Introduction

The robust controller design for an AVR system is among the significant interests of the researchers in the area of power system control. It is of paramount importance to maintain a constant voltage level to avoid equipment damage and poor voltage levels. If the voltage of a device violates the limits, the equipment may be unable to operate properly and may get damaged. The PID controller design has been one of the common methods used for controlling the AVR system. Tuning methods involving the Ziegler–Nichols method, soft computing techniques[1], optimization techniques are available in the literature [2,3,4]. In [5], a PID controller was tuned using particle swarm optimization for AVR system. Using a teaching and learning-based optimization algorithm, PID controller design was proposed in [6, 7] for the AVR system. A novel design method using coefficient diagram method (CDM) for controller design of an AVR system is presented in this paper. The CDM is an algebraic method which gives a robust controller so that the desired system performance is exhibited despite parameter variations [8,9,10,11]. A systematic controller design approach using CDM has been described in [12,13,14].

AVR Schematic

The block diagram of an AVR system [15]can be represented as shown in Fig. 1. The excitation system controls the generator terminal voltage to maintain the flow of the reactive power. The field circuit may be energized using dc generator positioned on the same rotor shaft of the synchronous machine. Advanced equipment utilize rotating rectifiers and the system is known as brush-less excitation system. The predominant means of control of reactive power is by controlling the excitation of a generator by means of AVR system. The basic operation of the AVR system is described as follows: Using a potential transformer the voltage magnitude of one phase is obtained, rectified and the disparity with the DC reference signal is assessed. The resulting error is amplified in order to regulate the field current of the exciter to further affect the exciter terminal voltage. Thus, if there is drop in the output voltage sensed by PT, the generator field current is increased that increases the generated EMF.

Fig. 1
figure 1

Block diagram representation of automatic voltage regulator

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The transfer function for the open loop of the schematic shown in Fig. 1 is given by

$$\begin{aligned} \mathrm{AVR}_\mathrm{open loop}(s)=\frac{K_{A}K_{E}K_{G}K_{R}}{(1+\tau _{A}s)(1+\tau _{G}s)(1+\tau _{E}s)(1+\tau _{R}s)} \end{aligned}$$
(1)

and the ratio of terminal voltage to the reference voltage is given by the transfer function

$$\begin{aligned} {\frac{V_{t}(s)}{V_\mathrm{ref}(s)}} = {\frac{K_{A}K_{E}K_{G}(1+\tau _{R}s)}{(1+\tau _{A}s)(1+\tau _{G}s)(1+\tau _{E}s)(1+\tau _{R}s)+K_{A}K_{E}K_{G}K_{R}}}. \end{aligned}$$
(2)

From (2), it is clear that the transfer function is a type 0 system.

AVR System with PID Controller

One of the widely used, popular and commercially used controllers is the PID controller. The PID controller is represented by

$$\begin{aligned} G_{\textit{C}}(s) = K_{\textit{p}}+\frac{K_{\textit{I}}}{s}+K_{\textit{D}}s \end{aligned}.$$
(3)

The transient response is improved by the zero added due to the derivative term and the integral control is used to improve or reduce steady-state error. In an example from [15] existing literature, the parameters involved in an AVR system are as given in Table 1.

Table 1 Parameters of an AVR System
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The transfer function \(G_\mathrm{cl}(s)\) of the closed-loop system is given by

$$\begin{aligned} {G_\mathrm{cl}(s)} = {\frac{K_{A}K_{E}K_{G}(K_{P}s+K_{I}+K_{D}s^{2})(1+\tau _{R}s)}{s(K_{P}s+K_{I}+K_{D}s^{2})(1+\tau _{A}s)(1+\tau _{E}s)(1+\tau _{G}s)(1+\tau _{R}s)+K_{A}K_{E}K_{G}K_{R}}}. \end{aligned}$$
(4)

A Brief Overview of Coefficient Diagram Method

There are mainly three approaches of control design theories, namely classical control approach, algebraic approach and modern control approach. The mathematical expression used are the transfer function, polynomials and the state space matrices, respectively. The transfer function becomes inaccurate when pole–zero cancelations occur. The state space involves extensive machine computations. In the polynomial approach, the numerator and denominator polynomials are handled independently; hence, there is no problem of pole–zero cancelations and also preserves the rigor of state space with the polynomial expression being equivalent to controllable or observable canonical form of state space [8]. Coefficient diagram method belongs to the polynomial approach. In this method, as the controller together with transfer function for the closed-loop system is partly specified, this is known as simultaneous approach and rest of the variables are obtained by design [8, 12]. In CDM, the design parameters are equivalent time constant, stability index and stability limit represented, respectively, as

  • \(\tau\)

  • \(\gamma _{i}\) and

  • \(\gamma _{i}^{*}\)

In CDM, the performance specifications are rewritten in terms of stability indices \(\gamma _{i}\) and equivalent time constant\(\tau\). Thus, the target characteristic equation is represented in terms of the aforesaid quantities that satisfies stability, performance and are co-related algebraically to controller parameters. A semi-log line diagram known as coefficient diagram is plotted using which the variation in coefficients of the characteristic polynomial can be observed and modified to obtain the desired system response.

Conditions for Stability and Instability in CDM

The schematic block diagram representation for CDM [8] is described as follows:

  • Reference input, r

  • Plant output, y

  • Control input to the plant, u

  • Disturbance signal, d

  • Numerator polynomial of the plant transfer function, \(B_{p}(s)\)

  • Denominator polynomial of the plant transfer function, \(A_{p}(s)\)

  • Reference polynomial in the numerator of the controller, F(s)

  • Feedback polynomial in the numerator of the controller, \(B_{c}(s)\)

  • Forward polynomial in the denominator of the controller, \(A_{c}(s)\)

\(B_{c}(s)\) and \(A_{c}(s)\) are designed to satisfy the desired transient response and F(s) is used to take care of the steady-state error. The output of the whole system is

$$\begin{aligned} y = \frac{B_{p}(s)F(s)}{A_\mathrm{cl}(s)}r+\frac{A_{c}(s)B_{p}(s)}{A_\mathrm{cl}(s)}d \end{aligned}$$
(5)

where \(A_\mathrm{cl}(s)\) is the characteristic equation of the system and is given by

$$\begin{aligned} A_\mathrm{cl}(s) = A_{p}(s)A_{c}(s)+B_{p}(s)B_{c}(s) = \sum \limits _{i = 0}^{n}a_{i}s^{i} \end{aligned}$$
(6)
$$\begin{aligned} \sum \limits _{i = 0}^{n}a_{i}s^{i} = a_{n}s^{n}+\cdots+a_{1}s+a_{0}. \end{aligned}$$
(7)

The definitions of \(\tau\), \(\gamma _{i}\) and \(\gamma _{i}^{*}\) are detailed as in[8]; the characteristic equation can be represented as [8, 12, 13]

$$\begin{aligned} A_\mathrm{cl}(s) = a_{0}\left[ \left\{ \sum _{i = 2}^{n}\left( \prod _{j = 1}^{i-1}\frac{1}{ \gamma _{i-j}^{j}}\right) \left( \tau s\right) ^{i}\right\} +\tau s+1\right]. \end{aligned}$$
(8)

The conditions used in CDM for determining stability are as follows [8, 9, 16];

The stability condition for third-order and fourth-order systems is derived from Routh–Hurwitz criterion

$$\begin{aligned} \gamma _{i}>\gamma _{i}^{*}\text {, }i = 1,2,\ldots ,(n-1). \end{aligned}$$
(9)

The sufficient condition of stability for higher-order systems is stated by Lipatov [9] and is given by

$$\begin{aligned} \gamma _{i}>1.12 \gamma _{i}^{*}\text {, for any }i\text {, where }i = 1,2,\ldots ,(n-1). \end{aligned}$$
(10)

For instability, the sufficient condition is given by

$$\begin{aligned} (\gamma _{i+1}\gamma _{i})^{0.5}<1\text {, for some }i\text {, where } i = 1,2,\ldots ,(n-2) \end{aligned}.$$
(11)

Selection of Stability Index \(\gamma _{i}\)

The selection of \(\gamma _{i}\) is very important in the design using CDM. The following properties suggested by Lipatov stability conditions [7, 9, 16] are helpful in the selection of \(\gamma _{i}\):

  • The ratio \(\frac{\gamma _{i}}{\gamma _{i}^{*}}\) may be used as a good measure to indicate degree of stability.

  • If all the \(\gamma _{i}\)s are greater than 1.5, the system is stable.

  • If all the \(\gamma _{i}\)s are greater than 4, all the roots of characteristic equation are real, negative and distinct.

  • Thus, value of stability index \(\gamma _{i}\) is usually selected in the interval of \(1.5\sim 4\).

  • In actual practice, a standard form discussed in the next section is recommended for the choice of \(\gamma _{i}\) in CDM which gives sufficiently robust and stable design in most of the cases.

  • The choice of stability indices can be relaxed as \(\gamma _{i}>1.5\) \(\gamma _{i}^{*}\) in order to impart more robustness by making some compromise on the stability and performance.

The standard form of CDM stated by Manabe is given by [17]

$$\begin{aligned} \gamma _{1} = 2.5,\quad \gamma _{n-1} = \gamma _{n-1} = \cdots = \gamma _{2} = 2. \end{aligned}$$
(12)

Controller Design Using CDM

The design steps [8] are summarized as follows:

  • Represent the numerator and denominator of the plant transfer function in polynomial form.

  • Reframe the given performance specifications in terms of design specifications of CDM.

  • Express assumed controller configuration in the polynomial form.

  • Determine the unknown parameters using Diophantine equation.

  • Plot coefficient diagram, cross-check the values of apposite coefficients apt to satisfy the performance specifications and make modifications in their values accordingly.

ITAE-Based Optimal Design of PID Controller

There are many performance indices used for optimal design namely ISE (Integral Square Error), IAE (Integral of Absolute Error), ITAE (Integral Time Absolute Error) and ITSE (Integral Time Square Error). It is known that ITAE has good selectivity, reduces weightage on large initial error and penalizes small errors that occur later in time response heavily [18]. Using ITAE as performance index the controller parameters of PID controller with nominal plant parameters have been obtained using MATLAB. The values obtained are

\(K_{P}\)

\(K_{I}\)

\(K_{D}\)

1.7789

1.9872

0.3836

ITAE-Based Optimal CDM Controller Design for AVR

For designing a CDM-based controller for AVR system, the controller structure has been chosen similar to the PID controller as

$$\begin{aligned} G_\mathrm{CDM}(s) = \frac{k_{2}s^{2}+k_{1}s+k_{0}}{l_{1}s}. \end{aligned}$$
(13)

In CDM-based controller design, there is additional advantage that the steady-state error can be independently taken care of and settling time specification can also be incorporated to fix up \(\tau\). To get the initial values for optimization, settling time 5 secs (thus \(\tau = 2\)), \(a_{0} = 1\) have been used, using (8) and standard Manabe form the initial values were deduced. Further according to the steady-state error requirement, \(k_{0}\) is evaluated as 0.1. The parameters tuned using ITAE are \(k_{2}\), \(k_{1}\) and \(l_{1}\). Using ITAE as performance index, the controller parameters of CDM controller with nominal plant parameters have been obtained using MATLAB. The values obtained are

\(k_{2}\)

\(k_{1}\)

\(l_{1}\)

0.1088

0.4222

0.3748

The step responses of the nominal plant with PID and CDM-based controllers are shown in Fig. 2.

Fig. 2
figure 2

Nominal plant step responses

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To investigate robustness of the two controller systems \(\pm \,5\%\) variation is applied to each of the plant parameters and step responses are obtained in each case. The responses of the sixteen cases of the perturbed plant are shown in Fig. 3, respectively, for PID-based and CDM-based AVR systems. The step responses depict that CDM-based system is more robust than PID system. Also, as the PID controller-based structure is devoid of pre-filter, the responses show steady-state error. However, the pre-filter of the CDM-based control structure resolves the steady-state error.

Fig. 3
figure 3

Perturbed plant step responses

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The coefficient diagrams were obtained for 16 plants only for variations \(\pm \,5\) percent in the 4 gains namely \(K_{A},K_{E},K_{G}\) and \(K_{R}\) of the perturbed plant family in both the controller systems as shown in Fig. 4.

Fig. 4
figure 4

Coefficient diagrams of the perturbed plant PID and CDM system

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For comparison of stability in both the cases, the ratio of stability index and stability limit in each case can be considered (Tables 2, 3).

Table 2 Ratio of stability index and stability limit PID system
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Table 3 Ratio of stability index and stability limit CDM system
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The comparison of the ratios of the stability index to stability limit in each case of the two systems depicts that the value of the ratios is larger for the CDM controller system than PID controlled AVR system. This indicates that the system with CDM controller is more stable as compared to the AVR system regulated by the PID controller.

Robustness Analysis Using Mikhailov’s Theorem

Mikhailov’s Stability Criterion

Another simple method to determine robust stability is applying Mikhailov’s stability criterion to the overbounding polynomial of the uncertain system.

Mikhailov’s stability criterion: A polynomial

$$\begin{aligned} P(s,q)=q_{0}+q_{1}s+\cdots +q_{n}s^{n}\text { , }q_{n}>0 \end{aligned}$$

is said to be stable if and only if its frequency plot \(P(j\omega ,q)\) begins on the positive real axis, excludes and makes counterclockwise encirclement of the origin with a phase increment of \(n\pi /2\) as \(\omega\) varies from 0 to \(\infty\).

The construction of overbounding polynomial that uses the monotonicity property of polynomials has been proposed by Kawamura and Shima [19]. The steps involved in determining robust stability using overbounding polynomial and Mikhailov’s theorem are

  1. 1.

    Check the monotonicity of the coefficient of P(s, q).

  2. 2.

    If coefficients contribute to form monotone polynomials of q then calculate maximum value and minimum value of each coefficient.

  3. 3.

    Construct the overbounding polynomial.

  4. 4.

    Obtain Mikhailov plot, of the image set of overbounding interval polynomial and determine robust stability according to Mikhailov’s stability criterion.

The characteristic polynomial in the PID controller-based AVR system is given by

$$\begin{aligned} A_{PID}(s) = s(1+\tau _{A}s)(1+\tau _{G}s)(1+\tau _{E}s)(1+\tau _{R}s)\nonumber \\&+(K_{P}s+K_{I}+K_{D}s^{2})K_{A}K_{E}K_{G}K_{R} \end{aligned}$$
(14)

and in the case of CDM controller-based AVR system, it is given by

$$\begin{aligned} A_\mathrm{CDM}(s) = l_{1}s(1+\tau _{A}s)(1+\tau _{G}s)(1+\tau _{E}s)\nonumber \\&(1+\tau _{R}s)+(k_{1}s+k_{0}+k_{2}s^{2})\nonumber \\&\quad K_{A}K_{G}K_{E}K_{R}. \end{aligned}$$
(15)

Simplifying (14), the characteristic polynomial in the PID controller-based AVR system is represented as

$$\begin{aligned} A_{PID}(s) = [a^{-}_{0},a^{+}_{0}]+[a^{-}_{1},a^{+}_{1}]s+[a^{-}_{2},a^{+}_{2}]s^2\nonumber \\&+[a^{-}_{3},a^{+}_{3}]s^3 +[a^{-}_{4},a^{+}_{4}]s^4+[a^{-}_{5},a^{+}_{5}]s^5, \end{aligned}$$
(16)

where the following definitions hold good

$$\begin{aligned} {a_{0}}&= {1.9872[K^{-} \quad K^{+}]}\nonumber \\ {a_{1}}&= {1.7789[K^{-}\quad K^{+}]+1}\nonumber \\ {a_{2}}&= {(0.3836[K^{-}\quad K^{+}]+[\tau _{A}^{-}+\tau _{G}^{-}+\tau _{R}^{-}+\tau _{E}^{-}\quad \tau _{A}^{+}+\tau _{G}^{+}+\tau _{R}^{+}+\tau _{E}^{+}])}\nonumber \\ {a_{3}}&= {([\tau _{AE}^{-}+\tau _{GR}^{-}+\tau _{AGR}^{-}+\tau _{EG_R}^{-}\quad \tau _{AE}^{+}+\tau _{GR}^{+}+\tau _{AGR}^{+}+\tau _{EG_R}^{+}])}\nonumber \\ {a_{4}}&= {([\tau _{AEG_R}^{-}+\tau _{EGR}^{-}+\tau _{AGR}^{-} \quad \tau _{AEG_R}^{+}+\tau _{EGR}^{+}+\tau _{AGR}^{+}])}\nonumber \\ { a_{5}}&= { [\tau _{AEGR}^{-}\quad \tau _{AEGR}^{+}]}. \end{aligned}$$
(17)

The characteristic polynomial in the CDM controller-based AVR system is represented as

$$\begin{aligned} {A_\mathrm{CDM}(s)} = {[b^{-}_{0},b^{+}_{0}]+[b^{-}_{1},b^{+}_{1}]s+[b^{-}_{2},b^{+}_{2}]s^2+[b^{-}_{3},b^{+}_{3}]s^3}\nonumber \\& {+[b^{-}_{4},b^{+}_{4}]s^4+[b^{-}_{5},b^{+}_{5}]s^5}, \end{aligned}$$
(18)

where the following definitions hold good

$$\begin{aligned} {b_{0}}&= {0.1[K^{-} \quad K^{+}]}\nonumber \\ {b_{1}}&= {0.4222[K^{-}\quad K^{+}]+0.3748}\nonumber \\ {b_{2}}&= {(0.1088[K^{-}\quad K^{+}]+0.3748[\tau _{A}^{-}+\tau _{G}^{-}+\tau _{R}^{-}+\tau _{E}^{-}\quad \tau _{A}^{+}+\tau _{G}^{+}+\tau _{R}^{+}+\tau _{E}^{+}])}\nonumber \\ {b_{3}}&= {(0.3748[\tau _{AE}^{-}+\tau _{GR}^{-}+\tau _{AGR}^{-}+\tau _{EG_R}^{-}\quad \tau _{AE}^{+}+\tau _{GR}^{+}+\tau _{AGR}^{+}+\tau _{EG_R}^{+}])}\nonumber \\ {b_{4}}&= {(0.3748[\tau _{AEG_R}^{-}+\tau _{EGR}^{-}+\tau _{AGR}^{-} \quad \tau _{AEG_R}^{+}+\tau _{EGR}^{+}+\tau _{AGR}^{+}])}\nonumber \\ { b_{5}}&= { 0.3748[\tau _{AEGR}^{-}\quad \tau _{AEGR}^{+}]}, \end{aligned}$$
(19)

where the parameters have been defined as

$$\begin{aligned} K&=K_{A}K_{E}K_{G}K_{R} \nonumber \\ \tau _{AEGR}&= \tau _{A}\tau _{E}\tau _{G}\tau _{R} \nonumber \\ \tau _{AEG_R}&=\tau _{A}\tau _{E}(\tau _{G}+\tau _{R}) \nonumber \\ \tau _{EG_R}&= \tau _{E}(\tau _{G}+\tau _{R}) \nonumber \\ \tau _{EGR}&=\tau _{E}\tau _{G}\tau _{R} \nonumber \\ \tau _{AGR}&= \tau _{A}\tau _{G}\tau _{R} \nonumber \\ \tau _{AE}&=\tau _{A}\tau _{E} \nonumber \\ \tau _{GR}&=\tau _{G}\tau _{R}. \end{aligned}$$
(20)
Table 4 Coefficients of overbounding characteristic polynomials
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The coefficients of the overbounding characteristic polynomials in both the cases are calculated and compiled in Table 4. Also, Mikhailov’s plot in each case for overbounding characteristic polynomial has been obtained to investigate robustness. The plot shown in Fig. 5 corresponds to AVR system with PID controller and CDM controller. The image set plot of PID system embraces the origin whereas for the same perturbation the CDM-based system clearly excludes the origin. It is clearly observed that CDM-based system is robustly stable as compared to PID controller-based system.

Fig. 5
figure 5

Mikhailov’s plot for PID controller-based and CDM controller-based AVR

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Conclusions

In this paper, a robust controller has been designed for an AVR using coefficient diagram method. The controller parameters have been obtained by minimizing ITAE. The results of AVR system controlled by the optimal CDM-based controller have been compared with the results of AVR system with a PID controller whose parameters also have been tuned using ITAE criterion. The step responses and the coefficient diagrams in both the cases have been plotted, the ratios of stability index to stability limit have been listed for each plant of the perturbed family of plants in both the cases. Also, to compare robust stability, overbounding interval polynomial has been constructed and the corresponding Mikhailov’s plot of the image set of the overbounding polynomial in each case has been obtained. The results depict superior performance of CDM controller-based AVR system than the conventional PID controller system.