strategies a new “forced dynamics control” technique (FDC) of electric drives employing a.c. motors based on feedback linearization [4] and enabling realization of various prescribed dynamic responses to speed demands is exploited.

The individual approaches to energy-efficient position control while respecting prescribed maneuver time between stops differ significantly. The problems of optimal trajectory planning based on the nonlinear continuous-time model and continuous state-space model of train movement are well presented in overview of Wang et al. [5]. The methods of solutions can be characterized as analytical solution and numerical optimization. The analytical methods have difficulties in finding general solutions due to complex nonlinear terms and consideration of the constraints. For the numerical optimization approaches the optimal solution is not always guaranteed too. The possible outcome is to solve the optimal trajectory planning problem as a mixed integer linear programming problem, which can be solved efficiently exploiting available solvers.

Combination of Model Predictive Control (MPC) and differential flatness is used by Pham et al. [6] for efficient energy management of an elevator employing PMSM supplied from microgrid* (solar panel* and* supercapacitor battery, backed-up with three-phase net)*. The flatness formulation of the considered system profiles consisting of stator currents and rotor speed are generated to minimize the stator resistances energy dissipation. Then, MPC is used to track given profiles while satisfying state input constraints. Simulation results using real system numerical data compare the energy efficiency of the designed method with method which combines the Maximum Torque per Ampere (MTPA) principles in connection with the trapezoidal speed profile.

The drive’s loss model is used by Blank et al. [7] for optimal trajectory planning. Iterative algorithm,* fmincon,* finds a constrained minimum of a scalar function of several variables to compute prescribed acceleration. The resulting trajectories* (constant for time-minimal interval, constant for half of maneuver time, sinusoidal, and linearly monotonously falling)* respect also prescribed jerk and reduce copper losses as well as losses of power electronics.

Minimization of the total dissipated energy taking into account Coulomb and viscous friction has been proposed by Xuejun et al. [8] for the drive employing d.c. motor with reduction gear. The optimal drive velocity and current functions are obtained as a function of the optimal zero crossing time when input torque of gear is changed from positive to negative. Comparisons of the dc motor drive consumption for optimized control algorithm and trapezoidal velocity function have revealed energy savings of optimized control algorithm, which were proportional to a moment of inertia.

For minimization of dissipated energy of battery powered mobile wheeled robot, C. H. Kim and B. K. Kim [9] analyzed and implemented three-step velocity control method. Two various forms of acceleration and deceleration profiles (constant and exponential) are investigated to find minimal energy consumption. As energy near-optimal solution the analysis reveals exponential acceleration profile completed with constant cruising speed and linear deceleration* (step acceleration-cruising speed-trapezoidal breaking)*. Objective function which respects drive’s energy is developed for three-step velocity control. Solution in real time finds efficient binary search algorithm. Suggested three-step velocity control extends the working time of mobile robot up to 30%.

A single-axis trajectory generator for point-to-point motion control was proposed by Thirachai et al. [10]. Generated profiles correspond to trapezoidal velocity function that can modify system parameters such as the destination, speed, and acceleration. A dead-beat controller is exploited when the current state variables are approaching the equilibrium point. Simulation results confirm that proposed system can provide finite steps and a zero steady-state error and showed that the overall performance is similar to a conventional method.

Direct robust adaptive controller for improvement of the tracking performance of high-speed train was developed by Luo and Xu [11]. The controller is designed using the back-stepping method to deal with nonlinearities and the parameterized uncertainties in the high-speed train dynamic model. Then, the projection algorithm is used to improve the robustness of the proposed control algorithm for slowly time-varying parameter. Theoretical analysis and the simulation results confirmed that the proposed algorithm can guarantee the stability of the designed closed loop system with the tracking error converging to a residual set exponentially.

For road vehicle employing PMSM Lu et al. in [12] solve problem of global energy optimization to obtain the optimal velocity reference. Dynamic programming taking into account efficiency model of the drive system and complex road conditions determines trajectory with minimum energy consumption. For control of speed of hybrid electric vehicle by controlling the throttle position Yadav et al. study performances of various controller* (five various controllers)* in [13]. Comparative analysis confirmed optimal performance of linear-quadratic regulator (LQR) for optimization of vehicle drive train efficiency.

To minimize energy expenditures of suburban trains De Martinis and Gallo [14] proposed new driving strategies using a railway simulation model as a subroutine. Their approach results in the definition of two optimized train speed profiles. The first model is based on energy saving approach while the second one is based on energy recovery strategies. Two scenarios for resulting speed profiles* (with coasting and without)* were proposed and tested on a real-scale case involving a suburban line for both models. Simulation results show a promising reduction in energy consumption with the optimized energy-efficient speed profile for a prescribed running time between two stops.

Sheepmaker et al. [15] studied optimal control of the train with respect to braking* (mechanical, mechanical combined with regenerative, and regenerative)*. Pseudo-spectral method is used to solve energy-efficient train control, which is modelled as an optimal control problem over distance. Results of analysis have shown that under regenerative breaking the optimal cruising speed is lower than without, the coasting regime is shorter, and the braking regime starts earlier. This led to extra energy saving what emphasizes the importance of kinetic energy utilization. The conclusion stresses substantial influence of regenerative braking on energy-efficient driving strategy and lower energy consumption.

Different control mechanism for a diesel-electric locomotive was used by Albrecht et al. [16] to design speed-holding control strategy for the train movement between two stations with nonconstant gradient and prescribed travelling time. Energy optimal type strategy based on discrete throttle settings alternates phases of coasting and maximum power to approximate the ideal minimum consumption strategy. The method has been successfully implemented in connection with special computation device for calculation of the optimal switching points and to provide in-cab advice to train drivers.

Detailed program for traction and brake applications capable of energy consumption minimization in the moving train along a given route for a given travel time was developed by Khmelnitsky [17]. The maximum principle is used to obtain the analytical information about optimal operation regimes and their sequences respecting also mixed and state constraints. Numerical algorithm exploiting this information then finds the optimal velocity profile. Designed control algorithm has capability of arbitrary restart due to outside conditions change and then recalculates the optimal velocity profile by constructing a new optimal profile for the remaining route length and travel time using the current speed as the initial one. Numerical examples confirm inherent accuracy and fast response of algorithm based on its analytical origin.

Contribution of this paper is the development and application of two energy near-optimal control strategies for the drives with rotational or translation movement and comparisons of their energy demands with modified strategies used for position control. To ensure the same conditions of comparison the regenerative braking is assumed for all presented strategies.

Designed control structure for simulation of energy saving control is shown in Figure 1 and consists of “Energy Optimal or Energy Near-Optimal Profile Generator” which is capable, from demanded position, ; prescribed maneuver time, ; and computed load torque, , of producing reference inputs: acceleration, ; speed, ; and position, for drive’s position control system. Block “Load Torque Computation” evaluates value of load torque as a function of drive’s speed using its individual coefficients for constant, ; viscous, ; and quadratic, components* (predetermined offline)*. The task of precompensator is to form control signal, , to ensure better tracking abilities of prescribed state variables. Design of precompensator is described in detail in Section 3.3 for vector controlled drive combined with FDC [18].