### Research papers on electrical drives

This equipment has passed the standard test of product checking center of engine and vehicle of Ministry of Railway and has finished the on board test of Guangzhou Metro Line 1. It has high performance-price ratio. This equipment has passed the standard test of product checking center of engine and vehicle of Ministry of Railway in April, and has finished the netting test in the new depot of Guangzhou Metro Line 4 in March, The trial operation in Beijing and Tianjin will be tried soon. It is used for applying AC power that linear electric motors need. It adopts a vector control method compensated the end effect and realize high-performance traction control of linear electric motors.

### International Journal of Advances in Electrical Machine and Power Electronics

The equipment was tested in the test wire of the new type linear electric motor transporting system in the city of Hohhot in the Nei Monggol Autonomous Region and has been working for 3 years. Starting from the nameplate data of the induction machine Fig. The mathematical model of the three phase squirrel cage induction machine IM in rotor field reference system operating at constant flux is Leonhard, :. By varying the flux component of the stator current, i ds , the magnetizing current response, i m r t , can have signifiant delay. Therefore, the standard state space form of the IM mathematical model can be obtained Fig.

The common objectives of the optimal control drive systems are: the smooth dynamic response, without oscillations; achieving the desired steady state; the fast compensation of the load torque; energy minimization. Taking into consideration the main objectives of the electrical drive system, the following Section describes the chosen quadratic performance criterion. The functional cost or the quadratic performance criterion is chosen such that the electrical input power, the energy expended in the stator windings of the electrical machines is minimized, and the cost of the control is weighted.

Therefore, the performance functional quadratic criterion Rosu, , Gaiceanu, , Athans, , is as follows:. The final condition is the required final state, x 1. The value of the final time is established by the feasibility condition, i. The optimal control problem taken into account in this chapter is with free-end point, fixed time and unconstrained.

### 1. Introduction

The restrictions of the magnitude for the control vector, and state vectors are managed through the design process, by an adequate choice of the weighting matrices. The existence and the unicity of the optimal control problem solution are based on the controllability and observability of the system 1.

In this way, the existence and the unicity of the optimal control problem solution are assured. The optimal control minimizing the Hamiltonian 4 is given by. By using the canonical system, the solutions of the costate y t and state vectors x t can be obtained:. The integration of the canonical system leads to the well-known matrix Riccati differential equation and the associate vectorial equation Rosu, , Gaiceanu, , Athans, et al.

## Tool of the Complete Optimal Control for Variable Speed Electrical Drives

The integration of these two equations is a very difficult task because the Riccati equation is nonlinear, its solution is recursive which can be calculated backward in time Gattami, et al. Moreover, the backward computation needs to know a priori the variation of the perturbation vector w t during the control interval [0, t 1 ]. In most electrical drives the last condition cannot be accomplished.

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A nonrecursive solution of the Riccati equation has been developed by using two linear transformations Rosu, et al. The first transformation changes actual time t into t 1 -t, that is time remaining until the end of the optimal process. In order to use only the negative eigenvalues for the computation of the fundamental matrix of the system 1 , the second transformation is made.

In the Appendix 3 the proper position of the eigenvalues and the eigenvectors of the linear system 1 is provided. By using the program sequence from Appendix 3, the stability of the system is assured and the corresponding matrix of the eigenvectors is obtained:. By introducing 15 and 16 transformations in 8 , the following differential system is obtained:.

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The solution of the new canonical system 21 can be obtained by using the following equation:. It is well-known that the positive exponentials produce the instability of the solution. For the most used electrical drive system this condition is hardly accomplished because the load torque cannot be known in advance, at the final time. A numerical solution of the optimal problem is proposed. The advantage of maintaining the sampled signal at the same value during the sampled period is that the zero order hold ZOH offers a brilliant solution to know the load torque at the final time. Therefore, under this assumption, the solution 22 is computed each sampled time, T, and the perturbation vector r has constant value during it:.

In this way, the first equation of the system 24 becomes:. The negative exponentials of the system 25 guarantee the stability of the system 1. The transversality condition 6 , in terms of m and n vectors, is as follows:. The costate vector at current time is as follows:. The solution of the optimal control problem has three components Fig. The optimal control, at any moment t , is given by Gaiceanu, , Athans, et al. In order to implement the optimal control system two electrical drive systems are combined: the conventional vector control and the optimal control drive.

Both electrical drive systems operate in parallel. The activation of one of them depends on the speed error: if the speed error is zero steady state operating regime , then the conventional vector control is active, else the optimal control drive is active.

Optimal control implementation consists of loading nine software modules developed by the author: nameplate motor data Fig. By using the Matlab code from Appendix 8 the graphical representation of the simulation results can be obtained Figures 4 The purpose of the SVPWM technique is to approximate the reference voltage vector U o u t by a combination of the eight switching patterns.

The implemented Matlab sequence for obtaining the pulse pattern of the optimal SVM modulator Appendix 2, Appendix 7 shown above is based on the developed methodology by the author Gaiceanu, In optimal control case, there is no speed overshoot, and smoothly state response speed and angle of the rotor magnetizing current have been obtained Fig. The conventional vector control consists of the rotor flux orientated drive system. The vector control is implemented in Simulink as following. The control system is in cascaded manner and consists of two major loops: the speed loop and the magnetic flux loop.

The torque control loop is the minor loop of the speed control one. By knowing the nameplate motor data Fig. The conventional vector control supposes the use of the two independent control axes: d -axis, along with the magnetizing flux direction, and q axis, in quadrature with d -axis. The most used field orientated control is that of the rotor orientated magnetizing flux Fig.

The magnetization flux and the torque can be controlled independently. Simulink implementation of the rotor field orientated vector control of the three-phase induction motor.

## Power Electronics and Drives

The inputs of the speed and flux regulators Simulink block Fig. More details can be obtained by using Fig. By using the field orientated mechanism implemented in Simulink Fig. Rejection means the submitted paper will not be published. Plagiarism is committed when one author uses another work without permission, credit, or acknowledgment. Plagiarism takes different forms, from literal copying to paraphrasing the work of another. Authors submitting to an IIETA journal should be aware that their paper may be submitted to CrossRef at any point during the peer-review or production process.

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## Power Electronics and Drives | Sciendo

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