The main objective of the chapter is to use Matlab software in order to develop the optimal control knowledge in relation to the biggest worldwide power consumers: electrical machines. These two major components, optimal control and electrical machines form an optimal drive system. The proposed drive system conducts to the energy efficiency increasing during transient regimes (starting, stopping and reversing), therefore reducing the impact upon the environment. Taking into account the world energy policies, the chapter is strategically oriented towards the compatibility with the priority requirements from world research programmes. This chapter will offer an original Matlab tool in order to implement a highly performant control for electrical drives. The optimal solution provided by the optimal problem is obtained by numerical integration of the matrix Riccati differential equation (MRDE). The proposed optimal control, based on energetic criterion, can be implemented on-line by using a real time experimental platform. The solution has three terms: the first term includes the reference state of the electrical drive system, the second term assures a fast compensation of the disturbance (i.e. the load torque in case of the electrical drives) by using feedforward control, and the third is the state feedback component. Therefore, there is a completed optimal control for linear dynamic systems. The simulation and experimental results will be provided. By using knowledge regarding the electrical machines, electrical and mechanical measurements, control theory, digital control, and real time implementation, a high degree of interdisciplinarity is obtained in the developed Matlab tool The optimal problem statement is without constraints; by choosing adequate weighting matrices, the magnitude constraints of control and of state variable are solved.
Matlab is a widespread software used both in academia and research centres. The author of this chapter facilitates the easy understanding and implementation of the highly efficient electrical drive systems by using Matlab software. The energy problem is one of the most important aspects related to the sustainable growth of the industrial nations. On the one hand, the industrial systems and the increased quality of life require every year a higher quantity of electrical energy produced; on the other hand, the same quality of life requires the generation and the harnessing of this energy with low pollution.
On the one hand, the electrical machines are the main worldwide energy consumer (Siemens), (Tamimi, et al., 2006). Therefore, the focus of the research on this system type is essential in order to minimize the energetical losses. A seemingly insignificant reduction of electricity consumed by each electrical motor has particularly important consequences at national/international level, primarily by reducing consumption of fossil fuels widely used to generate electricity, therefore contributing also to CO2 emissions reduction. Researches in this area are at different levels: I) direct conversion by redesigning the electric motors to obtain high efficiency; II) the power electronic systems by introducing static converters with diminished switching losses; III) control implementation of the conversion process by using the enhanced capabilities of new generation digital signal processor DSP (Veerachary M., 2002).
The efficiency of the motor is high on steady state operation, but very low in transients. On the other hand, the associated equipment for energy conversion purposes, i.e. power converters together with the control subsystem, assures both high static and dynamic performances (related to the settling time, error minimizing and so on), but they are not related to energy expenditure. The analysis of various technical publications shows a yearly increase of energy expenditure of the electrical motors, with a rate of 1,5% in Europe, 2,2% in USA, 3,1% in Japan; these values can even reach the rate of 13%, depending on the development of calculus technology (Matinfar, et al., 2005), (Gattami, et al.,2005), (COM, 2006).
Dynamic programming tool is very useful to obtain the optimal solution, but the values of the torque and speed must be known in advance (Mendes,
The optimal solution (Gaiceanu, 1997) provided in this chapter overcomes this type of scientific barrier (Rosu, 1985), making possible implementation of the optimal solution to any type of application, without knowing the speed and load torque in advance. As it is nonrecursive, determined on the basis of variational methods (by integrating the Matrix Riccati Differential Equation), the optimal solution can be calculated on-line without memorizing it from the final time to the initial time, as in the recursive method. Therefore, the nonrecursive solution can be implemented to any type of electrical motor with any load variation, for any operating dynamical regimes.
On the other hand, the problem of increasing power conversion efficiency in the drive systems is the subject of numerous worldwide studies. The Siemens Automation (Siemens) claims the optimization based on the speed controller which can be optimized in the time or in the frequency range. In the time range, the symmetrical optimum method can be used. In the frequency range, the optimizing is done according to the amplitude and phase margin rules, known from the control theory, but these are the conventional controls that are used in drives.
The interdisciplinarity of the chapter consists of using specific knowledge from the fields of: energy conversion, power converters, Matlab/Simulink simulation software, real time implementation based on dSPACE platform, electrotechnics, and advanced control techniques.
This chapter is focused on mathematical modelling, optimal control development on power inverter, Matlab implementation and analysis of the optimal control solution in order to minimize the electric input energy of the drive system. The Matlab on-line solution can be downloaded in real-time platforms, as dSPACE or dsPIC Digital Signal Controllers (Gaiceanu, et al., 2008).
The practical implication of using the proposed method is the on-line calculation of the optimal control solution. The main lines of the research methodology are:
The optimal control of the dynamical electrical drives problem formulation;
The optimal control solution;
The numerical simulation results;
The implementation of the optimal control.
The strategy of the optimal control electrical drive system is to use both the conventional control during the steady state and the developed optimal control during the transients. The conventional control is based on the rotor field orientated control using Proportional Integral controllers. If the speed error is different from zero, the software switch will enable the optimal control, else will enable the conventional control. Both types of control are developed in the next Sections.
2. Mathematical model of the variable speed electrical drives (VSED)
There are variable speed DC and AC drives. A DC drive system controlled by armature voltage at constant field is a linear, invariant and controllable dynamic system. Variable speed AC drives are mostly used in connection with induction and synchronous electrical machines. The modern drives contain the energy source, the power converters, the electrical machines, the mechanical power transmission and the load, linked by an adequate control. The well-known mathematical model of the vector controlled AC drive using three phase squirrel cage induction machine in rotor field coordinates supplied from the current inverter, operating at the constant flux (Leonhard, 1996) becomes a linear system (Gaiceanu, 2002). For the electrical drives having the power less than 50 kW, the power loss of the converter could be neglected. In order to compute the optimal solution for synchronous Drive system, the control in rotor field oriented reference frame of surface mounted permanent magnet synchronous motor (PMSM) drives is used. The decoupling of the control loops (torque and flux) is performed in rotor field reference frame. By setting a zero value of direct current stator component, the mathematical model of the PMSM becomes linear. Therefore, there is a common mathematical model of the electrical drives, described by the linear state space standard form representation:
where is the state vector,
Taking into consideration the most used electrical machine, the three-phase squirrel cage induction machine, the electrical drive based on it will be explained as follows. Starting from the nameplate data of the induction machine (Fig.1), the main parameters of the electrical motor can be found (Appendix 1).
The mathematical model of the three phase squirrel cage induction machine (IM) in rotor field reference system operating at constant flux is (Leonhard, 1996):
By varying the flux component of the stator current,
3. Optimal control problem statement
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.
3.1. The 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, 1985), (Gaiceanu, 2002), (Athans, 2006), is as follows:
in which, taking into consideration a starting of the IM, the initial conditions consists of the initial time
3.2. The solution of the optimal control problem.
The existence and the unicity of the optimal control problem solution are based on the controllability and observability of the system (1). The system (1) being controllable and completely observable, the weighting matrices are chosen such that
By using variational method, the Hamiltonian of the optimal control problem is given by
By using the canonical system, the solutions of the costate and state vectors can be obtained:
The boundary conditions of the canonical system (5) are:
-the initial state ;
-the transversality condition of the costate vector:
where the weighting matrix , and
The integration of the canonical system leads to the well-known matrix Riccati differential equation and the associate vectorial equation (Rosu, 1985), (Gaiceanu, 2002), (Athans, et al., 2006). 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., 2005), (Jianqiang,
A nonrecursive solution of the Riccati equation has been developed by using two linear transformations (Rosu, et al., 1998).
In order to use only the negative eigenvalues for the computation of the fundamental matrix of the system (1),
and by noting the canonical matrix (4×4) of the system (1) as:
the eigenvalues of the canonical matrix
By using the developed Matlab file from Appendix 3, the appropriate negative eigenvalues position placed on the main diagonal of the eigenvalues matrix,
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 knowing the inverse of the eigenvector matrix
the new vectors,
By introducing (15) and (16) transformations in (8), the following differential system is obtained:
Therefore, the canonical system can be written in terms of
By using the well-known relation
and by noting the new vector
the new form of the canonical system can be obtained:
The solution of the new canonical system (21) can be obtained by using the following equation:
As it can be observed the solution of the canonical system (22) solution contains both the positive, , and the negative exponentials, . It is well-known that the positive exponentials produce the instability of the solution. Moreover, in order to obtain the analytical solution of the system (22) the load torque (
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
The obtained solution of the canonical system (23) is:
The negative exponentials can be obtained by extracting
The negative exponentials of the system
The transversality condition (6), in terms of
Going back through the reverse transformations from
where the matrix
The solution of the optimal control problem has three components (Fig.3):
the state feedback , in order to assure the stability of the system;
the compensating feedforward load torque , for fast compensation of the main perturbation, i.e. the load torque;
the reference component , to track accurately the reference signal.
The numerical solution supposes the knowledge of the disturbance
4. Simulation and experimental results
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:
4.1. Optimal control implementation
Optimal control implementation consists of loading nine software modules developed by the author: nameplate motor data (Fig.1), determination of the motor parameters (Appendix 1), determination of the six sectors vector components to be used in Space Vector Modulation (SVM) (Appendix 2), standard space state form characterization of the three phase Induction Motor (Fig.2), the proper arrangement of the eigenvalues and eigenvectors (Appendix 3), optimal control implementation in MATLAB (Appendix 4), the main program (Appendix 5), graphical representation (Appendix 6), Space Vector Modulation –Matlab function
The SVM increases the DC-link voltage usage, the maximum output voltage based on the space vector theory is times higher than conventional sinusoidal modulation (Pulse Width Modulation). The purpose of the SVPWM technique is to approximate the reference voltage vector 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, 2002). 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.5, Fig.7).
4.2. Conventional vector control
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. The references of the control loop are: the desired angular velocity for the speed loop,
By knowing the nameplate motor data (Fig.1), the tuning of the PI speed and flux controllers parameters can be done (see the developed Matlab file shown in the Appendix 8). The conventional vector control supposes the use of the two independent control axes:
The inputs of the speed and flux regulators Simulink block (Fig.9) are the speed reference and the flux reference; the outputs are the
By using the field orientated mechanism implemented in Simulink (Fig.11), based on the input signals (IQ*,
The parameters of the speed and flux regulators are delivered by using the Matlab file developed in the Appendix 8.
In the Fig.12. the detailed decoupling and the voltage compensation between
The reference voltage components pair (VD*, VQ*) is used as input of the Pulse Width Modulation (PWM) transfer function block, shown in the Fig.13.
The performances of the speed control loop are shown in the Fig. 14, in which the step rotor speed reference (om_ref*) is applied and the actual rotor speed (om) follows the reference with zero steady state error. The performances of the implemented flux control loop are shown in the Fig. 15. Small flux overshoot (up to 11%) and zero steady state error are obtained (Fig.15). The IM starts after
Both types of control, i.e. conventional and optimal, have been implemented on DS1104 (Fig.17) controller board. The developed ControlDesk interface has two main functions: real time control and signals acquisition. The evaluation of the obtained experimental results reveals the system efficiency improvement through an important decreasing of the windings dissipation power and input power (Fig.18) in optimal control case. Therefore, the power balance provides an increasing of the system efficiency, thus improved thermal and capacity conditions are obtained.
The optimal control (7) of the electrical drive based on the induction machine (1) has been numerically simulated by using Z transform and zero order hold for a starting of a 0.75 [kW], 1480 [rpm] under a rating load torque of 4.77 [Nm]. The initial conditions of the optimal control problem are:
In order to obtain the optimal control, the nonrecursive solution has been used to avoid the main disadvantages of the recursive solution (that is, the solution can be calculated at the current time, not backward in time; the load torque can have any shape, either linear or nonlinear, in admissible limits, due to the fact that the solution is computed at each sample time).
A simplified solution is proposed by avoiding the constrained optimal control problem type. By using adequate weighting matrice
The optimal problem shown in this chapter has been formulated for a starting period. In the Fig.18, the energy evaluation for both control methods, conventional and optimal, is depicted. According to Bellman's theorem of optimality, the optimal problem can be extended also on the braking, or reversing periods. The components of the energy of the three-phase IM (Fig.20) are: W1 the input energy, W2 the output energy, WCu1 the energy expended in the stator windings, WCu2 the energy expended in the rotor windings, WFv the viscous friction dissipated energy, WpJ the accumulated energy in inertial mass, W the electromagnetic energy.
From the energetic point of view, it could be noticed that the output energy is the same in both controls (conventional and optimal) while the input energy decreases significantly in the optimal control case due to the decreasing of the stator copper losses, even if the rotor losses slightly increase in the optimal control case (Fig.20).
The proposed optimal control conducts to 8% electrical energy reduction, despite of using the conventional control. The conclusion is: decreasing the input energy for the optimal control system at the same time with maintaining the output energy in both systems, classical and optimal, the efficient electric drive system is obtained.
Due to using only the negative exponentials terms, the system is stable.
The optimal control can be applied in the applications with frequent dynamic regimes. The minimization of the consumed energy conducts to an increasing of the operational period of the electrical drives, or to the electrical motor overload allowance.
The optimal control could be extended to the high phase order induction machine applications, in a decoupled
Taking into consideration that this chapter is focused on developing and proving the performances of the optimal Electrical Drive Systems (EDS), an immediate effect is represented by the positive impact on the market as an alternative for electrical efficiency. The key of the proposed chapter is the integration of the developed Matlab tool by means of the high technology, i.e. the development of the systems with microprocessor. The chapter leads to drawing up specific research and development methods in the field of electrical drives. It has been underlined that by making EDS in an efficient manner, the CO2 emission reduction will be also obtained.
The achievement of this chapter is the opportunity to attract young MSc. and PhD students in R&D activities that request both theoretical and practical knowledge. The author will have new opportunities for developing international interdisciplinary research, technological transfer, sharing of experience and cooperation in the framework of the specific Research Programmes.
The Matlab file for tuning of the PI speed and flux controllers parameters in conventional control
This work was supported by a grant of the Romanian National Authority for Scientific Research, CNDI– UEFISCDI, project number PN-II-PT-PCCA-2011-3.2-1680.
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