Hội nghị toàn quốc về Điều khiển và Tự động hoá - VCCA-2011 VCCA-2011 Performance of robust controller for DFIM when the rotor angular speed is treated as a time-varying parameter Nguyen Tien Hung 1 , Ngo Duc Minh 2 1 Thainguyen University of Technology, email: h.nguyentien@tnut.edu.vn 2 Thainguyen University of Technology, email: ngoducminh@tnut.edu.vn Abstract: This paper describes the design of a robust current controller for doubly-fed induction machines (DFIM), in which the rotor angular speed is considered as an uncertain parameter. The robust controller is then synthesized to guarantee that the -norm of the closed-loop system is smaller than some given number for different frozen values of . Next, the robust performance of the robust controller with respect to other rotor angular speeds is investigated for both constant and fast parameter variations. Some simulation results are given to demonstrate the performance and robustness of the control algorithm. 1. Introduction In the literature, the control structure of DFIM including PI current controllers is described in [1], [2], [3], [4]. In some cases, the cross coupling term in the rotor equations that includes the mechanical angular speed is eliminated by adding a feed-forward term to the output of the q-axis controller [2], [5]. The rotor mechanical angular speed is treated as an scheduling parameter that is used for these compensators. In these situations the difficulties of the nonlinear dynamics of the doubly-fed induction machine are not taken into account, i.e., the model of the machine is linearized and it is assumed that the machine parameters required by the control algorithm are precisely known. Clearly, such controller designs might result in a closed-loop behavior that is highly sensitive to a change in operating conditions and/or parameters. In this work, a mixed loop shaping -design for the rotor current control loop at fixed frozen values of the rotor angular speed is presented first. Then the performance of the closed-loop system with controller designed for different frozen values of for other rotor angular speeds is investigated. The performance analysis is also extended for the case with the face of the stator voltage action. As a further investigation, the designed controller for a frozen values of is tested for a fast variation of the rotor speed along the whole parameter interval. 2. Preliminaries 2.1 Notations Let denote the space of square-integrable signals defined on the interval . A matrix is called symmetric if it is real and satisfies . The set of all symmetric matrices will be denoted by . A transfer function with a state-space realization will be denoted by 2.2 Linear matrix inequalities A linear matrix inequality (LMI) has the form (1) where denotes the vector of decision variables and . 2.3 The -norm Consider a linear input-output system that is described by (2) and whose transfer matrix is given by If is stable and if we choose the initial condition to be zero, defines a linear map on with a finite energy gain defined as It is well-known that the energy-gain of coincides with the -norm of the corresponding transfer matrix given by where stands for the largest singular value of the complex matrix matrix . 2.4 The bounded real lemma It is not possible to explicitly compute in terms of the realization matrices. Instead, one can 377 Hội nghị toàn quốc về Điều khiển và Tự động hoá - VCCA-2011 VCCA-2011 characterize stability of and the validity of the inequality (3) as an LMI in some auxiliary matrix and , which is one version of the celebrated bounded real lemma. Indeed, it can be shown [6] that is stable and that (3) holds if and only if and there exits some such that the Riccati inequality (4) is satisfied. By the Schur lemma (4), these conditions are equivalent to following system of LMIs [7] (5) This result is referred to as the bounded real lemma. Yet another application of the Schur lemma allows to rewrite these inequalities with into the following form [8], [9]: (6) Note that (6) are LMI constraints on and . This allows to determine the infimal for which (3) is true, and hence in turn the value , by minimizing over the constraint (6) which is a standard LMI problem. Let us now show how this procedure of analysis can be successfully generalized to synthesizing controllers. 2.5 performance A standard setup for control is presented in Figure 1, where represents the generalized disturbances, the controlled variable, the control input and the measurement output, while is a linear time-invariant system described as w z u y P K Figure 1. The interconnection of the system The goal in control is to find a stabilizing linear time-invariant (LTI) controller that minimizes the norm of the closed-loop system (7) where is lower linear fractional transformation of and , which is nothing but the closed-loop transfer function in Figure 1. 2.6 Sub-optimal control Let us now consider a generalized plant where weights are incorporated already as follows (8) If the linear time-invariant controller is expressed as (9) the closed-loop system admits the following state-space description: (10) where (11) In practice, the control problem is rather concerned with finding an LTI controller which renders stable and such that (12) holds true [11], where is a given number that specifies the performance level. This is the so-called sub-optimal problem. 2.7 controller synthesis Using the bounded real lemma for (12), the matrix is stable and (12) is satisfied if and only if the LMI 378 Hội nghị toàn quốc về Điều khiển và Tự động hoá - VCCA-2011 VCCA-2011 (13) holds for some . Unfortunately this inequality is not affine in and in the controller parameters which are appearing in the description of , , , . However, a by now standard procedure [8], [9], [12], [13] allows to eliminate the controller parameters from these conditions, which in turn leads to convex constraints in the matrices and that appear in the partitioning of (14) According to that of in (11) one then arrives at the following synthesis LMIs for -design [14]: (15) (16) (17) where and are basis matrices for the subspaces (18) respectively. Note that these inequalities are defined by open-loop system parameters only, and that they depend affinely on the design variables and . Hence we can directly minimize over these LMIs in order to compute the best possible level with (12) that can be achieved by a stabilizing controller. After having obtained and that satisfy (15)-(17) for some level , the controller parameters can be reconstructed by using the projection lemma [8]. This procedure for -synthesis is implemented in the robust control toolbox [15]. 2.8 Mixed sensitivity approach Figure 2a shows a simple feedback control system. This interconnection can be recast into a standard setup for control as depicted in Figure 2b. For this control configuration, engineers are usually interested in some specific transfer functions. In particular, is the sensitivity function which describes the influence of the external disturbance to the tracking error . is the complementary sensitivity function which describes the influence of the reference signal to the system output . Finally, is the transfer function from to the control input that indicates control activity [16]. ++ ¡¡ w y u z K G (a) ++ ¡¡ w z G u P y K (b) Figure 2. General feedback control configuration In general, performance of the closed-loop system that is specified by norm of the channel in (7) can be formulated as a multi-objective problem (see Figure 3). This leads to the minimization of (19) The multi-variable loop shaping with various specifications (19) is the so-called the mixed sensitivity design approach. ++ ¡¡ w y K u G 1 z 2 z 3 z z Figure 3. Mixed sensitivity control It is well-known in the literature that the transfer functions , , and need to be small in magnitude in order to achieve good command tracking and robust stability. However, the well- known constraint reveals that these requirements can not be achieved simultaneously over the whole frequency range. However, the use of frequency filters or weighting functions opens up the possibility to minimize the magnitudes of , , and over different frequency ranges [17]. Hence, in practice, instead of minimizing (19) one rather determines a stabilizing LTI controller that minimizes the cost function (20) 379 Hội nghị toàn quốc về Điều khiển và Tự động hoá - VCCA-2011 VCCA-2011 where , , are suitably chosen weighting functions (Figure 4). ++ ¡¡ w y u K G S W P W T W z Figure 4. Weighting functions. 3. The system representation In this paper, the mechanical angular speed is considered as a time-varying parameter. This particular choice is motivated by the fact that , which causes the system to be nonlinear, can be measured online. The value of the rotor angular speed varies by around the synchronous speed , i.e. (21) where is the nominal speed, is the variation of the rotor angular speed around its nominal value, and is the normalizing factor that maps the uncertain element into a normalized uncertain element such that . In the normal operation of the DFIM, the nominal speed is close to the synchronous speed . Hence, if we denote the ratio of the nominal speed and the synchronous speed by , i.e. we can write and (22) Here, is a scaling factor that allows to present the variation of the rotor speed around the synchronous speed . From that point, the deviation of the rotor speed by from the nominal speed can be expressed as (23) The representation of in (23) provides a flexible choice of the rotor speed range in the controller design for the DFIM. As a result, the system matrices presented in [18] can now be rewritten as follows: where and (24) in which (25) (26) The DFIM model [18] reads as (27) where (28) (29) In (28) and (29), and represent the input and output signals of the disturbance channel corresponding to the time-varying parameter . Equations (27), (28), and (29) in combination with the output equation in [18] can now be expressed as (30) (31) where is an unity matrix, is an zero matrix, . is also called the perturbation block. Since the two last rows of the matrices and are zero, let , , and we can write (32) where , and are two vectors, is a matrix. 380 Hội nghị toàn quốc về Điều khiển và Tự động hoá - VCCA-2011 VCCA-2011 Equations (30) and (31) can be easily simplified as (33) where Let be the transfer function with the state-space realization (33), i.e. (34) The system can then be generally described by (35) where is the transfer function mapping , and is the transfer function from to . The linear fractional transformation (LFT) representation of the system is depicted as shown in Figure 5. w  z    s v r v r y rc G Figure 5. LFT representation of the system 4. control of the system In this section we start with -synthesis for mentioned frozen values of the rotor speed . Then the performance of the LTI controller designed for a fixed value of is evaluated with other constant values of as well as with a fast variation of the rotor speed along the parameter range. 4.1 The control configuration With the LFT representation of the plant as shown in Figure 5 we can now derive a standard control structure for the synthesis of an -controller as depicted in Figure 6. Here, is the LTI part of the plant as given in (34), is the uncertainty block as given in (32), is the controller that is to be designed. + ¡ w  z    s v r v r e ref r i rc K rc G r y Figure 6. Structure of the closed-loop system in design In this configuration, is the reference input, is the controller output, is the controlled output, and is the controller input which is equal to the tracking error. In this case, the transfer function from the reference input to the tracking error will be . The transfer function from reference inputs to controlled outputs is denoted by , i.e. (36) 4.2 loop shaping design The interconnection of the system used for the controller synthesis is shown in Figure 7. The external control input consists of stator voltages and reference rotor currents . The controller output is . The controller input or tracking error is . The controlled variable is . Note that the components of the external control inputs are considered as disturbances and their influences on the controlled outputs must be reduced as much as possible. The weighting function is used to shape the function which is corresponding 381 Hội nghị toàn quốc về Điều khiển và Tự động hoá - VCCA-2011 VCCA-2011 to the transfer function from the reference input to the tracking error . is kept large over the low frequency range for tracking. The weighting function is used to shape the transfer function from the external control input to the controlled output . The selection of the weighting function is not only intended to keep the closed loop bandwidth at a desired value, but also to reject the effects of the components and on the controlled outputs as discussed above. Note that a large bandwidth corresponds to a faster rise time but the system is more sensitive to noise and to parameter variations [16]. + ¡¡ + ¡¡ ref rd i ref rq i rc w sd v sq v rn G rd v rq v rc K rd i rq i rcd e rcq e rtd W rtd z rtq W rtq z rsd W rsd z rsq W rsq z rc z Figure 7. The interconnection of the system The standard control problem is to find a stabilizing LTI controller at fixed frozen values of such that the -norm of the channel is smaller than a given number . at fixed frozen values of . The set of 620kW DFIM parameters is applied for the controller synthesis. During the controller design stage, a trial-and-error-repetition technique is used in order to achieve the desired performance specifications by adjusting the weighting functions. The design steps are repeated until we are able to meet the required performance specifications. Finally, the following weighting functions were obtained: (37) (38) For the chosen frozen value of (at under- synchronous speed), the controlled system with current controller with the above given weighting functions achieves a norm of 0.36. 4.3 Simulation results with the current controller Figure 8 shows the frequency responses of the controlled system with the current controller and the inverse of the weighting functions , and . Figure 8a,b show the relevant magnitude plots of the complementary sensitivity and sensitivity functions of the closed-loop system with the performance requirements specified by and . The blue-thick curve shows the response of the output with respect to the reference inputs . This curve corresponds to the transfer function (see equation (36)). Similarly, the red-thick curve shows the response of the output with respect to the reference inputs and it corresponds to the transfer function . Meanwhile the black-solid curve shows the influence of the reference input on the output corresponding to the transfer functions , and the green-solid curve shows the influence of the reference input on the output corresponding to the transfer functions . The inverse of the weighting functions , and (see Figure 7) are depicted by dotted lines A, and B in Figure 8a, while the inverse of the weighting functions , and are depicted by dotted lines C, and D in Figure 8b, respectively. The influences of the stator voltage on the controlled outputs and controller inputs are show in Figure 8c,d with the same color and line styles. 10 0 10 1 10 2 10 3 10 4 10 5 10 6 -120 -100 -80 -60 -40 -20 0 20 40 Magnitude (dB) Closed-loop performance of reference inputs to outputs Frequency (rad/sec) A B (a) 10 0 10 1 10 2 10 3 10 4 10 5 10 6 -150 -100 -50 0 50 100 Magnitude (dB) Closed-loop performance of reference inputs to control errors Frequency (rad/sec) C D (b) 10 0 10 1 10 2 10 3 10 4 10 5 10 6 -140 -120 -100 -80 -60 -40 -20 0 Magnitude (dB) The eff ects of stator voltages to outputs Frequency (rad/sec) (c) 10 0 10 1 10 2 10 3 10 4 10 5 10 6 -140 -120 -100 -80 -60 -40 -20 0 Magnitude (dB) The eff ects of stator voltages to control errors Frequency (rad/sec) (d) Figure 8. Performance of the controlled system with current controller in the frequency domain for . It is clear in Figure 8 that the sensitivity and complementary sensitivity functions are below the inverse of the performance weighting functions. The bandwidths corresponding to the channels and are about rad/s. The gains of the frequency responses of the stator voltages to controlled outputs and controller inputs are all smaller than -10db. This indicates that the controlled system 382 Hội nghị toàn quốc về Điều khiển và Tự động hoá - VCCA-2011 VCCA-2011 has good disturbance rejection with respect to the stator voltage . The overshoots of the channels and are about 15%. Moreover, the gains corresponding to the frequency responses of the channels and are smaller than -22db. This means that the cross-coupling interaction between and remains quite small, or in other words, the rotor current components can be considered to be no influence on one another. As a result, the characteristics of electrical torque and power factor responses are not deteriorated. 0 1 2 3 4 5 x 10 -3 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 Closed-loop performance of reference inputs to outputs time (s) i rd ref  i rd i rd ref  i rq i rq ref  i rq i rq ref  i rd (a) 0 1 2 3 4 5 x 10 -3 -0.2 0 0.2 0.4 0.6 0.8 1 Closed-loop performance of reference inputs to control errors time (s) i rd ref  e rcd i rd ref  e rcq i rq ref  e rcq i rq ref  e rcd (b) 0 0.002 0.004 0.006 0.008 0.01 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 The effects of stator voltages to outputs time (s) v sd ref  i rd v sd ref  i rq v sq ref  i rq v sq ref  i rd (c) 0 0.002 0.004 0.006 0.008 0.01 -0.05 0 0.05 0.1 0.15 0.2 0.25 The effects of stator voltages to control errors time (s) (d) Figure 9. Performance of the controlled system with current controller in the time domain for . Figure 9 shows the time responses of the controlled system for a step input. The solid line in Figure 9a shows the response of the output with respect to the reference input and it corresponds to the transfer function in equation (36). The dashed line shows the response of the output with respect to the reference input and it corresponds to the transfer function in equation (36). The dotted curve shows the influence of the reference input on the output corresponding to the transfer function , and the dash-dotted curve shows the influence of the reference input on the output corresponding to the transfer function . The solid line in Figure 9b shows the response of the control error with respect to the reference input . The dashed line shows the response of the control error with respect to the reference input . The dotted curve shows the influence of the reference input on the control error , and the dash- dotted curve shows the influence of the reference input on the control error . The influences of the stator voltages on the controlled outputs and control errors are also show in Figure 9c,d with the same line styles. 0 1 2 3 4 5 x 10 -3 -0.2 0 0.2 0.4 0.6 0.8 1 time (s) Closed-loop performance of reference inputs to outputs i rd ref  i rd ( m = 0.63  s ) i rd ref  i rq ( m = 0.63  s ) i rq ref  i rq ( m = 0.63  s ) i rq ref  i rd ( m = 0.63  s ) i rd ref  i rd ( m = 0.9  s ) i rd ref  i rq ( m = 0.9  s ) i rq ref  i rq ( m = 0.9  s ) i rq ref  i rd ( m = 0.9  s ) (a) 0 1 2 3 4 5 x 10 -3 -0.2 0 0.2 0.4 0.6 0.8 time (s) Closed-loop performance of reference inputs to control errors (b) 0 0.002 0.004 0.006 0.008 0.01 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 time (s) The effects of stator voltages to outputs (c) 0 0.002 0.004 0.006 0.008 0.01 -0.05 0 0.05 0.1 0.15 0.2 0.25 time (s) The effects of stator voltages to control errors v sd ref  e rcd ( m = 0.63  s ) v sd ref  e rcq ( m = 0.63  s ) v sq ref  e rcq ( m = 0.63  s ) v sq ref  e rcd ( m = 0.63  s ) v sd ref  e rcd ( m = 0.9  s ) v sd ref  e rcq ( m = 0.9  s ) v sq ref  e rcq ( m = 0.9  s ) v sq ref  e rcd ( m = 0.9  s ) (d) Figure 10. Performance of the controlled system with current controller for frozen value for . Note that the current controller is designed with a fixed frozen value of the rotor angular speed . Hence, the obtained performance is not guaranteed for the whole region of variation of . However, we can further investigate the performance of the closed-loop system with the controller designed for the frozen value for other angular rotor speeds. In the following investigation, we consider the performance of the controlled system with the rotor speed variations by from the rotor nominal speed ( ), i.e. . In order to do so we synthesized two controllers for the frozen parameter values using the same weighting functions as in (37) and (38). Then we plot the time responses of the closed- loop system with the controller designed for the frozen value applying for the case where and , respectively. The time responses of the closed-loop system with the local controllers designed for the frozen values and are also plotted on each figure for the purpose of comparison of the achieved performance among these controllers. Figure 10 shows the performance of the closed-loop system at with two controllers designed for the frozen value and , respectively. The thick-solid lines are related to the designed controller for the frozen value . The thin-lines are related to the local controller designed for . As can be seen from Figure 10a, the time responses of the 383 Hội nghị toàn quốc về Điều khiển và Tự động hoá - VCCA-2011 VCCA-2011 outputs and with respect to the step change of the reference inputs and , respectively, of the closed-loop system with the controller designed for the frozen value are maintained for the frozen values if compared with that in Figure 9. In addition, these curves are almost the same with that of the local controller designed for . The same conclusion can also be drawn for the curves related to the time responses of the control errors , with respect to the step change of the reference inputs , (Figure 10b), and the influences of the stator voltages , on the controlled outputs , (Figure 10c) and control errors , (Figure 10d), respectively. However, the remarkable difference in the performance among these controllers is indicated by their cross- coupling interactions. The effects of the stator voltages and to the outputs , (Figure 10c) and to the control errors , (Figure 10d), respectively, in the case of the controller designed for the frozen value are larger than that of the local controller designed for . 0 1 2 3 4 5 x 10 -3 -0.2 0 0.2 0.4 0.6 0.8 1 time (s) Closed-loop performance of reference inputs to outputs i rd ref  i rd ( m = 1.17  s ) i rd ref  i rq ( m = 1.17  s ) i rq ref  i rq ( m = 1.17  s ) i rq ref  i rd ( m = 1.17  s ) i rd ref  i rd ( m = 0.9  s ) i rd ref  i rq ( m = 0.9  s ) i rq ref  i rq ( m = 0.9  s ) i rq ref  i rd ( m = 0.9  s ) (a) 0 1 2 3 4 5 x 10 -3 -0.2 0 0.2 0.4 0.6 0.8 time (s) Closed-loop performance of reference inputs to control errors i rd ref  e rcd ( m = 1.17  s ) i rd ref  e rcq ( m = 1.17  s ) i rq ref  e rcq ( m = 1.17  s ) i rq ref  e rcd ( m = 1.17  s ) i rd ref  e rcd ( m = 0.9  s ) i rd ref  e rcq ( m = 0.9  s ) i rq ref  e rcq ( m = 0.9  s ) i rq ref  e rcd ( m = 0.9  s ) (b) 0 0.002 0.004 0.006 0.008 0.01 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 time (s) The effects of stator voltages to outputs v sd ref  i rd ( m = 1.17  s ) v sd ref  i rq ( m = 1.17  s ) v sq ref  i rq ( m = 1.17  s ) v sq ref  i rd ( m = 1.17  s ) v sd ref  i rd ( m = 0.9  s ) v sd ref  i rq ( m = 0.9  s ) v sq ref  i rq ( m = 0.9  s ) v sq ref  i rd ( m = 0.9  s ) (c) 0 0.002 0.004 0.006 0.008 0.01 -0.05 0 0.05 0.1 0.15 0.2 0.25 time (s) The effects of stator voltages to control errors v sd ref  e rcd ( m = 1.17  s ) v sd ref  e rcq ( m = 1.17  s ) v sq ref  e rcq ( m = 1.17  s ) v sq ref  e rcd ( m = 1.17  s ) v sd ref  e rcd ( m = 0.9  s ) v sd ref  e rcq ( m = 0.9  s ) v sq ref  e rcq ( m = 0.9  s ) v sq ref  e rcd ( m = 0.9  s ) (d) Hình 11. Performance of the controlled system with current controller for frozen value for The performance of the closed-loop system at with two controllers designed for the frozen value and , respectively, is shown in Figure 11. The thick-solid lines are related to the designed controller for the frozen value . The thin-lines are related to the local controller designed for . Similarly to the previous simulation, the time responses of the closed-loop system with the controller designed for the frozen value and the local controller designed for corresponding to the step change of the reference inputs , , and to the effect of stator voltages , are almost the same, except their cross-coupling interactions. In the case of the controller designed for the frozen value , the influences of the stator voltages and to the outputs , (Figure 11c) and to the control errors , (Figure 11d) are larger than that of the local controller designed for . Obviously, the performance of the the controller designed for the frozen value for other rotor angular speeds are not maintained because of the cross-coupling interactions between the stator voltages , and the outputs , as well as the stator voltages , and the control errors , . This may cause a large tracking error for the controlled system since the stator voltages and are the input disturbances. 0 1 2 3 4 5 x 10 -3 690 700 710 720 730 740 750 760 d component of the rotor currents time (s) Ampere  m = 0.63 s  m = 0.9 s  m = 1.17 s (a) 0 1 2 3 4 5 x 10 -3 0 50 100 150 200 250 300 350 400 q component of the rotor currents time (s) Ampere  m = 0.63 s  m = 0.9 s  m = 1.17 s (b) 0 1 2 3 4 5 x 10 -3 690 700 710 720 730 740 750 760 d component of the rotor currents time (s) Ampere  m = 0.63 s  m = 0.9 s  m = 1.17 s (c) 0 1 2 3 4 5 x 10 -3 0 50 100 150 200 250 300 350 q component of the rotor currents time (s) Ampere  m = 0.63 s  m = 0.9 s  m = 1.17 s (d) 0 1 2 3 4 5 x 10 -3 690 700 710 720 730 740 750 760 770 d component of the rotor currents time (s) Ampere  m = 0.63 s  m = 0.9 s  m = 1.17 s (e) 0 1 2 3 4 5 x 10 -3 0 50 100 150 200 250 300 350 q component of the rotor currents time (s) Ampere  m = 0.63 s  m = 0.9 s  m = 1.17 s (f) Hình 12. Performance of the controlled system with current controllers designed for (a, b), for (c, d), and for (e, f) at different constant values of . In order to evaluate the performance of the closed- loop system with controller designed for different frozen values of for other rotor angular speeds in the face of the stator voltage action, we performed the simulations with the set value of and the set value of as shown in Figure 12. The time responses of the (Figure 12a) and (Figure 12b) components of the rotor currents achieved by the 384 Hội nghị toàn quốc về Điều khiển và Tự động hoá - VCCA-2011 VCCA-2011 designed controller for the frozen value are plotted by the solid curves. While the dashed and and the dash-dotted curves show the performance of this controller for the value , and , respectively. These figures reveal that the tracking errors of the and components of the rotor currents achieved by this controller are increased for and . This is because of the cross-coupling interactions between the stator voltages , and the outputs , as well as the stator voltages , and the control errors , become larger for bigger rotor angular speeds as presented in the previous simulation. Figures. 12c and 12d show the time responses of the and components of the rotor currents achieved by the designed controller for the frozen value (solid curves) for the value (dashed curves), and (dash-dotted curves), respectively. Figures. 12e and 12f show the time responses of the and components of the rotor currents achieved by the designed controller for the frozen value (solid curves) for the value (dashed curves), and (dash-dotted curves), respectively. These figures reveal that performance of the controller designed for a frozen value of is not guaranteed for these other values of . 0 0.002 0.004 0.006 0.008 0.01 0 100 200 300 400 500 600 700 800 d component of the rotor currents time (s) Ampere  m = 0.63 s  m = 0.9 s  m = 1.17 s (a) 0 0.002 0.004 0.006 0.008 0.01 0 100 200 300 400 500 600 700 800 d component of the rotor currents time (s) Ampere  m = 0.63 s  m = 0.9 s  m = 1.17 s (b) 0 0.002 0.004 0.006 0.008 0.01 0 50 100 150 200 250 300 350 400 d component of the rotor currents time (s) Ampere  m = 0.63 s  m = 0.9 s  m = 1.17 s (c) 0 0.002 0.004 0.006 0.008 0.01 0 50 100 150 200 250 300 350 400 d component of the rotor currents time (s) Ampere  m = 0.63 s  m = 0.9 s  m = 1.17 s (d) 0 0.002 0.004 0.006 0.008 0.01 200 250 300 350  m time (s) rad/s (e) 0 0.002 0.004 0.006 0.008 0.01 200 250 300 350  m time (s) rad/s (f) Hình 13. Performance of the controlled system with three current controllers designed for , , and , respectively, with fast variations of the rotor speed. For further investigation, a simulation with the controller designed for a frozen values of for a fast variation of the rotor speed along the whole parameter interval is carried out. We consider three local controllers designed for the frozen values , , and as above. The parameter trajectory is given by the step response of the rotor speed. Figures. 13a,c,e show the behaviors of the and components of the rotor currents when the rotor angular speed increases from 70% to 130% of the nominal speed of the rotor , where (rad/s), . Conversely, the behaviors of the and components of the rotor currents when the rotor angular speed decreases from 130% down to 70% of the nominal speed of the rotor are shown in Figures. 13b,d,f. These figures reveal that the controllers do not guarantee tracking during the fast parameter transition. The control error increases along the parameter trajectory and reaches the largest value at the end of it. 5. Conclusions This paper briefly recapitulated the theory of the mixed loop shaping -design for the rotor current controller for DFIMs at some fixed frozen values of the rotor angular speed. The performance of these current controllers has been investigated for different values of the mechanical angular speed varied by % from the rotor nominal speed. The simulation results showed that the performance of the controller designed for a frozen value of was not completely guaranteed for other rotor angular speeds. An important point that is needed to be emphasized in this particular case is that the performance of the controller is considerably changed for fast parameter variations. In order to get better performance level for the controlled system, the designed controller has to adapt to changing of the rotor angular speed. 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The 10th International Conference on Control, Automation, Robotics and Vision, ICARCV 2008, Hanoi, Vietnam, (0), 2008. Dr. Ngo Duc Minh was born in Lang son, Vietnam, in 1960. He received the B.S. degree from Thainguyen University of Technology in 1982 in Electrical Engineering, M.S. degree from Hanoi University of Technology in 1997 in Electrical Engineering and in Industrial Information Technology, and Ph.D degree from Hanoi University of Technology in 2010 in Atutomation Technology. He is currently a vice-chair of the Education department of Thainguyen University of Technology. Dr. Minh’s interests are in the areas of high voltage technology, hydrolic power plant, power supply, control of electric power systems, FACTS, BESS, AF, PSS equipments, new and renewable energy technologies, distribution power systems. Nguyen Tien Hung was born in Thainguyen, Vietnam. He received the B.S. degree from Thainguyen University of Technology in 1991 and M.S. degree from Hanoi University of Technology in 1997, both in Electrical Engineering. He is currently a Ph.D candidate at Delft Center for Systems and Control (DCSC), Delft University of Technology, the Netherlands. His main research interests include topics in robust control, linear parameter varying control of nonlinear systems, gain-scheduling design, and their applications in electrical systems. 386 . the mechanical angular speed is eliminated by adding a feed-forward term to the output of the q-axis controller [2], [5]. The rotor mechanical angular speed is treated as an scheduling parameter. changing of the rotor angular speed. In that sense, the rotor angular speed can be adopted as a gain-scheduling parameter. Tài liệu tham khảo [1] G. Tapia A. Tapia and J. X. Ostolaza. Reactive. investigated. The performance analysis is also extended for the case with the face of the stator voltage action. As a further investigation, the designed controller for a frozen values of is tested