A Fractional Order Power System Stabilizer Applied on a Small-Scale Generation System

: In this paper, a Fractional Order Power System controller (FOPSS) is designed, and its performance and robustness are experimentally evaluated by tests in a 10 kVA laboratory scale power system. The FOPSS design methodology is based on the tuning of an additional design variable, namely the fractional order of the controller transfer function. This design variable is tuned aiming to obtain a tradeoff between satisfactory damping of dominant oscillating mode and improved closed-loop system robustness. For controller synthesis, transfer function models were estimated from data collected at selected operating points and subsequently applied for the controller design and for obtaining upper bounds estimates on the operating-point depends on plant uncertainties. The experimental results show that the FOPPS was able to obtain a robust performance for the considered set of the power system operating conditions.


Introduction
Due to the continuously growing electrical energy demand, electric power systems need to be operated close to their stability limits [1]. Consequently, new research on advanced and intelligent control techniques is required to improve the system stability margins. In particular, the dominant electromechanical oscillation mode must be well-damped for the range of allowed operating conditions to preserve system stability and to assure safe operation. To accomplish this task, Power System Stabilizers (PSS) are used to improve the damp of dominant oscillation mode in the interconnected power system [2]. Research on new advanced control for PSS performance improvment is a very active field in control system engineering. Therefore, many studies, including on adaptive control [3], gain-scheduling fuzzy control [4] and robust control [5], to name a few, have been carried out in the last two decades. A promising robust control strategy whose potential is now being investigated in different engineering applications is Fractional Order Control. Fractional order controllers can be seen as a natural extension of conventional control techniques [6]. Considerable attention in fractional order controllers have been carried out in recent years, mainly because of their additional design parameters gives a more flexible degree of freedom on the design task when compared to conventional The remainder of this paper is organized as follows: Section 2 presents a brief introduction to fractional order systems. Section 3 presents the tuning method for the classical power system stabilizer and the fractional order power system stabilizer. In Section 4, after a brief description of the 10 kVA laboratory power system, the identification tests for local model identification is presented and discussed. Section 5 presents the design of both fractional PSS (FOPSS) and conventional PSS. In Section 6, a robustness assessment of the designed controllers regarding a set of operating pointing dependent plant uncertainties is performed. Section 7 presents and discusses the results of experimental tests performed in a small-scale power plant. Finally, in Section 8, the conclusions of this study are presented.

Background of Fractional Order System in the Frequency Domain
A fractional order operator may be described as where q is a rational number, defining the fractional order, and a and t are the limits for the derivative and the integral operators. This fractional order operator definition is known as Caputo's definition [10,11].
Energies 2018, 11, 2052 3 of 20 Because the Laplace transform is widely used for control analysis and design, it is usually useful expressing the fractional order operator in its frequency domain version, by using the Laplace´s transform of Equation (2), as follows where null initial conditions have been assumed in Equation (3).

Phase and Gain Contributions Due to Fractional Order Lead-Lag System
A fractional order system is defined in the following form, By using s = jω, the corresponding magnitude and phase are given by: 20 log 10 (jωT 1 + 1) α = 20 log 10 |(jωT 1 + 1)| α = 20α log 10 1 2 + (ωT 1 ) 2 , ∠(jωT 1 + 1) α = αarctg(ωT 1 ), It can be seen, from Equations (5) and (6), that both magnitude and phase contributions can be controlled by setting the value of the fractional order parameter α. Therefore, besides the time constant T 1 , the fractional order parameter α allows for an additional degree of freedom for the design of damping controllers. This property will be exploited in the next sections to design fractional order based PSS.

Classical Tuning Method for Power System Stabilizer
To tune the conventional PSS compensator as well as the fractional compensator, GEP(s) represents the dynamics from Automatic Voltage Regulator (AVR) set point increment to the corresponding increment on the electrical torque component via excitation system. In this work, this transfer function was obtained by using the experimental identification method previously presented in Section 4. The classical Lead-Lag compensator tuning method was obtained based on the methodology proposed in the seminal paper by Larsen and Swann [12]. This standard method is based on phase compensation design followed by PSS gain tuning by using the root-locus method. The Lead-Lag Classical transfer function is given by Equations (7) and (8): Step 1: Use the obtained electromechanical natural frequency value shown previously in Figure 1; find the phase needed to accomplish phase compensation of Lead-Lag Compensator; and choose the value of the parameter T 1 . Typical values of the parameter T 1 are within 0.2-1.5 s [2], applying Equations (9) and (10): Energies 2018, 11, 2052 4 of 20 Step 2: Find the desired Torque damping coefficient to tune KPSS gain, applying Equation (11): Step 3: Find the Lead-Lag compensator gain, applying Equations (12) and (13): Energies 2018, 11, x FOR PEER REVIEW 7 of 22

Fractional Order Compensator Tuning Method
The tuning of the FOPSS parameters is accomplished by using a frequency domain approach which consists in extending the classical PSS design to calculate the parameters of the compensator, raised to a fractional power value. The fractional power parameter enables shaping the desired closed-loop frequency. Therefore, to design FOPSS, this section presentes a sequence of four steps: Steps 1-3 are to find parameters of Lead-Lag compensator, while, in Step 4, the gain tuning is performed: Step 1: Find natural electromechanical frequency modes with a low damping ω n .
Step 2: Choose T 1 parameter value, and then find T 2 parameter value by using Equation (14): Energies 2018, 11, 2052 5 of 20 Remark 1: In Step 2, the phase condition imposes a single equation having two unknown variables, namely the values of time constants T 1 and T 2 . As a possible choice, the value of T 1 has been fixed at the same value obtained for the conventional PSS and T 2 has been calculated using Equation (14).
Step 3: Applying Equations (15) and (16), the fractional order power α is found: Step 4: After parameters T 1 , T 2 and α have been calculated, the value of KPSS gain is then obtained by specifying a desired damping coefficient ξ d and using the modulus condition, as follows: Finally, the transfer function of Fractional Order PSS (FOPPS) is obtained in the following fractional order Lead-Lag [13,14] form:

Fractional Order Approximation
After its design, real-world implementation of the fractional order controller must be done. This task is carried out by searching for an integer order dynamic system which approximates the dynamic behavior of the designed fractional order controller in a specified frequency range. In this work, the approach proposed in [14], which is based on pole-zero network synthesis is adopted. This approach can be outlined by the following steps [14]: Step 1: Get the exact frequency response of fractional order system.
Step 2: Select an appropriate order for the numerator and denominator of the approximating integer order system, in a range of frequencies of interest (for PSS design, this range can be, for instance, the range of frequencies containing the frequencies of the target electromechanical oscillating modes).
Step 3: Find the values of the parameters (values of poles and zeros) of the approximating integer order model by using the invfreqs function, in the Matlab R2015a software.
Step 4: Verify if the obtained controller satisfies the desired performance behavior. If the control objective was not reached, then return to Step 2 and select other values for the numerator and denominator order or modify the range of frequencies of interest.

Small-Scale 10 kVA Power System
Controller evaluation was performed in a small-scale power system comprised of a synchronous generating unit (10 kVA, 220 VRMS, 60 Hz), a transmission line model, transformers, as well as measurement and control subsystems (Figure 2a). The system has a typical machine-infinite bus bar power system configuration ( Figure 2b). The synchronous generator was specially built so that its physical parameter values are similar, in per unit values, to that of large synchronous generators. An inductive branch represents a typical short transmission line. Table 1 presents the corresponding values for some nominal values of physical parameters of the laboratory power system. The synchronous generator unit is equipped with a fast-actuating field excitation system based on a Energies 2018, 11, 2052 6 of 20 DC-DC buck converter. The automatic voltage regulator (AVR) is of proportional type, as described in [4], by the same research group of this paper.

Experimental Environment
The control and drive panel is installed in an industrial standard panel where the components responsible for the activation and small-scale 10 kVA power generation system control are installed. This panel comprises the components responsible for the drive, measurement of current and voltage signals necessary for the system operation. Figure 2 presents a control panel design scheme and a panel photograph are shown. The Exciter actuator is a DC-DC buck power converter that controls the synchronous machine rotor current directly, the microcontroller that the embedded systems control laws of AVR and PSS is a Microchip dsPIC 30F4011.
In Figure 3, the AVR and PSS graphical man-machine interface (MMI) is shown. The MMI that operates the digital AVR and PSS has a set of keys to send commands directly to the instrument and numerical and graphical displays to show several parameters and signals during operation.

Experimental Environment
The control and drive panel is installed in an industrial standard panel where the components responsible for the activation and small-scale 10 kVA power generation system control are installed. This panel comprises the components responsible for the drive, measurement of current and voltage signals necessary for the system operation. Figure 1 presents a control panel design scheme and a panel photograph are shown. The Exciter actuator is a DC-DC buck power converter that controls the synchronous machine rotor current directly, the microcontroller that the embedded systems control laws of AVR and PSS is a Microchip dsPIC 30F4011.
In Figure 3, the AVR and PSS graphical man-machine interface (MMI) is shown. The MMI that operates the digital AVR and PSS has a set of keys to send commands directly to the instrument and numerical and graphical displays to show several parameters and signals during operation.

System Identification Tests
The power system presents a nonlinear dynamic behavior which depends on its operating condition. To represent the relevant dynamic around a set of representative operating state, a set of discrete-time linear Autoregressive with Exogenous Input (ARX) are constructed, having a structure giving by Equation (21), where q −1 is the unitary delay operator and A(q −1 ), and B(q −1 ) are, respectively, the denominator and the numerator polynomials whose value of the parameters vectorθ = [b 1 , b 2 , ..., b nB , a 1 , a 2 , ..., a nA ] T is estimated by using a standard Least Mean Square parameter estimation approach [15]. The obtained ARX models are then subsequently applied for both controller design and robustness analysis tests.
Operating points corresponding to three different level of the generator loading levels "LOW", "MEDIUM" and "HIGH", corresponding, respectively, to generated active power P e = {0.3, 0.5, 0.65} p.u. For all the selected operating conditions, the magnitude of the generator terminal voltage was kept close to its nominal value (VT equal to 1.00 p.u.), while the value of generated reactive power was kept close to 0 p.u. Figure 4 shows the results of the measured active power response for small pulse disturbance applied in superposition to the AVR setpoint value. The operating point of this test corresponds to the MEDIUM loading condition (P e = 0.5 p.u.). A dominant oscillation mode, having a natural period around T OSC = 0.8 s, can be observed. This dominant oscillation mode presents very low damping and, therefore, a Power System Stabilizer (PSS) must be designed to provide sufficient damping for this dominant mode. By using the estimated value of T OSC , a suitable value for the sample time value must satisfy T S << T OSC and, therefore, a sample time value of T S = 0.060 seconds has been chosen for both data acquisition and control objectives.
p.u.). A dominant oscillation mode, having a natural period around TOSC = 0.8 s, can be observed. This dominant oscillation mode presents very low damping and, therefore, a Power System Stabilizer (PSS) must be designed to provide sufficient damping for this dominant mode. By using the estimated value of TOSC, a suitable value for the sample time value must satisfy TS << TOSC and, therefore, a sample time value of TS = 0.060 seconds has been chosen for both data acquisition and control objectives. A Pseudo Random Binary Signal (PRBS) was designed to excite uniformly the system modes in the range of frequencies from 0.5 to 3 Hz, which includes the nominal range for the local mode A Pseudo Random Binary Signal (PRBS) was designed to excite uniformly the system modes in the range of frequencies from 0.5 to 3 Hz, which includes the nominal range for the local mode in power studies. Small amplitude (0.01 p.u.) for the PRBS signal was selected in order not to disturb the terminal voltage regulation considerably. A set of equations to implement the PRBS signal as in [15], is given by Equations (22)- (27): The estimated spectrum obtained by processing the collected data is shown in Figure 5, where can be observed that the system has a dominant oscillating mode whose natural frequency is close to 1.2 Hz, which is poorly damped without using a PSS damping controller.
Using the recorded input-output data, considering the PRBS signal as the system input and measured active power deviation as the system output, fourth order ARX models, having the structure given in Equation (21), were estimated for the power system operating around LOW, MEDIUM and HIGH loading conditions, corresponding, respectively, to following levels of generated electric active power: P e = {0.30, 0.50, 0.65} p.u. These identified local models are named P 1 , P 2 , and P 3 , respectively. The corresponding values for parameters of the local models P 1 , P 2 and P 3 , were obtained using standard Least Mean Square (LMS) estimation techniques [15] and are provided in Table 2. The estimated spectrum obtained by processing the collected data is shown in Figure 5, where can be observed that the system has a dominant oscillating mode whose natural frequency is close to 1.2 Hz, which is poorly damped without using a PSS damping controller. Using the recorded input-output data, considering the PRBS signal as the system input and measured active power deviation as the system output, fourth order ARX models, having the structure given in Equation (21), were estimated for the power system operating around LOW, MEDIUM and HIGH loading conditions, corresponding, respectively, to following levels of generated electric active power: Pe = {0.30, 0.50, 0.65} p.u. These identified local models are named P1, P2, and P3, respectively. The corresponding values for parameters of the local models P1, P2 and Medium loading (P e = 0.5 p.u) P 2 B(q −1 ) Coefficients P e = 0.5 p.u.
High loading (P e = 0.65 p.u) In Figure 6, it can be observed that a satisfactory fitting has been obtained between the output estimated by using the local models P 1 , P 2 , and P 3 and the real output measurements. This confirms that the identified models are accurate enough for control design purposes as these models were able to capture the frequency of the target oscillation mode and, therefore, the model is suitable for PSS control design. Additional standard validation tests based on correlation analysis, not shown, confirmed the adequacy of the identified model.
In Figure 6, it can be observed that a satisfactory fitting has been obtained between the output estimated by using the local models P1, P2, and P3 and the real output measurements. This confirms that the identified models are accurate enough for control design purposes as these models were able to capture the frequency of the target oscillation mode and, therefore, the model is suitable for PSS control design. Additional standard validation tests based on correlation analysis, not shown, confirmed the adequacy of the identified model. In Figure 7, the magnitude frequency response obtained from the estimated local models P1, P2, and P3 is shown. It can be observed that, for all three models, the value of natural frequency of the dominant mode is around 8 rad/s. Furthermore, the damping for this target mode changes substantially with the operating condition. Such variability makes the design of robust PSS controller a quite challenging task. In Figure 7, the magnitude frequency response obtained from the estimated local models P 1 , P 2 , and P 3 is shown. It can be observed that, for all three models, the value of natural frequency of the dominant mode is around 8 rad/s. Furthermore, the damping for this target mode changes substantially with the operating condition. Such variability makes the design of robust PSS controller a quite challenging task.  The corresponding eigenvalues computed for the local models P1, P2 and P3 are provided in Table 3, where ξi(j) and ωi(j) are the respective values of relative damping and the natural frequencies of the i-th eigenvalue, λi, i = 1, 2, 3, 4, calculated for each identified local models Pj, j = 1, 2, 3. It can be observed that the relative damping of dominant the eigenvalue is very low, indicating that the power system presents reduced stability margins.  The corresponding eigenvalues computed for the local models P 1 , P 2 and P 3 are provided in Table 3, where ξ i (j) and ω i (j) are the respective values of relative damping and the natural frequencies of the i-th eigenvalue, λ i , i = 1, 2, 3, 4, calculated for each identified local models P j , j = 1, 2, 3. It can be observed that the relative damping of dominant the eigenvalue is very low, indicating that the power system presents reduced stability margins. A flowchart that describes all five steps applied to identify the plant models is presented in Figure 8. In Step 1, the frequency band where lies the natural electromechanical modes is selected. In Step 2, the PRBS signal is designed. In Step 3, the PRBS signal in the AVR control loop is applied to excite the system. In Step 4, the ARX model structure is selected. In Step 5, the continuous model is obtained, by applying the inverse z transform ZOH method, using sampling time equal to 0.06 s. The PRBS has 5 cells, the 3 th and 5 th cells are used to feedback the XOR port.
Apply the PRBS at the eletrical power system Select the structure to identify the system, using the ARX model Selected structure ARX 4-4-1, which the numerator and denominator has 4 th order, further one delay.
Applying inverse Ztransform in identified system System identification method

Using the ZOH method
The PRBS signal is added in the loop of the Automatic Voltage Regulator (AVR), aiming at excitate the eletromechanical modes.

Design of the Fractional Order PSS and Tests in the 10 kVA Laboratory Power System
For design purposes, in the following, the nominal model P0 has been taken to be P0 = P2, namely the local model identified for the MEDIUM loading operating condition of the power system (see Table 2). The rationale for this choice is that the MEDIUM loading shows to be the most frequent operating condition for the studied laboratory power system. The design objective is to increase the damping of the observed dominant oscillation mode, without changing substantially its natural frequency. Hence, a desired damping ξd = 0.2 has been specified for the dominant oscillating mode. A Fractional Order PSS (FOPSS) (having the structure given by

Design of the Fractional Order PSS and Tests in the 10 kVA Laboratory Power System
For design purposes, in the following, the nominal model P 0 has been taken to be P 0 = P 2 , namely the local model identified for the MEDIUM loading operating condition of the power system (see Table 2). The rationale for this choice is that the MEDIUM loading shows to be the most frequent operating condition for the studied laboratory power system. The design objective is to increase the damping of the observed dominant oscillation mode, without changing substantially its natural frequency. Hence, a desired damping ξ d = 0.2 has been specified for the dominant oscillating mode. A Fractional Order PSS (FOPSS) (having the structure given by Equation (28)), has been designed. The parameters of the FOPSS controller, namely KFOPSS, T 1 , T 2 and α (the fractional order), were computed by using the algorithm steps already presented in Section 3 of this paper and their obtained values are provided in Table 4.
For performance comparison purposes, a conventional PSS (having two similar first-order Lead-Lag blocks) has also been designed by using the same nominal design specifications as stated above for the fractional order PSS design. This conventional PSS has been designed by using a standard classical technique [12,16]. Such classical design consists of performing a phase compensation step (allowing to obtain the values for the parameters N (number of Lead-Lag blocks) and time constants T 1 and T 2 ), followed by an adjustment of the PSS gain value, KPSS, by using the root-locus technique. This KPSS last tuning step has been performed by taking care for not to reduce the damping of the remaining system's modes substantially. The values of the parameters for the conventional PSS are also provided in Table 4. To summarize, the proposed methodology is presented as a simplified flowchart in Figure 9.
A flowchart where all steps applied to identify the plant models is presented in Figure 7. In a the sequence of choosing the model to tune the controller, the following technique is applied. Firstly, in Step 1, the electromechanical low damping mode related to the desired frequency low damping ω n is identified. In Step 2, the T 1 time constant values linked to the zero of the FOPSS is chosen as 0.2 s. In Step 3, the value of T 2 time constant associated with the FOPSS denominator is computed and its value is 0.0783 s. In Step 4, the relation for computing the fractional parameter, α, carried out by Equation (16) is presented. In sequence, Steps 5 and 6 are related to obtaining the K FOPSS value of 0.2429. Finally, to allow for real-world FOPPSS implementation, its infinite dimensional transfer function (Equation (28)) must be approximated by using a finite dimensional (integer order) transfer function. This finite order approximation is held in the frequency band ω ∈ [0.01, 100] rad/s and is selected second to implement the structure of compensator. This order is chosen due to the great fit obtained in the frequency band comparing structures of higher order given by Step 7.  To accomplish this task, a frequency domain interpolation method, proposed in Reference [14], has been applied to obtain the corresponding finite order FOPSS controller approximation given by Equation (29) (29) In Table 5, the eigenvalues computed for the nominal closed-loop nominal system are provided for both: the Conventional PSS and the FOPSS controllers. It can be observed that, in the nominal operating condition of the design, both controllers were able to satisfactorily improve the relative damping of the dominant oscillating mode as expected. To accomplish this task, a frequency domain interpolation method, proposed in Reference [14], has been applied to obtain the corresponding finite order FOPSS controller approximation given by Equation (29): C FOPSS (s) = 0.158s 2 + 3.017s + 13.59 s 2 + 15.54s + 55.95 .
In Table 5, the eigenvalues computed for the nominal closed-loop nominal system are provided for both: the Conventional PSS and the FOPSS controllers. It can be observed that, in the nominal operating condition of the design, both controllers were able to satisfactorily improve the relative damping of the dominant oscillating mode as expected. For the digital implementation of the FOPSS control law by using a DSPIC microcontroller chip, the FOPSS approximated transfer function, C FOPSS , given by Equation (21), has been discretized by using the Tustin method, with T S = 0.06 seconds sample time interval. Such discretization allows for representing the FOPSS control law by using a canonical RST standard structure given by Equation (30). Table 6 provides the corresponding values obtained for the coefficients of the digital law in Equation (22), for the designed FOPSS controller. Similar discretization steps were also performed for the designed conventional PSS, and the corresponding coefficients of the digital control law (in the form of Equation (30)) are provided in Table 6.

Robust Performance Assessments for the Designed Fractional Order Power System Stabilizer
To assess the robust performance of the designed Fractional Order Power System Stabilizer (FOPSS), a recently developed robust performance evaluation methodology [17] was applied. Such methodology is based on Robust Bode plots, which makes it both easy to understand and easy to apply. For the allowed frequency of interest, robust performance boundaries are established for allowed and forbidden regions in the magnitude and phase Bode Plots for the loop nominal loop transfer function L 0 FOPSS (s) = C FOPSS (s)P 0 (s), where C FOPSS (s) and P 0 (s) are, respectively, the designed FOPSS (its finite-dimensional approximation, given by Equation (29)) and the nominal plant model used in the controller design.

Robust Performance Boundaries Computation
In [18], it has been proven that a necessary and sufficient condition assuring the robust performance of a closed loop system composed by an uncertain plant model P(s) = P 0 (s) (1 + W 2 (s)∆(s)) in unit negative feedback with a designed controller C(s) is given by, where P 0 (s) is the nominal plant model, ∆(s) is a multiplicative uncertainty satisfying ∆(s) ∞ < 1, By using the inequality in Equation (31) and by defining the nominal loop transfer function as being L o (s) = C(s)P o (s), the following pair of coupled inequalities can be derived [17], considering ∀ω ∈ 0 ∞ .
where the dependency on ω has been omitted for the sake of simplicity. By specifying a range of frequencies of interest, the inequalities in Equations (32) and (33) are used to numerically computing bounds on the allowed and forbidden regions for |L o (jω)| and arg(L o (jω)). These calculated bounds are then superimposed into the Bode plot (magnitude and phase) of L o (jω), for analysis purposes. Any observed intersection of |L o (jω)| plot (or arg(L o (jω))) with the computed robustness boundary regions given by Equations (32) and (33) mean that the designed controller fails to satisfy the robust performance for the considered set of uncertainties. Therefore, the above described robust analysis methodology was applied to evaluate the robust performance of the designed FOPSS against the uncertainties due to the plant operational conditions.

Selecting the Nominal Plant Model P 0 (jω)
The nominal plant model used to perform robust performance analysis is the same as the one chosen for the FOPSS control design, namely P 0 = P 2 (the local model identified for the MEDIUM loading operating condition). Such local model was chosen because it is the most frequent operating condition in the 10 kVA Laboratory Power System.
It can be observed that the plot of |W 1 (jω)| always remains under the plot of 1/S o (jω), for the considered range of frequencies, as desired, while the |W 2 (jω)| plot has a frequency dependent upper bound for the operating point dependent set of considered plant uncertainties.

FOPSS Robust Performance Assessment
By using the inequalities in Equation (24) and (25), the corresponding forbidden regions have been computed for the robust performance assessment of the designed FOPSS controller. In Figure 11, the Bode plot for the designed nominal loop transfer function L0 FOPSS (s) = CFOPSS(s)Po(s) is presented. In Figure 11, the regions in which the robust performance of the closed-loop system is violated (forbidden regions) are indicated by dark and gray colors. It can be observed that, for the range of frequencies of the controller design, namely   [1, 100] rad/s, the plot, does not intersect the forbidden regions in the considered frequency range. This result confirms that FOPSS is robust against the set of acknowledged operating point dependent uncertainties. In Figure 11, the plot is also presented, for the designed conventional PSS. The conventional PSS is also robust while showing a lightly reduced robustness margin in comparison to the fractional order PSS.

FOPSS Robust Performance Assessment
By using the inequalities in Equation (24) and (25), the corresponding forbidden regions have been computed for the robust performance assessment of the designed FOPSS controller. In Figure 11

FOPSS Robust Performance Assessment
By using the inequalities in Equation (24) and (25), the corresponding forbidden regions have been computed for the robust performance assessment of the designed FOPSS controller. In Figure 11, the Bode plot for the designed nominal loop transfer function L0 FOPSS (s) = CFOPSS(s)Po(s) is presented. In Figure 11, the regions in which the robust performance of the closed-loop system is violated (forbidden regions) are indicated by dark and gray colors. It can be observed that, for the range of frequencies of the controller design, namely   [1, 100] rad/s, the plot, does not intersect the forbidden regions in the considered frequency range. This result confirms that FOPSS is robust against the set of acknowledged operating point dependent uncertainties. In Figure 11, the plot is also presented, for the designed conventional PSS. The conventional PSS is also robust while showing a lightly reduced robustness margin in comparison to the fractional order PSS. In Figure 11, the regions in which the robust performance of the closed-loop system is violated (forbidden regions) are indicated by dark and gray colors. It can be observed that, for the range of frequencies of the controller design, namely ω∈ [1, 100] rad/s, the plot, L FOPSS o (jω) = |C FOPSS (jω)P o (jω)| does not intersect the forbidden regions in the considered frequency range. This result confirms that FOPSS is robust against the set of acknowledged operating point dependent uncertainties. In Figure 11, the plot L PSS o (jω) = |C PSS (jω)P o (jω)| is also presented, for the designed conventional PSS. The conventional PSS is also robust while showing a lightly reduced robustness margin in comparison to the fractional order PSS.

Experimental Field Tests
In this subsection, the results obtained by carrying out experimental tests on the 10 kVA small-scaled power system are presented and discussed. The signal corresponding to open-loop tests (without using any damping controller) are shown in blue color dashed line. For the closed-loop cases, black color dash-dot line is used for the signals corresponding to the conventional PSS performance while red color solid line is used to signals corresponding to the performance of the fractional order PSS (FOPSS).
In Figure 12a, the performance behavior of the active power deviation obtained for a low load level operating condition (active power close to 0.1 p.u.) is shown, while, in Figure 12b, the corresponding control effort is shown. The applied disturbance pulse signal has a short-duration (around 60 ms) and small amplitude (0.05 p.u.). This disturbance test signal was applied in superposition to the setpoint value of the AVR Controller (around 1.0 p.u.). It can be observed in Figure 12a that the FOPSS shows effective damping of the dominant electromechanical oscillation. Furthermore, it can also be observed that this performance is obtained with a control effort lightly reduced in comparison to conventional PSS effort (see Figure 12b).

Experimental Field Tests
In this subsection, the results obtained by carrying out experimental tests on the 10 kVA small-scaled power system are presented and discussed. The signal corresponding to open-loop tests (without using any damping controller) are shown in blue color dashed line. For the closedloop cases, black color dash-dot line is used for the signals corresponding to the conventional PSS performance while red color solid line is used to signals corresponding to the performance of the fractional order PSS (FOPSS).
In Figure 12a, the performance behavior of the active power deviation obtained for a low load level operating condition (active power close to 0.1 p.u.) is shown, while, in Figure 12b, the corresponding control effort is shown. The applied disturbance pulse signal has a short-duration (around 60 ms) and small amplitude (0.05 p.u.). This disturbance test signal was applied in superposition to the setpoint value of the AVR Controller (around 1.0 p.u.). It can be observed in Figure 12a that the FOPSS shows effective damping of the dominant electromechanical oscillation. Furthermore, it can also be observed that this performance is obtained with a control effort lightly reduced in comparison to conventional PSS effort (see Figure 12b).  Figure 13a presents the closed-loop system response for the power system at the high loading operation point 0.6 p.u. The FOPSS performance had almost the same behavior as the previously analyzed operating condition. It can be observed that the FOPSS controller was able to provide effective damping of the oscillations. Figure 13b presents the corresponding controls effort, showing that the FOPSS effort remains lower than the effort provided by the conventional PSS.  Figure 13a presents the closed-loop system response for the power system at the high loading operation point 0.6 p.u. The FOPSS performance had almost the same behavior as the previously analyzed operating condition. It can be observed that the FOPSS controller was able to provide effective damping of the oscillations. Figure 13b presents the corresponding controls effort, showing that the FOPSS effort remains lower than the effort provided by the conventional PSS.

Cost Function Evaluation
The obtained results were also corroborated by an analysis of the performance indices  (34) Figure 14 shows the ISE for all the six investigated operating points related to the active power deviation response (Figure 14a) and the corresponding cost function for control effort (Figure 14b). It can be observed, in Figure 14a, that the ISE value obtained for the FOPSS controller is smaller than the ISE value obtained when applying the conventional PSS controller for most of the tested operating condition, from low to high level of active power generation. In fact, Figure  14a shows that the control effort provided for both (PSS and FOPSS) controllers are quite similar. These results indicate that the FOPSS may be regarded as competitive for real-world applications in large generator units. Figure 14b reveals that the improved performance obtained by the FOPSS is achieved without using an excessive control effort. Figure 14b presents the cost function applied related to the system control signal at all six operation points. At an operating point corresponding to a low level of loading (0.3 p.u), the ISE obtained for the FOPSS controller shows to be slightly above the one obtained for the conventional PSS. This result might be due to some time-variable uncertainties that happen during experimental tests, including variations on the value of the RMS voltage of the larger power system. Thus, FOPSS has an improved dynamic behavior of active power deviation.

Cost Function Evaluation
The obtained results were also corroborated by an analysis of the performance indices (34) Figure 14 shows the ISE for all the six investigated operating points related to the active power deviation response (Figure 14a) and the corresponding cost function for control effort (Figure 14b). It can be observed, in Figure 14a, that the ISE value obtained for the FOPSS controller is smaller than the ISE value obtained when applying the conventional PSS controller for most of the tested operating condition, from low to high level of active power generation. In fact, Figure 14a shows that the control effort provided for both (PSS and FOPSS) controllers are quite similar. These results indicate that the FOPSS may be regarded as competitive for real-world applications in large generator units. Figure 14b reveals that the improved performance obtained by the FOPSS is achieved without using an excessive control effort. Figure 14b presents the cost function applied related to the system control signal at all six operation points. At an operating point corresponding to a low level of loading (0.3 p.u), the ISE obtained for the FOPSS controller shows to be slightly above the one obtained for the conventional PSS. This result might be due to some time-variable uncertainties that happen during experimental tests, including variations on the value of the RMS voltage of the larger power system. Thus, FOPSS has an improved dynamic behavior of active power deviation.

Conclusions
This paper reportes the design and experimental tests of a fractional order based power system stabilizer (FOPSS). The FOPSS can be seen as a natural extension of the conventional Lead-Lag based PSS structure. In this case, the fractional order parameter provides an additional degree of freedom which may be exploited to obtain performance robustness improvements.
Regarding the damping of the target dominant oscillation mode, it was found, by performing experimental tests in a laboratory scale power system, that the FOPSS presents a reduced loss of performance in comparison to its conventional counterpart, for the set of allowed operating conditions. Furthermore, the improved robustness of the FOPSS has been obtained using a similar control effort to the one provided by using a conventional PSS. The obtained experimental results for the power system studied in this paper shows that the AVR modes (frequency around 16 rad/s) have slightly greater damping than the damping values obtained for the same mode, in comparison to the designed conventional PSS. However, further additional theoretical studies must be performed to affirm that this behavior holds in general case. A possible FOPSS drawback is its more demanding computational implementation which may demand high order digital filters to obtain a suitable approximation of the fractional order dynamics. An advantage of fractional order approach is its ability to provide improved performance due to its additional tuning parameters. However, to the best of the author's knowledge, this is a challenging problem from a complex analysis domain to which a formal solution remains to be provided. According to Reference [14], thus far, such statement has only been evidenced using practical tests in rig systems. The experimental results presented in the present paper seem to empirically corroborate this statement, as it can be observed that, for the studied power system, the FOPSS provided presented a robust performance with control effort not higher than the control provided by the conventional PSS.

Conclusions
This paper reportes the design and experimental tests of a fractional order based power system stabilizer (FOPSS). The FOPSS can be seen as a natural extension of the conventional Lead-Lag based PSS structure. In this case, the fractional order parameter provides an additional degree of freedom which may be exploited to obtain performance robustness improvements.
Regarding the damping of the target dominant oscillation mode, it was found, by performing experimental tests in a laboratory scale power system, that the FOPSS presents a reduced loss of performance in comparison to its conventional counterpart, for the set of allowed operating conditions. Furthermore, the improved robustness of the FOPSS has been obtained using a similar control effort to the one provided by using a conventional PSS. The obtained experimental results for the power system studied in this paper shows that the AVR modes (frequency around 16 rad/s) have slightly greater damping than the damping values obtained for the same mode, in comparison to the designed conventional PSS. However, further additional theoretical studies must be performed to affirm that this behavior holds in general case. A possible FOPSS drawback is its more demanding computational implementation which may demand high order digital filters to obtain a suitable approximation of the fractional order dynamics. An advantage of fractional order approach is its ability to provide improved performance due to its additional tuning parameters. However, to the best of the author's knowledge, this is a challenging problem from a complex analysis domain to which a formal solution remains to be provided. According to Reference [14], thus far, such statement has only been evidenced using practical tests in rig systems. The experimental results presented in the present paper seem to empirically corroborate this statement, as it can be observed that, for the studied power system, the FOPSS provided presented a robust performance with control effort not higher than the control provided by the conventional PSS.