Adequacy of Lyapunov Control of Power Systems Considering Modelling Details and Control Indices

Nonlinear excitation controllers for power systems are considered as promising replacements for their linear counterparts, such as power system stabilizers (PSSs). However, Lyapunov based nonlinear controllers currently available in literature have been derived using simplified third order machine dynamics and have not been tested on realistic power system models, and the costs associated with these controllers have also not been studied. The paper aims to assess the performance of existing Lyapunov controllers for a detailed benchmark power system model. Two new Lyapunov based nonlinear controllers have been derived using IEEE recommended subtransient machine model, and their performance and costs have been compared with existing methods. The controllers have been evaluated using metrics for both small signal stability and transient stability. It has been proven theoretically (via stability analysis) and demonstrated via simulations that proposed Lyapunov controllers are more viable options for control of power system oscillatory dynamics. It has also been demonstrated that detailed subtransient model is required for accurate estimation of states via dynamic state estimation for use in the realization of Lyapunov controllers.

Defined as ( Defined as (x q − x l )/(x q − x l ). K q2 Defined as (x q − x q )/(x q − x l ). V t Generator terminal voltage, in p.u. ps p Controller states (including washout filter). ( . ) ( .) − (.) 0 ; State variable transformation. x, μ State and input vectors respectively. y, n x Measurement vector, State vector dimension. R a Armature resistance, in p.u.; K S PSS gain. K A , n μ ST1A gain, Control vector dimension.

V, I
Bus voltages and bus current injection vectors resp. N Number of nodes in the power system network. T CH , T SV Time constants of the turbine and governor resp. P CH , P SV Steam chest power output and valve position resp. P C , R D Governor set point and regulation constant resp. PSS Power system stabilizer; CCT Critical clearing time. AVR Automatic voltage regulator.

I. INTRODUCTION
E XCITATION Control of multi-machine power systems has been of interest to researchers and engineers as it directly decides the system stability. The design of an excitation controller has the following objectives [1]: a) to be optimally adapted to the system mathematical model, b) to be decentralized so that control decisions rely only on local measurements, c) to be immune to changes in system set points and operating conditions, d) not be affected by large disturbances, and e) robust to measurement sensor anomalies. Commonly used auxiliary linear controllers, such as PSSs, are not perfect solutions to these challenges. A multi-machine power system is non-linear in nature and any disturbances are likely to significantly change the operating point of the system. For linear controllers based on linear control theories, linearization of the system is required in the neighborhood of a pre-determined system equilibrium or a set of equilibria. Such controllers are only valid in such a neighborhood and any significant change in the system may cause stability problems. Additionally, linear state feedback controllers may be sluggish as they require linearization of the system dynamics and their optimality depends on the adequacy of the system dynamic model [1]- [14]. Recently, with the systematic development of nonlinear control theory, nonlinear controllers have been proposed as a better solution.
Various nonlinear controllers proposed in power system literature may be broadly divided into two categories: methods based on normal forms [1]- [5] and methods based on Lyapunov functions [6]- [10]. Normal form-based methods involve transforming the system using a curvilinear coordinate system, while Lyapunov based methods involve finding a positive nonincreasing energy-like function of the system states. Although significant development has been made in control design using both the techniques, there are still a few drawbacks of the existing methods. Firstly, most non-linear controllers are derived from a third-order model of a synchronous machine. Such model is a simplified representation of generator dynamics and cannot characterize subtransient dynamics of a machine. A controller derived from such a model may have unexpected effect on a power system and may even affect local stability at the machine [11], [12], [13]. To the best of the authors' knowledge, only one paper has shown the importance and necessity of using detailed models for deriving nonlinear controllers based on normal forms [14], while a corresponding study on Lyapunov control is not available in power system literature. The Lyapunov controllers available in literature have only been implemented and simulated on a simple network modelled using third-order machine models. This makes it unclear if these controllers are capable of being applied to realistic power systems. Also, a comparative study of existing Lyapunov controllers is not available in literature. Similarly, the performance of such controllers considering costs of control and instrumentation chain anomalies has not been studied. Finally, it is assumed that states required for realization of these controllers [1]- [10] are directly available as measurements which may be a gross approximation.
This paper aims to address the aforementioned drawbacks by finding answers to the following questions: 1) Are Lyapunov controllers based on third-order machine model compatible with (or work with) a realistic network model with detailed machine models? 2) How do different nonlinear controllers perform compared to each other?
3) What is the impact of model order on the accurate estimation of machine states for use in the realization of these controllers? 4) Does the performance of Lyapunov control improve if controllers are derived from sixth-order model? 5) Are the controllers (derived from third or sixth-order machine model) realizable in practice considering costs of control and instrumentation chain anomalies? And is computation an issue in realizing these Lyapunov controllers derived from third or sixth order models? In this paper, two representative Lyapunov controllers have been tested using the 68-bus benchmark power system [15]. The proposed controllers are based on the subtransient machine dynamics and focus on both small signal and transient stability. Specifically, the following metrics have been evaluated and compared for all the tested controllers: modal analysis, an index for transient stability (integrated variation of per-unit rotor energy), critical clearing time (CCT), control cost and voltage regulation. The principle contributions/advantages of this work are as follows: 1) Two new decentralized nonlinear excitation controllers based on detailed IEEE 2.2 model have been proposed for control of oscillatory dynamics and overall stability improvement. One controller is based on feedback linearization (namely LYPF1) while the second (namely LYPF2) is based on nonlinear optimal control. Compared to all other Lyapunov controllers LYPF2 uses reduced control effort while increasing the stability margins. Additionally, it does not contain any derivative term in the control law unlike other Lyapunov controllers.
2) The compatibility, performance, and control efforts of the Lyapunov controllers based on simplistic and detailed machine models has been thoroughly studied and compared, to answer the above questions. 3) Estimated states via dynamic state estimation (DSE) have been used in realizing all the controllers to eliminate the gross approximation about the availability of machine states as direct measurements. It has also been established that DSE based on detailed machine models is necessary to accurately estimate the machine states for use in the Lyapunov controllers based on any model order. 4) The asymptotic stability of the system with the derived controllers has been proven analytically. Theoretical findings are complimented by performing eigen-value analysis of a test system with the derived controllers. 5) The feasibility of the application of the Lyapunov controllers including proposed ones (LYPF1 and LYPF2) has also been discussed by comparing their computational requirements and real-time tests. The rest of the paper is organized as follows: Section II reviews synchronous machine models and Section III briefly introduces the controllers being studied. Section IV presents the concise derivation of the two new controllers. Section V presents the stability analysis of the proposed controllers. Section VI introduces the detailed case studies and presents the comparison results and inferences followed by conclusions in Section VII.

II. POWER SYSTEM MODEL
The dynamics of a conventional power system are dominated by generator behavior. Depending upon the application, a synchronous generator can be represented by mathematical models at different complexities with certain assumptions and modelling approximations. The model complexity of a synchronous machine is reflected by the numbers of damper coils on its rotor [16], [17]. The sixth-order model is the recommended model as per IEEE [11], with a field winding and a damper coil on d-axis and two damper coils on q-axis. It is known as sub-transient model, or Model 2.2, described using set of equations given in Appendix-A [18]- [20].

III. LYAPUNOV CONTROLLERS BASED ON SIMPLISTIC MODELS
Unlike normal form-based controllers, Lyapunov function based controllers do not have a uniform derivation methodology. It varies with different choices of Lyapunov function. It is required that any Lyapunov function must be radially unbounded and positive definite while the time-derivative of the function along system trajectory is required to be negative definite.
The two Lyapunov controllers based on simplistic third-order machine model, as studied and compared in this paper, are obtained from [6], [7], with the first controller given by: and T d0i C 2i = V −1 ti V qi . The second controller given by: , where 5, 5, 5, 1 and 156 are chosen respectively for q 1i , q 2i , q 3i , ρ 1 and ρ 2 . θ i and β i meets adaptive law: The controllers (1) and (2) are referred to as LYP1 & LYP2, respectively.
Remark 1: In MATLAB implementation, use of derivative block should be avoided in Simulink as it is treated as a zero gain when linearized [21]. Instead, it can be replaced by an approximate lead compensator transfer function Ns/(Ns + 1) (N is a large positive number).

A. Feedback Linearization Based Lyapunov Control
Feedback linearization involves transformation of a detailed nonlinear model into a linear equivalent paradigm through a change of variables while exactly preserving the system nonlinearity in the transformed model [22]- [27]. To study the effect of machine model on control design, a Lyapunov function based controller has been derived from a sixth-order machine model. The derivation is similar to the derivation of LYP1 in [6]. For the i th machine, if the deviations in generated power, rotor speed and terminal voltage from their respective nominal values are selected as output, then: The Lyapunov function is chosen to be: As V i is a sum of quadratic terms, it can be easily verified that it is positive definite and radially unbounded. The timederivative of V i (y i ) along the power system can be simply written as: In order to achieve global asymptotic stability, the trajectory of derivativeẏ i is constructed as a linear control system written as [6]:ẏ where The virtual input v i equals the time-derivative of terminal voltage. The next step is to calculate the time-derivative of terminal voltage V ti , which satisfies (7).
As V di and V qi are algebraic functions of time, for convenience, they are denoted as b(t) and c(t) and V 2 ti is denoted as a(t). The derivative is: Substituting (A9.1) and (A9.2) from Appendix-A to replace b(t) and c(t).
By substituting the virtual input and the constructed trajectory of derivative, the expression ofV i is: where It is required thatV i must be negative definite in order to achieve global asymptotic stability. The gains vector [ k 1i k 2i k 3i ] = [ 17 −18 65 ] (obtained via pole placement technique [24]) ensuresV i is negative definite. Hence, the final control expression is: with S 1i and S 2i as expressed as earlier. This Controller (11) is referred to as LYPF1.

B. Nonlinear State-Feedback Optimal Control
The Affine representation of the subtransient dynamics (A1)-(A10) is given by (12) after transforming the state and control variables asx The voltage deviationṼ t in (34) acts as an 'exogenous input' to a nonlinear system (34).
Remark 2: An 'exogenous input' to a system is an input whose magnitude cannot be altered by design. If an exogenous input is measurable, it is possible to cancel out or minimize its effect by incorporating it in a control law.
where,μ i =μ i +μ i ·μ i corresponds to feedback control law whereasμ i design incorporatesṼ t in the control law to stabilize this deviation and thereby mitigates its impact on the system dynamics by minimizing H 1 (quadratic cost function given by (14)).
The optimal control value function (Lyapunov function candidate) [22], [23] (17) used to derive the feedback control law μ i for the dynamical system (16) after cancelling the effect of V ti viaμ i design is defined as Differentiating (17), The optimal control signalμ i is derived from (18) by using the Pontryagin's minimum principle based stationary conditions [23]. These stationary conditions are given by (19) whereas the corresponding control law derived from it is given by (20). This formulation is a nonlinear equivalent of a linear quadratic regulator (LQR).
, a i is the i th element of the state vectorx i and γ is the tunable gain vector (Appendix-B) [23]. The controller effectiveness gets enhanced by including the terminal voltage deviation in the feedback law (15), (23) as an exogenous/pseudo control input (unlike [25], [26], wherein voltage and frequency are used) and it enhances the system dynamic performance. Therefore, V ssi is nonzero only during dynamic conditions. The DC component in V ssi is filtered out by a washout filter [15]. The controller of (25) is referred to as LYPF2 (Fig. 1).

V. SYSTEM STABILITY WITH PROPOSED CONTROLLER
The proof of asymptotic stability for the LYPF1 follows similar procedure as given in Section-IV of [14], and is omitted here. In the case of LYPF2 controller the following lemma is defined.
Theorem 1: The control law (23) guarantees the asymptotic stability of the decentralized dynamics (12) and thereby guarantees the asymptotic stability of the overall power system irrespective of the operating point.
The affine representation of the system dynamics of all machines of the power system may be combined using (12). Therefore, ⎡ In compact form the above equation (29) can be represented as follows.Ẋ Where, The composite representation of the overall system dynamics (30) resembles (12). Therefore, the asymptotic stability of the system (30) is proved with Theorem 1 while using the same sequential steps as used for the decentralized dynamics (12). It is further validated by performing the eigenvalue analysis of the overall system (detailed in Section VI.)

A. System Description and Controller Realization
The proposed controllers were tested on a 16-machine, 68 bus IEEE benchmark test system, as shown in Fig. 2. In the study, it is assumed that all machines have the controllers enabled as shown in Fig. 3. Low frequency oscillations in power systems are global as they involve generators, loads, and substantial portion of the power network. Each synchronous machine contributes to these low frequency oscillations in varying degrees [15]. Therefore, each machine is equipped with a controller to dampen out these oscillations [28], [29]. All these controllers need the interior machines states as inputs to derive the control effort. These states are estimated locally from terminal bus phasor measurements using a dynamic state estimator (DSE) [27]- [31] as shown by the schematic in Fig. 3. [15] presents a study in small signal stability when the system is controlled by PSS-AVR (see Appendix-B for PSS equations). MATLAB-Simulink platform has been

B. Dynamic State Estimation and Model Order
Implementation/realization of control laws (1), (2), (11) and (25) requires accurate estimation of machine states as they are not measurable directly. Generator bus PMU measurements serve as inputs for the dynamic state estimator (Fig. 3). The procedure to estimate the synchronous machine states from generator bus PMU measurements uses a decentralized methodology [31]. In the DSE methodology, generator bus voltage phasor (both angle and magnitude) are used as pseudo-inputs to make the estimation process decentralized. The unscented Kalman filter based DSE is used to derive the estimates of the states dynamically [31], [32]. If a simplistic model (A1)-(A2), (A11)-(A14) of the synchronous machine is used in the DSE, the estimated states do not follow the actual trajectory of these states due to model inadequacy whereas if the detailed subtransient model (A1)-(A10) is used in the DSE, the estimated states track the actual time trajectory of the machine states. For illustration, the dynamic estimate of the machine speed of the 13 th generation unit of the test system (Fig. 2) considering simplistic and detailed order models is shown in Fig. 4. In this case study, a fault was simulated at bus 54 of the test system at t = 1s and cleared by opening circuit breakers on line 53-54 at t = 1.18s.

C. Base Case and Lyapunov Control
The base case refers to the case of the power system at equilibrium with no external disturbance applied. It is a basic test of compatibility and demonstrates the influence of difference in model complexity. The system is expected to have a new equilibrium if controllers derived using third-order machine model are applied. Fig. 5. shows the graphs of system frequency in time domain when the system is controlled by LYP1, LYP2, LYPF1 and LYPF2. In the case of LYP1 and LYP2 there is a transition to a new steady state value. Such transitions in frequencies occur because of mismatch in assumptions of system model and thus in assumed system equilibrium. Taking E fd as an example, since the power system is modelled with sixth-order machine models, subtransient machine dynamics define the dynamics of the system. Therefore, at equilibrium, the initial excitation voltage E fd for subtransient model is However For LYPF1 and LYPF2, the system remains at pre-calculated equilibrium as expected since there is no mismatch in model complexity.

D. Modal Analysis
Power system oscillations contain multiple frequency components (modes). Each eigenvalue corresponds to a mode. Modes where electromechanical states (i.e., rotor speed and rotor angle) have high participation are electromechanical modes. Among them, poorly damped low frequency oscillations (0.2 ∼ 2Hz) (particularly inter-area oscillations [18], [33]) pose a serious threat for secure and stable operation of an interconnected power systems. Unless well damped, such oscillations hamper the transfer of large amounts of power through transmission lines [11], [12]. With their low frequency, it takes a long time for them to settle down and thus at least 10% damping ratio [15] is desired. The eigen-value plot of the system with and without Lyapunov controllers and PSS is shown in Fig. 6. All the Lyapunov controllers improve the damping ratios of all low frequency modes including inter-area modes. Table I lists the damping ratios and frequencies of inter-area modes if the system is controlled by PSS, LYP1, LYP2, LYPF1 and LYPF2. The corresponding eigenvalue plot of inter-area modes for illustration is shown in Fig. 7.
As shown in Table I and Figs. 6, 7, all controllers can handle the instability (negative damping ratio) in open loop system. However, every nonlinear controller exhibits a much higher damping ratio but a relatively lower modal frequency with respect to inter-area modes. A brief explanation of why this happens is as follows. The studied nonlinear controllers work by changing the excitation of the rotor field windings and thereby modulate the developed electrical torque. These nonlinear controllers modulate the field winding excitation and therefore the synchronous torque produced due to interaction of stator and rotor magnetic fields. In response to a change in the dynamic states of the machine due to a disturbance, the nonlinear controllers respond immediately, and provide fast damping to any electromechanical mode of the system. The modal damping ratio increases, while modal frequency decreases significantly. Modal damping ratios in the case of LYPF2 are lower than other nonlinear controllers but much higher than PSS whereas inter-area mode frequencies remain almost same as PSS i.e., LYPF2 improves the damping Furthermore, a participation factor analysis [17], [18] was also conducted to measure the sensitivity of a mode to a state. In nonlinear systems, the generic second-order expression for modes-in-states participation is given by (31) [12].
where, p 2ik represents the participation of k th mode in i th state and p i 2jl represents the participation of combination of j th and l th modes in i th state. Assuming the system is not extremely stressed and the individual components of the initial vector x 0i = x i (0) values are statistically independent, then the contribution from the second term in the above expression is marginal [12], [13]. For relative ranking of the principal participating states in particular modes, the second term in the generic second order expression of participation factors can be ignored. Therefore, the participating factor p 2ik has been used to determine the ranking of the principal participating states in a particular inter-area mode as shown in Table II. Table II lists the top 5 normalized participation factors (NPFs) and corresponding states in each inter-area mode for all the controllers. It should be noted that all of the Lyapunov controllers -LYP1, LYP2, LYPF1 and LYPF2 exhibit high participation factors from rotor speed and rotor angles (electromechanical states). However, LYP1, LYP2 and LYPF1 also have significant participation factors from controller/non electromechanical states, with LYP2 having highest participation from a controller state. In other words, the three feedback linearization based Lyapunov controllers significantly modify the mode-shapes of interarea modes via high participation of electrical/controller states and large control effort. As any type of Lyapunov control or PSS control that we have studied in this paper is after all a type of excitation control (which controls the excitation voltage of the machine, which then influences the developed torque, which in turn influences the rotor angle), ideally excitation control shouldn't have a direct influence on the inter-area modes from controller/electrical states, and this is true for both PSS and LYPF2. But this is not true for LYP1, LYP2 and LYPF1, where we see direct influence from control/electrical states in the inter-area modes, and which also leads to reduction in their modal frequencies to values which are usually not observed for interarea modes (that is, frequencies less than 0.2 Hz) [16]. It should be noted here that the change in modal properties depend upon control parameters, total control effort, the feedback variables and state participation factors [28]. The overall damping of modal oscillations in the power system is decided not only by the frequencies of the dominant modes, but also by the damping ratios of those modes [17]. Thus, the reduction in frequencies of the inter-area modes in the case of LYP1, LYP2 and LYPF1 has no apparent adverse effect on control performance because this reduction is well compensated by a significant increase in damping ratios. Nevertheless, the associated modification of the inherent nature of the inter-area modes is not desirable as it can have an adverse effect on the stability of the system in the long run.

E. Total Control Cost
The total cost is a sum of the control effort and state deviations [25]. The control effort (CE) for each machine is measured as the square of deviation in control input, which is E fd , from its steady state value. State deviation (SD) for each machine is measured as the sum of square of deviations in the seven states (δ, ω, E d , E q , ψ 1d , ψ 2q , E dc ) from their respective nominal values. In this case study, a 1% step change in input mechanical torque was applied at t = 10s as a disturbance to each machine in the system. The deviations were sampled at 100 Hz and are accumulated for 15 seconds (from t = 10s to t = 25s). A sample plot of control effort with different controllers for machine 1 is shown in Fig. 8   . For a small disturbance, 15 seconds are adequate for the system to settle down to a new steady state. Table III lists the control effort (32) and state deviation (33) for each controller.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. where, , and k − andk are the sampling instants at the start and the end, respectively, of the accumulation of CE and SD.
It may be observed from the above table that all the nonlinear controllers have a higher control cost than the conventional PSS. This is because nonlinear controllers tend to respond vigorously in order to maintain stability when subjected to either small or large disturbances ( [1], [2], [14], [26]). On the other hand, nonlinear controllers behave differently with respect to state deviations. LYP2 exhibits very low state deviations. LYPF2 with AVR leads to a moderate state deviation that is the closest to that of PSS and AVR. LYP1 and LYPF1 have similar basis exhibit the highest state deviations. The cumulative costs of control in the case of LYPF2 with AVR are much lesser than other nonlinear controllers but higher than traditional PSS (Table III). It is important to note that too high control costs (in the case of LYP1, LYP2 and LYPF1) may cause the over-fluxing of generator transformer, which is not desirable.
It was found that such high state deviations result from rotor angle excursions. As LYP1 and LYPF1 do not include rotor angle in feedback, the controller has no control over rotor angle. Thus, any disturbance leads to new steady state rotor angles and the controllers are not able to restore them back to nominal values. This, in turn, significantly increases the accumulated rotor angle deviation over time compared to other controllers.

F. Critical Clearing Time and Transient Stability Measure
Critical clearing time (CCT) refers to the longest time a fault can be tolerated by a system without losing stability. To find CCT, a three-phase symmetric fault was simulated by adding a large admittance to a bus node in system admittance matrix. The fault was simulated at bus 54 and cleared by opening circuit breakers on line 53-54. Faults of different time lengths were simulated until a CCT was found. System instability manifested itself as loss of synchronism, that is, non-zero rotor speed slips and irrevocable separation of machines into two mutually asynchronous groups. Table IV lists CCTs of each controller. Furthermore, slips and terminal voltages of machine 1, 8 and 13 and inter-area power flow (from NETS to NYPS through intertie 60-61) when subjected to a fault described above that lasts 200ms are shown in Figs. 9-11. The three machines are chosen as they are electrically the closest and most remote from fault location, respectively (see Fig. 2).
For better understanding of the above CCT results an index W tk (34) is defined as integrated variation of per-unit total energy exchanged between rotors of the generation units and the system has also been used for the transient stability evaluation (TSE) of the controllers. A lower value of TSE index indicates that the transient swings are smaller and TSE margins are better.
where τ sim = 30s is the observation period, W t is the summation of per-unit energy exchanged and k n is a normalization constant. W t in (34) is calculated by using the rotor slip speeds (with center-of-inertia as frame of reference) of all the individual The corresponding values of W tk are listed in Table IV. The disturbance in this case study lasts for 180ms (which is less than the least CCT amongst all the controllers other than LYP2). For further evaluation, faults were initiated near buses 47 and 60 of the test system (Fig. 2) and the faults were cleared by opening the breakers of the faulted lines. CCTs for these fault scenarios were obtained using the aforementioned procedure and the results have been presented in Table V. With LYP1, LYPF1 and LYPF2 controllers the CCT improves significantly compared to PSS and the results are consistent for any disturbance scenario.

G. Voltage Regulation
It is vital for a power system to maintain terminal generator voltages at their nominal values in post-disturbance condition. In this case study, post-fault-clearing generator terminal voltages were studied for the critically cleared fault at bus 54. The postfault terminal voltage of machine 1, 8 and 13 are shown (Fig. 12). Note that the controllers LYP1 and LYP2 settle down to a new steady state.
The controller is deemed effective if the post-disturbance voltage regulation of the generator buses is very small. The percentage voltage regulation of the i th generator bus VR i is given by (36). The voltage regulation indices; average (VR avg ) and maximum (VR max ) voltage regulations of the generator buses were used to quantify and study the effect of the studied controllers including proposed ones on the voltage regulation under various power disturbance scenarios. Three different disturbance scenarios were used to assess the impact of controllers on the voltage regulation. In the first disturbance scenario, a fault was simulated at bus 54 and cleared by opening circuit breakers on line 53-54. In the second disturbance scenario, a fault was simulated at bus 27 and cleared by opening circuit breakers on line 53-27. In the third disturbance scenario, the loads at bus 61 and bus 47 were increased by 10% and 7% respectively. The corresponding regulation indices are given in Table VI. As inferred from the tabulated results, controllers LYP1, LYPF1 and LYPF2 have good voltage regulation whereas with LYP2, the voltage regulation is relatively poor. Similar results were obtained by simulating multiple disturbance scenarios.  where, V ti0 and V ti∞ are respectively the pre-disturbance and post-disturbance bus voltages of the i th generator.
Controllers LYP1, LYPF1 and LYPF2 have good voltage regulation (as illustrated in Fig. 12, Table VI), owing to the terminal voltage term in their feedback loops. However, LYP2 does not control the terminal voltage on its own. LYP2 requires switching to a voltage controller using a so-called 'membership function' in post-disturbance condition [7]. Fig. 11 also corroborates the above findings/inferences.

H. Implementation and Measurement Noise
A general schematic of the implementation of these nonlinear controllers is shown in Fig. 3. Realization of these nonlinear controllers in practice (field) requires a dynamic state estimator (DSE) [27], [31] to derive the unmeasurable states from the terminal measurements of the generation unit. A noisy measurement acts as an impediment in the realization of any control law particularly when derivative terms feature in it. As derivative terms feature in the control law of the LYP1 and LYPF1, the presence of the DSE for realization of these nonlinear controllers brings additional benefits as a noise filter. Assuming a 1% total vector error in the measurements as per IEEE standards [34], [35], the performance of all these controllers and PSS was evaluated as illustrated in Fig. 13. Since the DSE was used as the input stage of these controllers their performance remains almost unchanged. The performance of the PSS, however, deteriorates in the case of a noisy measurements, probably due to the noise accentuation by the washout filter.
All the aforementioned test cases were simulated considering 0.1% total vector error (TVE) in the PMU measurements which is less than 1% TVE as recommended by IEEE standard C37.118.1-2011 for PMUs [34]. For comparison, results have also been obtained considering 0.5%, 1% and 2% TVEs in the PMU measurements. The performance of the controllers does not change even when TVE is as high as 1% TVE, as the DSE estimates are accurate for measurement noises less than or equal to 1% TVE. However, the control performance is affected by inaccurate estimates which happens when the noise/bad-data (TVE) in the acquired phasor measurements is 2%. With 2% TVE the estimated state trajectories are less accurate as shown in Table VII in which the corresponding root mean square errors in estimated states of the 13 th generation unit of the test system are tabulated. In the case of LYP1, LYP2 and LYPF1, the cost indices increases slightly for TVEs greater than 1%. However, the effect in the case of LYPF2 is too marginal to make any substantial impact on the performance of the controller. The costs shown in Table VIII illustrates the impact of the impact of the less accurate estimates on the performance of the controllers. Due to presence of derivative terms in the control laws, the increase in the cost indices with LYP1, LYP2 and LYPF1 controllers is more than LYPF2 controller. Lastly, bad-data in the measurements can be detected and corrected using the bad-data detection algorithm [15]. That way the impact of the noise/bad-data can be alleviated to ensure that it does not affect the controller performance.

I. Robustness to Varying Operating Conditions
The Lyapunov control laws (1), (2), (11) and (25) get adapted to the varying power system conditions. Therefore, these control designs remain valid for any operating condition. For three operating conditions, a 1% step change in input mechanical torque was applied at t = 10s as a disturbance to each machine in the system. The deviations were sampled at 100 Hz and are accumulated for 15 seconds (from t = 10s to t = 25s). The cost indices were used to assess the performance of Lyapunov control designs for three different operating cases. Case-I is the base case (same case discussed in Section-VI., sub-Section E.). In case-II, the real-power load on bus 61 of the test system (Fig. 2) was increased by 50 MW. In case-III, the real-power load on bus 47 of the test system was increased by 70 MW. As observed from the tabulated data of control indices (Table IX), the controller performance does not change (or the change is quite marginal) upon change in the operating conditions. Therefore the performance of all these controllers does not change with change in operating conditions.

J. Computational Requirements and Feasibility
For nonlinear Lyapunov design, the complete power system simulation (along with the DSEs and the controllers at each generation unit of the test system) was performed in real-time (RT) simulation environment. A 100s simulation takes an average running time of 29.4s, 32.8s, 32.9s, and 38.7s for the LYP1, LYP2, LYPF1 and LYPF2 respectively on a personal computer with i7-4970S-CPU, 2 GHz Intel-processor and 8 GB RAM. Additionally floating-point operations per second (FLOPS) [36] were also computed to assess the feasibility of their real-time application. All these Lyapunov designs use less than hundred thousand FLOPS. Therefore, their computational requirements can be met by an ordinary DSP processor with processing speed in MHz. Moreover, all these controllers are decentralized, rely on local measurements and are therefore not affected by the scalability of the system/application. OP5600 Opal-RT multiprocessors were also used to execute the test case scenario in real-time Fig. 14. In this test case scenario, a fault is simulated at bus 54 of the IEEE 68 bus test system for 200ms and the fault is cleared by opening circuit breakers on line 53-54. QNX platform based Opal-RT multiprocessor was used to emulate the power system and network dynamics and therefore worked as a real-time station and a Linux Redhat OP5600 core implemented the proposed LYPF2 algorithm. The host command station with RT-lab (personal computer with i7-4970S CPU, 2GHz processor and 8GB RAM), real-time station (RTS) and Redhat core2 (implementing LYPF2 scheme) interact between themselves through 100Mbit/s ethernet connection. Built-in DAC/ADC modules act as interface between the two Opal-RT cores and it was virtually impossible for the LYPF2 controller to distinguish between the emulated and the actual plant. Tektronix-MDO4054B-3 oscilloscope was used to capture the RT results (Fig. 14).

K. Potential for Field Application
The case studies presented in the paper establish the theoretical applicability of these Lyapunov based control methods. However, their dependencies must be considered for their field applicability. All these controllers are a direct application of dynamic state estimator (DSE), and therefore their practical realization also depends on how swiftly DSE is adopted by the industry. Since DSE is a fairly new technology, it requires reasonable time to be adopted by the power system industry. DSE has been widely researched in the field of power systems, and shows very promising potential for industry application. ABB Research Ltd. has filed a patent on "parallel computation of DSE" implying the adoption of DSE technology by the industry [37]. Similar patents on DSE technology were filed recently [38], [39]. The practical impediments and associated drawbacks of DSE have been studied /addressed in the recent power system literature [29], [40]. Therefore, with a high potential for DSE technology, the Lyapunov control for power systems discussed in the paper has a potential to be adopted for oscillatory control of future power systems.

L. Discussions
For the various case studies presented, the five questions that the research aims to answer may be revisited. Firstly, compatibility problem between the models used to design and test the controller does exist since even though LYP1 and LYP2 do not fail, they lead to a new system equilibrium, which may not be desired. Accordingly, one obvious advantage of controllers derived from sixth-order model (LYPF1 and LYPF2) over that derived from third-order model (LYP1 and LYP2) is that the controllers derived from sixth-order model are compatible with a detailed power system model, and hence, can be more realistically used for power system control.
Secondly, the nonlinear controllers have two advantages over PSS in the case study. All nonlinear controllers have a much higher damping ratio in inter-area modes than PSS and three nonlinear controllers (LYP1, LYPF1 and LYPF2) have much longer CCT for the faults simulated as compared to PSS. On comparing normal form-based control with Lyapunov based control, it can be observed that Lyapunov based controllers have better control performance as they have higher CCTs and better voltage regulation (please see [14] to compare the CCT and voltage regulation of normal form-based control with those presented in this paper).
Thirdly, the realization of all these Lyapunov controllers requires accurate estimation of machine states via dynamic state estimator. It has been established that detailed synchronous model is required to estimate these states accurately as estimates based on simplistic model are not accurate. Therefore, a detailed synchronous model is needed to perform DSE for realization of any Lyapunov controller.
Fourthly, without filtered measurements the performance (of all the controllers) is bound to deteriorate particularly in the case of controllers with derivative terms. However, the effect is alleviated by the DSE filter at the input of these nonlinear feedback controllers. It should be noted that amongst all the controllers, LYPF2 is least affected as it does not contain a derivative term in its feedback law. Moreover, the computation times for the Lyapunov controllers derived from sixth order model (LYPF1 and LYPF2) are marginally higher than their counterparts (LYP1 and LYP2) derived from third order model. However, all these controllers use less than hundred thousand FLOPS. Therefore, computation time is not an issue as their computational requirements can be met by a megahertz (MHz) processor.
Finally, all the three Lyapunov controllers based on feedback linearization and adaptive backstepping (LYP1, LYP2 and LYPF1) have drastic impact on the position of interarea modes because of which the frequencies of these modes are so much decreased that these modes are difficult to be classified as interarea modes. Moreover, the total costs of control for the three controllers are also very high (as can be seen in Table III). This means that all the three controllers increase the synchronizing torque by a large extent (as explained in Section VI. D.) which can cause over-fluxing issues in practice. The proposed LYPF2 controller, on the other hand, has modal frequencies similar to that of the linear PSS controller and its total costs of control are of the same order as PSS, and at the same time LYPF2 can provide much higher damping to interarea modes and its transient stability indices (like CCT and W tk (Table IV)) are much better than PSS. Thus, the proposed nonlinear controller provides the best features of both linear and nonlinear control and can be used safely without changing the inherent nature of a power system. This is an advantage which may prove to be critical for field implementation of Lyapunov control (and nonlinear control in general) in power systems. It should be noted that the adoption of the Lyapunov methodologies for oscillatory control in power systems depends on how swiftly DSE is adopted by the power system industry.

VII. CONCLUSION
In this paper, the control performance of various Lyapunov based nonlinear controllers was studied for the benchmark NETS-NYPS test system. It was found that nonlinear controllers derived from third-order machine models are not compatible with detailed power system models and, hence, nonlinear controllers should be derived from a detailed subtransient model of a machine. It was also found that nonlinear controllers can have an advantage in modal behavior, CCT, voltage regulation and robustness to noise (due to DSE) over a PSS. Also, Lyapunov based controllers show better performance as compared to normal form [14] based methods. The drawback of existing Lyapunov controllers has also been identified in terms of severe impact on interarea modes, very high costs of control and unrealistically high synchronizing torques. A new derivative free Lyapunov controller (LYPF2) has been proposed to address these drawbacks and serve as practical solution for field implementation of nonlinear control.
APPENDIX-A IEEE 2.2 Model with Turbine Governor Dynamics [41]: where V di = −V ti cos(δ i − θ i ), V qi = V ti sin(δ i − θ i ) and A9) However, R ai usually is much smaller compared to x qi and x di , and the above equation gets simplified as follows.
With (A8), (A9.1) and (A9.2), torque T ei can be calculated as: The third-order machine model is a simplified model where subtransient dynamics are neglected by removing the damper windings. (A1)-(A2) still hold for this model. The remaining equations for the model are as follows [19].
Furthermore, automatic voltage regulators (AVRs) through excitation control regulate the generator terminal voltage at its nominal value. In this paper, it is assumed that all AVRs are assumed to be the standard static exciters (ST1A), with equations given as follows [15].
. . , Y GN ), Y Gj = (R ai + jx d ) −1 when i th generator is connected to j th network node, otherwise Y Gj = 0. Y N is the network shunt admittance whereas Y L is load admittance matrix [15], [16].

Tuning of gain vector γ
Arbitrary choice of controller gain may not produce the desired results. Therefore, the gain vector γ is tuned using a meticulously justified gradient based tuning method (detailed in the chapter 7 of [23]) given by (B.1) below.
, α 1 = 7.6, α 2 = 1, Q (x i ) =x T i I n xx i , R = I n µ . I n x and I n µ are the identity matrices of orders n x and n μ respectively. E dci dynamics is absent in 68 bus NETS-NYPS power system (Fig. 2) as all the generation units are turbo-generators (i.e., x d = x q ). Therefore, the state vector dimension reduces to 7. For 9 th generation unit the value of the gain vector γ in steady state is given below:  [15], [33], [28]- [40] T wiṗs1i = p s1i ,