^{1}**Mr.Roshan S. Gaidhane, ^{2}Prof. M. A. Gaidhane**

Priyadarshini College of Engineering Nagpur, India Department of Electrical Engineering

### Abstract—

In this paper, the direct torque control (DTC) is employed for fast and slow torque and flux control of induction motor coupled to an inverter (Inv-IM). This paper describes a combination of direct torque control (DTC) and space vector modulation (SVM) for an adjustable speed sensor-less induction motor (IM) drive. The motor drive is supplied by a two-level inverter. The inverter reference voltage is obtained based on input-output feedback control, using the IM model in the stator – axes reference frame with stator current and flux vectors components as state variables. We first model the DTC of Inv-IM as a hybrid system (HS). Then, we abstract the continuous dynamics of the HS in terms of discrete events. We thus obtain a discrete event model of the HS. And finally, we use Supervisory Control Theory of discrete event system (DES) to drive Inv-IM.

**Index term s- direct torque control (DTC), discrete event system (DES), inverter coupled induction motor**

(INV-IM), supervisory control theory (SCT).

**I. INTRODUCTION**

The main advantage of Induction motors (IM) is that no electrical connection is required between the stator and the rotor, they have low weight and inertia, high efficiency and a high overload capacity [1]. There exist several approaches to drive an IM. Induction motor control methods can be broadly classified into scalar control and vector control. In scalar control, V/F control is the important control technique, it is the most widespread, reaching approximately 90% of the industrial applications. The structure is very simple maintaining a constant relation between voltage and frequency and it is normally used without speed feedback, hence this control does not achieve a good accuracy in both speed and torque responses mainly due to the fact that the stator flux and the torque are not directly controlled. Vector control is a technique that can reach a good accuracy, but its main disadvantage is the necessity of a huge computational capability and of a good Identification motor parameters.The method of Field acceleration overcomes the computational problem of vector controllers by achieving some computational reductions [2][4]. And the technique of Direct Torque Control (DTC) been developed by Takahashi [5][6][7][8] permits to control directly the stator flux and the torque by using an appropriate voltage vector selected in a look-up table. And the technique of Direct Torque Control (DTC) been developed by Takahashi [5][6][7][8] permits to control directly the stator flux and the torque by using an appropriate voltage vector selected in a look-up table. The conventional DTC drive contains a pair of hysteresis comparators, a flux and torque estimator and a voltage vector selection table. The torque and flux are controlled simultaneously by applying suitable voltage vectors, and by limiting these quantities within their hysteresis bands, de-coupled control of torque and flux can be achieved. However, as with other hysteresis bases systems, DTC drives utilizing hysteresis comparators suffer from high torque ripple and variable switching frequency. . The torque and flux are controlled simultaneously by applying suitable voltage vectors, and by limiting these quantities within their hysteresis bands, de-coupled control of torque and flux can be achieved. However, as with other hysteresis bases systems, DTC drives utilizing hysteresis comparators suffer from high torque ripple and variable switching frequency. The most common solution to this problem is to use the space vector modulation depends on the reference torque and flux. The main advantages of SVM-DTC are minimal torque response time and the absence of coordinate-transform, voltage modulator block, controllers such as PID for flux and torque for these advantages, DTC is the control method adopted in this paper.

We propose a three-step method to model the DTC of Inv-IM. In a first step, we model the DTC of Inv-IM as a

hybrid system (HS) with a discrete event dynamics defined by the voltage vectors used to control IM; and a

continuous dynamics defined by continuous equations on the stator flux vector(Φs) and the electromagnetic

torque(Г).

Hybrid system in the sense that it consist of discrete component (inverter) and the continuous component (induction motor).

In a second step, we abstract the continuous dynamics of the HS in terms of discrete events. Some events are used to represent the entrance and exit of the torque Γ and the amplitude Φs of in and from a working point region.

And some other events are used to represent the passage of the vector Φs between different zones. By this

abstraction, the continuous dynamics of the system IM is described as a discrete event system (DES). In a third step, we use Supervisory Control Theory (SCT) to drive Inv-IM.

**II. INVERTER AND ITS DISCTERT EVENT MODEL**

The inverter (Fig.1) is supplied by a voltage Uo and contains three pair of switches for i = a, b, c. The input of the inverter is a three-bit value (Sa Sb Sc) where each Si can be set to 0 or 1. A value 0 of Si sets to (close, open), and a value 1 sets it to (open, close). The output of the inverter is a voltage vector Vs that drives IM. The inverter vector s is determined by Equation (1), and thus, depends uniquely on Uo and (Sa Sb Sc).

(1)

The inverter can produce eight vectors k = 0, 1,…7)corresponding respectively to the eight possible

values(0=000 to 7=111) of (Sa Sb Sc).from equation (1),we obtain easily equation (2) that computes the eight

vectors , k=1,2,..7. Note that v0 and v7 are null. Figure (2) represents the six non-null voltage vectors in

the D-Q axes which represent the stationary reference frame fixed to the stator.

The inverter can be modeled by a 8-state automaton who’s each state qk (k = 1… 7) means: “Vk is the current voltage vector”. To adopt the terminology of hybrid systems, the term mode will be used as a synonym of state. The transition from any mode q⋆ to a mode qk occurs by an event Vk which means “starting to apply Vk”.

**III. INDUCTION MOTOR AND ITS CONTINUOUS MODEL**

The induction motor is a continuous system because its behavior is modeled by algebraic and differential equatons on two continuous variables, the stator flux (Φs) and the electromagnetic torque(Γ).

**A. Model of torque and flux**

With DTC, the voltage vector VS generated by the inverter is applied to the IM to control the flux Φs and the torque Γ. Let us first see how and Φs, Γ can be expressed. In a Stationary reference frame, the flux vector Φs is governed by the differential Eq. (3), where Rs is the stator resistance and Is is the stator current vector. Under the assumption that , is egligible w.r.t VS (realistic if the amplitude of Φs is sufficiently high); we obtain Eq. (4) which approximates the evolution of Φs from Φso after a delay t.

*B. evolution of flux and torque*

Eq. (4) implies that the application of a vector voltage generates a move of the end of Øs in the direction of . Note that consists of a radial vector (parallel to ) and a tangential vector (orthogonal to ). Increases (resp. decreases) the flux Øs (i.e., the amplitude of ) if it has the same (resp. opposite) direction of . rotates clockwise (resp. counterclockwise) if the angle from to is +π/2 (resp. –π/2). From Equation (5). We deduce that increases (resp. decreases) the torque Г if the angle from to is π/2 (resp. –π/2).

Figure 3 illustrates the evolution of when have the same direction as and the angle from to is +90 degrees. Therefore, in this example both the flux Øs and the torque Г increase. As proposed in [5] to divide the possible global locus of into the six zones Z1, Z2… Z6 of Fig. 4. Table I shows how the flux magnitude Øs and the torque Г evolve when is in Zi (i = 1,.6) under the control of each of the eight vectors (k = 0, 7, i – 2 · · i+3), where indices are defined modulo 6 (and not modulo 8). Symbols ↑, ↓ and = mean “increases”, “decreases” and “is constant”, respectively. We see that under the control of i-2, i-1, i+1, i+2, 0 and 7, the evolution of Øs and Г is known. But vectors i and i+3 are problematic because they can both increase and decrease the torque Г in the same zone Zi, depending if Øs is in the first or the second 30 degrees of Zi. This problem will be called non-determinism of the six-zone division.

*C. solving the non-determinism of six-zone division *

In this two approaches were proposed to solve the non-determinism of the 6-zone division. The first approach is based on the observation that the non-determinism occurs when is in a zone Zi while one of the control vector i or i+3 is applied A solution is to leave non-determinism as soon as it appears, by applying a control vector different from i and i+3. We suggest to select the control vector to be applied among the four control vectors i-2, i-1, i+1, i+2 because these four vectors permit to obtain all the combinations of the evolution of (Øs, Г) (see Table I).

A second approach to solve the non-determinism is to use twelve zones by dividing each of the six zones Zi into two zones Zi,1 and Zi,2 comprising the first and the second 30 degrees, respectively [13], [1]. Figure 5 represents the twelve-zone division. In each zone Zi,j and under the control of i-2, i-1, i+1, i+2, 0 and 7, the evolution of Øs and Г is thus the one already indicated in Table I or Zi. Table II shows the evolution of Øs and Г in zone Zi,j under the control of i and i+3.

**IV. MODELING OF IM AS DES BY ABSTRACTING ITS CONTINUOUS DYNAMICS**

Let us show how the continuous dynamics of IM presented in Sect. III is abstracted in terms of discrete events. The first abstraction consists in translating by events the entrance and exit of (ØS, Г) in and from a working point region. The second abstraction consists in translating by events the passage of the vector between orientation zones.

*A. Abstracting the entrance and exit of (Øs, Г)*

Let Ø wp and Гwp the flux magnitude and the torque defining the targeted working point. That is, the aim of control will be to drive IM as close as possible to (Øwp, Г wp). We define a flux interval [Øwp-, Øwp+] centered in Ø wp, and a torque interval [Гwp, Гwp] centered in Гwp. We partition the space of (Øs, Г) into sixteen regions Ru,v for u, v = 1, 2, 3, 4, as shown in Fig. 6. The objective of the control will be to drive IM into the set of regions{Ru,v : u, v = 2, 3}(shaded in Fig. 6) and to force it to remain into this set. We define the event that represents a transition from Ru,v to Ru′,v for any v, and theevent that represents a transition from Ru,v to Ru,v′ for any u. Since only transitions between adjacent regions are possible, the unique possible events are the following: if u<4, if u>1, if v<4, if v>1. With the above abstraction, the evolution of (Øs, Г) can be described by a 16-state automaton, whose states are noted (u, v) and correspond to the sixteen regions Ru,v, u, v = 1, 2, 3, 4. The transitions between states occur with the events defined above , , , .

*B. Abstracting the passage of between orientation zones*

In Sec. III-B and III-C, we have shown how to partition the global locus of into six or twelve zones (Figs. 4 and 5). This partitioning is very relevant, because we have seen that from the knowledge of the current zone occupied by .

We can determine the control vector to be applied for obtaining a given evolution of (Øs, Г) (Tables I and II). With the 6-zone partition, we define the event Zii′ that represents a transition from Zi to Zi’. Since only transitions between adjacent zones are possible, the unique possible events are the following: , ,

where i-1and i+1 are defined modulo 6. We can thus abstract the evolution of by a 6-state automaton, whose states are noted (i) and correspond to the zones Zi, i = 1…6. The transitions between states occur with the events defined above: , , We can use the same approach with the 12-zone partition, by defining

the event that represents a transition from Zi,j to Zi′,j′ . Since only transitions between adjacent zones are possible, the unique possible events are the following: , , , , where i-1 and i+1 are defined modulo 6. We can thus abstract the evolution of by a 12- state automaton, whose states correspond to the zones Zi,j , i = 1 ···12 and j = 1, 2.

*C. modeling IM as a DES*

In Sec. IV-A, we have shown how to abstract the evolution of , Г) by a 16-state automaton. In Sec. IV-B, we have shown how to abstract the evolution of by a 6-state or 12- state automaton. In the sequel, we consider uniquely the 6- state automaton because it reduces the state space explosion which is inherent to the use of automata. As we have seen in Sect. III-C, the 6-zone partition necessitates to apply a control vector different from Vi and Vi+3 when Øs is in Zi. We will explain in Sect. V how this requirement can be guaranteed by supervisory control of DES. Let us see how the two automata (16-state and 6-state) are combined into an automaton Mk that abstracts the behavior of IM when a given control vector Vk is applied by the inverter.

A State of Mk is noted (u, v, i)k since it is a combination of a state (u, v) (corresponding to Ru,v) and a state I (corresponding to Zi ). Mk can therefore have at most 6 × 16 = 96 states (u, v, i)k, (u, v = 1…4, i = 1…6). By interpreting Table I, we determine the transitions of Mk as follows, where Vk is the control vector currently applied by the inverter. From state (u, v, i)k of Mk :

- The event can occur when u < 4 and ØS increases, i.e., when k is equal to one of the following values: i-1, i+1, i, 0 if i is odd, if i is even. This event leads from (u, v, i)k to (u+1, v, i)k.
- The event can occur when u > 1 and ØS decreases, i.e., when k is equal to one of the following values: i-2, i+2, i+3,0 if i is dd. This event leads from (u, v, i)k to (u-1, v, i)k.
- The event can occur when v < 4 and Γ increases, i.e., when k is equal to one of the following values: i+1, i +2, i, i+3. This event leads from (u, v, i)k to (u, v+1, i)k.
- The event can occur when and v > 1 and Γ decreases, i.e., when k is equal to one of the following values: i-2, i-1, i, i+3. This event leads from (u, v, i)k to (u, v-1, i)k.
- The event can occur when rotates clockwise, i.e., when Γ increases, i.e., when k is equal to one of i+1the following values: i+1, i+2, i, i+3. This event leads from (u, v, i)k to (u, v, i+1)k.
- The event can occur when rotates counter clock-wise, i.e., when Γ decreases, i.e., when k is equal to one of the following values: i-2, i-1, i, i+3. This event leads from (u, v, i)k to (u, v, i-1)k. Due to the non-determinism of Sect. III-B, the events depending on the evolution of Γ ( , , , ) are potential but not certain when k is equal to i or i+3.

**V. USE OF SCT TO DRIVE IM**

*A. introduction to SCT*

In supervisory control, a supervisor Sup interacts with a DES (called plant) and restricts its behavior so that it respects a specification. Sup observes the evolution of P (i.e., the events executed by the plant) and permits only the event sequences accepted by S. To achieve its task, Sup will disable (i.e., prevent) and force events. The concept of controllable event has thus been introduced, meaning that when an event e is possible, then Sup can disable it if and only if e is controllable; e is said uncontrollable if it is not controllable [9]. We will also use the notion of forcible event, meaning that when an event e is possible, then Sup can force e to preempt (i.e., to occur before) any other possible event, if and only if e is forcible; e is said unforcible if it is not forcible [14]. A method has been proposed to synthesize Sup automatically from P, S and the controllability and forcibility of every event [9].

*B. The Plant Inv-IM Modeled as a DES*

The plant to be controlled is the system Inv-IM (i.e., inverter with IM). In Section II, we have modeled the inverter by an automaton A with 8 states qk (k = 0…7) Corresponding to the 8 control vectors Vk, respectively. And in Section IV-C, when a given VK is applied by the inverter to the IM, we have modeled the evolution of IM by an automaton MK that can have at most 6 × 16 = 96 states (u, v, I)k, (u, v 1,…4,i=1,…6).Therefore, the system Inv-IM can be modeled y replacing in A each mode qk by the automaton Mk . The transition from any state (u, v, i)* to a state (u, v, i)k occurs by an event Vk . The obtained automaton, noted P , can therefore have at most 8 × 96 = 768 states. The initial state is (1; 1; 1)0,

that is, initially: the flux and the torque are in Region R1,1, the flux vector is in zone Z1 and the null control vector Vo is applied. The set of marked states is { (u; v; i)k : u, v = 2, 3},because the objective of the control is to drive Inv-IM into the set of regions {Ru,v : u, v = 2, 3} (i.e., the set of states { u; v; i k : u, v = 2, 3}), and then to force it to remain into this set. For the purpose of control, we define an undesirable event Null meaning that the flux or the torque has decreased to zero, and a state E reached with the occurrence of Null .We will see later how Null and E are necessary. Therefore, the automaton P has actually at most 769 (768+ the state ), and its alphabet Σ is:

*C. control architecture*

We propose the control architecture illustrated in Figure 7. The interaction between the plant and the supervisor is realized through two interfaces A and G.

A: In Sect. II, we have modeled the inverter by an automaton executing the events Vk, where k = 0,.…7. The interface A translates every event Vk generated by the supervisor into (Sa Sb Sc), which is the 3-bit code of

k. And the inverter translates (Sa Sb Sc) into the control vector Vk, which is applied to IM.

G: In Sect. IV, we have modeled IM by an automaton executing the events Z, Г, Ø. and in Sect. V-B, we have added an event Null in the model of IM. The interface G generates these events. Sup observes the events Z, Г, Ø, Null. Since these events are generated by IM through G, Sup has no control on them. Hence, these events are uncontrollable and unforcible. Sup generates the events VK, and thus, has all control on them. Hence, these events are controllable and forcible.

**VI. CONCLUSION**

In this control method, a much better behavior of the DTC-SVM performance is presented , achieving one of the main objectives of the present work, which was to control the torque by reduce the torque ripple and consequently improve the motor performance compared to classical DTC. This control method shows better results in high as well as in low speed also as shown in results for low speed. As a future work, we intend to

improve our control method by using a hierarchical control and a modular control, which are very suitable to take advantage of the fact that the event based model of the plant has been constructed hierarchically and modularly.

**VII. SIMULATION AND RESULTS**

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