The following cases are considered. This is to show the initial conditions in the pre-disturbed conditions. The results are shown in Fig. Therefore, the generator initially decelerates and its speed as well as its frequency is reduced. It is clear from the transient power-angle characteristics eq. Therefore, as expected, the disturbance initially causes reduction in the power angle due to the input power reduction. The electrical power is also reduced. The damping power - which is mathematically a scaling of the speed deviation by the damping coefficient see eq.

In addition, the electrical active power and the power angle are in-phase. The original value of the damping coefficient of the system is assumed to be 0. The new values are 0. The value of the damping coefficient associated with each curve is shown in the figures. Time s Fig. The higher the damping coefficient, the better the response lower amplitudes and settling time. It is also shown that the frequency of the oscillations is insensitive to the damping coefficient.

The case of zero damping is associated with sustained oscillations of constant amplitude critical or marginal stability. In addition, positive damping is associated with some types of loads. In comparison with the data of case 1, in this case only the inertia constant of the generator is changed. In this case, two additional values are considered which are 5 sec, 20 sec, and sec.

The simulation results are shown in Fig. This is attributed to the increase in the K. E stored in the rotating system with the increase of the inertia constant. This means that the energy needed to alter the speed of the generator. Consequently, the amplitude decreased with the increase in the inertia while restoration of the normal K. E takes more time, which results in increasing the settling time and reducing the frequency of the speed oscillations. E energy with increase in the inertia, it is clear that the increase of the inertia increases the amplitudes of the angle oscillations.

In addition, the settling time is increased and the frequency of the angle oscillations is reduced. Same comments can be given to the electrical power due to its tight relation to the power angle. This is of great importance in keeping the system synchronized. Therefore, with higher inertia, a disturbance is unlikely to move the system to the extremes state in comparison with a system with lower inertia. This is because the damping power is directly proportional to the speed deviations. Therefore, increasing the damping ratio of low inertia generators can be a good measure for enhancing its stability.

Mandour, M. EL-Shimy, F. Bendary, and W. Bendary, W. The grid coupling strength is inversely proportional to the value of the network connecting the generator to the infinite bus. In the considered simplified model, the equivalent reactance network is the external reactance xe. If xe is small, then strong coupling is achieved while high value of xe is referred to weak coupling. With the SMIB system data as given in case 1, only the external reactance is given some new values. These considered values are 0. Consequently, the values of the initial power angle, the transient emf, and the synchronizing power are changed.

These values are shown in Fig. This figure shows that increasing the electrical power is associated with an increase in the power angle and a decrease in the synchronizing power. In addition, Table 4. Therefore, the synchronizing power decrease with either an increase in the generator loading or decrease in the network coupling strength or both of them. Actually, Fig. As previously stated and also shown in Fig.

The phrase better refers to lower the amplitude of oscillations, and settling time. For example, in Fig. This required increase in the emfs mandates an increase in the design rating of the generator excitation system. In addition, for a given design, increasing the emf reduces the active power production capability. This is because the active and reactive power of a generator are dependent quantities and related to the generator MVA rating by. From a practical point of view, the reactive power needed for compensating long lines is not only provided by the centralized generators.

Instead, distributed reactive power compensators are installed along the long lines for reactive power support, voltage control, and voltage stability enhancement. The transient reactance equals to 0. The p. In this example, it is assumed that the line outage disturbance is small enough for the linearized model to be valid. Therefore, before and after the outage of the line, the transient emf will not be changed; however, the transient power-angle characteristics will be changed as shown in Fig.

In that figure, the curve number I refers to the pre-outage characteristics while the curve number II refer to the post-outage characteristics. Since, the outage of the line results in an increase in the transfer reactance xe between the generator and the infinite bus, the maximum power of curve II is low in comparison with curve I. This shows that the line outage can be simulated by an equivalent sudden change in the mechanical power.

Numerical values and calculation methodology are shown below. Based on Fig. The operating point corresponding to point b cannot be stable due to the mismatch between mechanical and electrical power. The initial change in generator input power. Therefore, the considered disturbance can be simulated in the same way as the previous cases. Therefore, the final steady state angle is expected to be high in comparison with the initial steady state angle.

The response verifies this issue. In addition, the sudden increase in xe causes sudden decrease in the electrical loading on the generator. Consequently, it is expected that the frequency will increase just after the line outage. In this case, the mechanical power input remains unchanged during the transient process. The disturbance in this case can be simulated by an equivalent sudden change in the power angle. When the line is outage disconnected, the transient power curve changed to curve II and the operating point moves to point b on curve II. It is assumed that the line will return to service after a very short duration.

Now, the system characteristic return to curve I and at the instant of line reconnection, the operating point moves to point d on the curve I. Due to the power mismatch, a transient process acts and afterward the system operating point settles at the initial operating point. The overall changes in the system due to the disturbance are null; however, the difference acts as a disturbance which is equivalent to the momentary outage of the line. In this case, the data of the system of case 5 is considered such that one of the lines is temporary disconnected and the equivalent sudden angle change equals to 10 degrees 0.

The original simulink model is modified as shown in Fig. Since the transient process will mainly act on the curve. The response of the system for momentary disturbances is called zero-input response while the response for sustained disturbances is called forced response. By the end of this numerical example, it is worthy to be noted that the conclusions derived from the analysis of the presented simple system and simplified models are still valid for larger system and more complex models.

This is will be demonstrated later in this textbook. The equation presents a linearized standard second order differential equation and its solutions will be presented in the following part. The solutions corresponding to various zero-input and forced responses will be considered. From the block diagram of Fig. Referring to the state equation 4.

It is known that the response depends on the roots of the characteristic equation. Three different cases can be realized as shown in Fig. The general solution in this case is. In this case, as shown in eq. This situation, of course, is corresponding to an unstable response. This situation, of course, is corresponding to a stable response. This situation is corresponding to an unstable response. The situations described in case 1 are also valid in this case.

The roots of the characteristic equation take the form:. In this case, the TD response is oscillatory. And the response time constant is given by:. The frequency of the oscillations takes the form. The stability of the system in this case is governed by the signs of the real parts of the eigenvalues according to the following situations. The time domain response of this situation is harmonic oscillations with exponentially rising amplitude i.

The TD response of this situation is stable damped harmonic oscillations as shown in Fig. The system in this case is called critically stable or in bifurcation. Assume the per unit damping power coefficient is 0. It is required to obtain and plot the functions describing the motion of the rotor i. Hence, oscillatory solution governed by 4. The plots of the solutions take the form shown in Fig. This solution is in confirmation with the TD solution obtained with the Simulink.

## Dynamic equivalent method of interconnected power systems with consideration of motor loads

Analyze the results. Expressions describing the TD response of the SMIB as subjected to sudden changes in the mechanical power inputs can also be derived. The system response in this case is a forced response in which the disturbance is sustained. The linearized equation of motion of the system can be rewritten as:. The characteristic equation of the system is given by 4. This equation can be written as and the output takes the form. The s-space of this model can be obtained using the Laplace transform such that.

This is shown in eq. The reader may use the following data for applying the time domain solutions. Assume a per unit damping power coefficient is 0. While the system is operating under steady state condition, the breakers of one of the lines are suddenly open and remain open.

Obtain and plot the functions describing the motion of the rotor i. Solution hints. The disturbance can be simulated by as a step change in the mechanical power input. The pre-disturbance transient power-angle characteristics take the form. The post-disturbance approximated transient power-angle characteristics take the form,. Since the system dynamics occurs along the post-disturbance power angle characteristics.

Then, the synchronizing power is calculated as:. The responses are then shown in Fig. In addition, it is shown that the shape of the response can be easily predicted from the eigenvalues of the system. Although the considered system is just SMIB and the adopted model is very simple, the derived conclusions are general regardless of the system size and the model complexity. Generally, the roots of the characteristic equation of a power system are the eigenvalues of the state matrix A. The time response of a dynamic system is dependent on the values of the eigenvalues.

The eigenvalues of a state matrix A can also be found by solving:. In fact, expansion of the determinant of 4. The stability of the system is determined by the eigenvalues as follows:. A real eigenvalue corresponds to a non-oscillatory mode, such that: a. A negative real eigenvalue represents a decaying mode.

The larger its magnitude ,the faster the decay. A positive eigenvalue represents aperiodic instability. Complex eigenvalues occur in conjugate pairs, and each pair corresponds to an oscillatory mode. The real component of the eigenvalues gives the damping, and the imaginary component gives the frequency of oscillation. Such that: a. A negative real part represents a damped oscillation.

A positive real part represents oscillation of increased amplitude. Stable if the real part of all eigenvalues are negative i. Unstable if the real part of at least one of the eigenvalues is positive i. Bifurcation o if the real part of at least one of the eigenvalues is zero i. Therefore, the d-axis flux density is assumed to be constant during transient. Note that the transient emf is proportional to the direct axis flux linkages. Consequently, the field circuit electromagnetic dynamics are neglected. The linearized classical model can be considered as an elementary model for the analysis of the damping and synchronizing power of synchronous generators.

Higher order models of the synchronous machines can simulate the field circuit dynamics14,15,16, The following simplifying assumptions are considered in the following model, 1. The effects of the damper windings amortisseur effects are neglected. These effects include additional damping; however, selecting an appropriate damping coefficient is a possible way of considering the amortisseur effects in a simple way.

This assumption is quite acceptable when the rotor speed deviations are small. The electromagnetic transient of the stator windings are neglected. Concepts of synchronous machine stability as affected by excitation control. IEEE Transactions on power apparatus and systems, 88 4 , Power system oscillations. It can be shown that neglecting these transients is compensated by neglecting the electromagnetic transients associated with the power network. Therefore, modeling the power network components by time-independent parameters is also acceptable. In addition, the phasor diagram can be used to define the stator equations Fig.

The machine saturation is neglected. For generality, a salient-pole machine model is considered. Given the stated assumptions, the machine model becomes a third order model instead of a second order model. The additional differential equation describes the field circuit dynamics. This model is as follows. Note that the generator field voltage Efd equals to 1. The linearized form of this model is defined by the Heffron-Phillips Constants as follows,.

The block diagram representing the linearized model of the SMIB system with field circuit dynamics included is shown in Fig. As shown in the figure, additional control systems can be included. These control systems include the prime mover and the excitation control systems as well as additional stabilization controls. In addition, the model can now simulate the machine terminal voltage.

The equations representing these constant can be found through the linearization of the machine model represented by equations 4. In order to do so, linearization of equations 4. The constant K1 represents the change in electrical torque for a change in rotor angle at constant flux linkage in d-axis i. Equations 4. Therefore, equation 4. The constant K2 represents the change in electrical torque for a change in d-axis flux linkage at constant rotor angle i. It is clear that K3 represents the impedance factor that correlates the steady state and transient reactances while K4 represents the demagnetizing effect of change in rotor angle i.

Using 4. The definitions 4. The results are:. Given that. Therefore, the negative value of K5 and the positive value of K6 are expectable. These variables are determined based on the mathematical models presented in Appendix 1.

## Download Dynamic Security Of Interconnected Electric Power Systems Volume 1

The value of Vo is supposed to be a known quantity as it is the infinite bus voltage magnitude while the other variable can be estimated as follows. Referring to Fig. Generally, the synchronizing power coefficient is defined by:. With the field circuit dynamics neglected i. With the field circuit dynamics considered, by replacing in 4. This reduction is due to the demagnetizing effect of armature reaction. The reduction in Ps is expected to have a negative impact on the settling time and the frequency of oscillations.

Since the values of K1, K2, and K3 are usually positive, then it is possible to say that the due to the demagnetizing effect of armature reaction adds a positive damping DAR. Since then or , then 4. Both coefficients are rotor- speed dependent and they are quadrature to each other as shown in Fig. Example Simulation and analysis of the linearized model of the SMIB system considering field circuit dynamics. The presented linearized third-order model is implemented on the Simulink as shown in Fig. The parameters of the considered system and main variables of the considered system are given as follows such that the reactances, voltages, and power values are represented in p.

The system data are typical values of generators in nuclear power plants The steady state initial conditions and the K-constants as well as some other parameters are consequently determined using the presented model. Their values of the K-constants are,. The main objective of this example is to show the effect armature reaction damping.

Since D is set to zero, the performance of the classical model was —. This is demonstrated by each of the following disturbances:. A sudden change in the power angle by 10 o Fig. All simulations show clearly the damping effect of the field circuit dynamics.

The model can provide many other simulations and analysis that may be implemented by the interested readers. An excitation control system is composed of three main components. These components are the field winding, exciter i. The regulator is nonlinear due to the presence of the Efd limiter. The excitation control system model can be integrated with the 3 rd order linearized model of the machine as shown in Fig. The main objective of this model is to investigate the impact of the excitation control system on the damping of the system. There are many other applications in which the model of Fig.

Measurement-based estimation of the system inertia is an example that will be considered later in this chapter. Since measurement-based equivalence is a target of this book, no more detailed mathematical modeling will be given to the system of Fig. Example Simulation and analysis of the linearized model of the SMIB system with excitation control system. The system of Fig. The combined transfer function of the exciter and the regulator is given the simple form,. With the gain of G s is set to zero, the effect of the excitation control system is deactivated.

In this case, the effect of the effect of the field circuit dynamics is considered as shown in Fig. In this example, it is assumed that the terminal voltage regulator is fast enough for neglecting the time delay effect of its time constant TR. The simulations of the system with each of the mentioned gains are performed. The Simulink model is shown in Fig.

It is clear from the. Reducing the AER gain improves the stability. This is shown in Fig. In both situations, the system is stable; however, the stability suffers from two issues: 1 The steady state error in the terminal voltage is high and its value is increased with the increase in the gain KA. Therefore, the stabilization gain or damping enhancement caused by the reduction of the AER gain is counteracted by the high steady state errors and the long settling time.

The impact of the gain KA can be demonstrated and explained using the root locus of the system which is shown in Fig. It is clear from the figure that the critical value of the gain is Higher values of the gain result in system instability while with lower values the system is stable but its damping is poor. Appendix 2 provides an overview of the extraction of linear transfer from a Simulink model and the linearized analysis of transfer function as well as gain selection based on the root locus analysis. This AER caused instabilities depend on the system parameter, the operating conditions, and the AER structure as well as its parameters.

For example, the responses of Fig. PSSs can also provide a tool for enhancing the system performance. Tuning of PSSs and the control design issues are out of the scope of this chapter; however, stabilization of the system of Fig. Generally, the main objective of PSSs is to artificially produce a damping torque, which in-phase with the rotor speed. This torque is called a supplementary stabilizing signal and the controllers used to generate these signals are known as power system stabilizers. The frequency or acerbating power may be also used as input signals to the PSS.

The transducer measures and converts the input signal to voltage. The lead-lag stages are usually composed of identical a number of phase compensators. The function of the lead-lag stages is to provide an overall phase lead over the frequency range of interest to compensate for the lag produced in the generator-excitation system.

The minimum number of phase compensators is one while the maximum known number is three while the most popular designs contain two compensators. The output signal of the lead-lag stages is then amplified by an amount Ks and sent through a washout stage. The output limiter prevents the stabilizer output signal from interfering with the action of the voltage regulator during severe system disturbances.

With an AER gain of 50, the system of Fig. In this example, PSS is used for system stabilization. The parameters of the PSS in this example are selected based on the experience; however, the parameters should be optimized for the best additional damping, performance enhancement, and stability. A simplified PSS is considered where the transducers and high pass filters are assumed to be fast enough for their transfer functions to be neglected.

In addition, the effect of the PSS output limiter is neglected. One stage lead-lag compensator is considered. The parameters of the PSS are shown in Fig. Although the PSS parameters in this example are roughly estimated i. Better performance can be obtained by PSS optimized tuning; however, this issue is out of the scope of this chapter. Time s a. Time s b Fig. With the speed expressed in p.

The damping power coefficient D relates this power to the p. The transfer function of the linearized swing equation can be represented by the block diagram of Fig. The damping power term is shown by dotted lines as it can be omitted from the block diagram if the machine electromagnetic transients for example, the dynamics of the field circuit and damper windings as well as other dynamics associated with the controls.

The electrical changes in the generator load are instantaneously reflected as changes in the electrical power. The load also responds to the changes in the frequency and voltage changes. Two popular static load models are usually used for load response modeling; the exponential load model and the ZIP load model. The exponential load model can be represented by 4. Kpf and Kqf represents respectively the active and reactive load sensitivity to frequency variations. The relation between these coefficients is,. From frequency sensitivity point of view, equations 4. Given that in p.

For focusing the analysis on the load frequency response, the damping constant D will be neglected in the following analysis. Based on the presented load reaction to frequency variations, the swing equation can then take the form of equation and can be represented by Fig. Based on block reduction theory, the block diagram of Fig. Therefore, the steady state speed is the speed at which the load change is exactly compensated by the frequency-dependent load change.

In addition, Fig. The inverse Laplace can be easily determined by partial fraction and the use of the Laplace transform table23,24 i. As shown, a sudden increase in the load causes steady state reduction in the speed while a sudden reduction in the load causes a steady state increase in the speed or frequency. The torque-speed characteristics for a governor-less generator can be determined based on the torque-power relation,. The integration constant can be obtained from the steady state initial conditions i.

Hence, in p. This relation shows that the torque-speed relation in a governor-less system is linear. In addition, this relation is correct over a limited at rated speed see Fig. As shown, an increase in the frequency results in a reduction in the mechanical torque as the input power is constant. Since power systems do not contain bulk power storage and the load also is a stochastic variable, then the total power generated should be sufficient to provide a power balance at every instant.

The dilemma is that the time constants associated with the mechanical power changes are significantly larger in comparison with most load variations; however, this problem is solved by the proper operation of dispatchable generators as explained in the following. Consider a system with n-dispatchable generators. The daily load forecast is used as a main input data to the system operators. The continuous load forecast curve is transformed into a discrete function with a sampling time in the order of 30 min or 60 min. Then the load level at each time step is used to allocate the power over the generators operating in the PV-mode.

Usually security-based economic dispatch is used to determine the appropriate load sharing between the PV-generator. Up to that point, the power balance requirement cannot be achieved due to the mismatch between the forecasted load curve and the generation as well as other mismatch sources such as forecast errors, power losses, and random events e. Therefore, the remaining generator is set to operate as a slack generator. The slack generator is not actually has a specific load sharing demand. Instead, this generator operates as a power balancer for securing the power balance requirement and consequently keeping the system operating at an acceptable frequency.

The described operation of the system can only be possible with the use of speed governors. Two main types of speed governors are available according to the desired operational function of a generator. PV-generators are required to share the load. Therefore, they are equipped with speed governors with droop characteristics.

A slack generator is required to balance the system. Therefore, it is equipped with an isochronous governor. The ideal steady state speed-power characteristics of these types of governors are shown in Fig. It can be seen from the figure that a droop governor reacts to changes in the frequency by opposite changes in the mechanical power. These two types. In addition, Appendix 4 provides an overview of the fundamentals of speed governors. In load frequency response analysis, the generator model is represented by Fig. Therefore, Fig.

There many designs of steam and hydro turbines. Three popular designs are considered in the following analysis. These designs are non-reheat steam turbines, reheat steam turbines, and hydro-turbines. The non-reheat steam turbine model shown in Table 4. The model shown in Fig. The typical parameters of the turbine transfer functions are shown in Table 4. The results show that with a gain of 0. Higher values of the gain result in oscillatory performance while at a gain of 5. Therefore, proper design of the gain for a desired load-frequency performance is essential for stable and satisfactory dynamic performance of the governor.

The most important characteristics of the isochronous governors are that the steady state errors in the speed or frequency are zero. This is due to the reset action of the integrator, which results in continuous control actions of the governor till the zero error in the speed is achieved.

Time sec Fig. Speed droop governors The function and performance of an isochronous governor are satisfactory when the generator is operating off the grid or for grid balancing function; however, in a grid-connected operation, more than one isochronous governor generator results in problems in the load sharing between the generators and the overall stability of the system.

Unless they have exactly the same settings, if two generators having isochronous governors operate in parallel they will fight each other to set the system to the individual settings of each of them. Since the optimal economic secure operation of generators requires specific optimal settings to be allocated to each generating unit. Therefore, using more than one isochronous generator is practically impossible. Stable load sharing between parallel units requires that a change in a generator frequency is to be compensated by its power output. In other words, an increase in the frequency should be compensated by a reduction in the power output and vice versa.

This action can be achieved by the use of governors having speed-droop or speed-. The ideal steady state characteristics of a speed-droop governor are shown in Fig. The value of R defines the droop of the governor as shown in Fig. The droop defines the steady state change in the frequency due to a steady state change in the output power. Example Load-frequency response of generator with a speed-droop governor. In this example, the considered disturbance and the parameters of system of Fig. The block diagram of Fig. The chosen value of the droop is high for clearly showing the steady state errors.

Two values for the governor gain are selected to show the critical damped and oscillatory damped responses. These values are 0. The results show that with either value of the governor gain, there is a steady state error in the frequency, output power, and valve position. In addition, this steady state error equals to the droop. It can also be easily seen that the settling time and overshoots associated with the generator equipped with a droop governor are much lower in comparison with the generator equipped with an isochronous governor.

The addition of the droop constant to the governor control loop reduces the effect of the integrator and improves the dynamic performance; however, the steady state performance is degraded due to the steady state error. Appendix 3 describes these special characteristics of hydro-turbines and the transient droop compensator. Therefore, speed-droop governors alone cannot restore the power system frequency to the pre- disturbance level. This is can be explained based on Appendix 4 and Fig. It is clear from Fig. The governor will not restore the frequency to the original value; however, this is can be achieved by increasing the reference power Pref settings of the governor by the use of the speed-changer shown in Fig.

Consequently, the droop characteristics move upward as shown in Fig. Therefore, the speed-changer provides a supplementary control to the governor for adjusting its the settings. Une The load sharing between generators can also be provided by the droop governors and the speed-changers.

If two generating units operating in parallel for supplying a load and both of them equipped with isochronous governors with a very slight difference in their speed settings design parameter , then one of the units will try to carry the entire load and the other will shed all of its output. Therefore, neither load sharing nor control of sharing is possible when units operating in parallel have isochronous governors. This is can also be explained by considering two isochronous units coupled together on the same load and the speed settings are not the same..

Since there cannot be two different speeds or frequencies on one system, one unit will have to decrease its actual speed and the other unit will have to increase its actual speed to an average speed between the two units. The governor on the unit that decreased speed will move to increase steam to try to correct for the decrease in speed, and the governor on the other unit that increased speed will move to decrease the steam to try to correct for the increase in speed.

### Dynamics and stability of conventional and renewable energy systems

The result will be that the unit with the higher speed setting will continue to take all of the load until it reaches its power limit, and the other unit will shed all of its power and become motored driven by the other unit. Therefore, the system will become unbalanced when isochronous units coupled together. Consequently, in the power generation system, only one generator can be equipped with an isochronous governor while the rest of generators are to be equipped with droop governors. With droop governors, the load sharing is possible as shown in Fig. The system frequency can be restored by adjusting the speed changers or the reference power settings.

Consequently, the speed- power relation is changed such that the frequency is restored. When a generator is paralleled with a large utility grid, it is important to consider that: 1. The utility will act as an isochronous generator. Consequently, a simple isochronous unit cannot be paralleled to the utility. If the governor speed reference is less than the utility frequency, the utility power will flow to the generator and motor the unit. On the other hand, if the governor speed is even fractionally higher than the frequency of the utility, the governor will go to full load in an attempt to increase the interface bus frequency.

Since the definition of a utility is a frequency which is too strong to influence, the generator will remain at full load. Droop governors provide the solution to this problem. The droop causes the governor speed reference to decrease as load increases. This allows the governor to vary the load with the speed setting since the speed cannot change.

With the speed-changer considered, the block diagram of Fig. Example Load-frequency response of generator with a speed-droop governor and load reference control. In this example, the example of Fig. The Simulink model of the system is shown in Fig. Based on the previous discussion, it is expected that the droop governor is not capable of restoring the system frequency; as shown, there are steady state frequency errors associated with the change in the power input to the generator. The solution to this issue is demonstrated in the next example.

Example Load-frequency response of generator with a speed-droop governor and load reference control — AGC for restoring system frequency. For adjusting the reference power Fig. The value of Kperf is set to 0. The simulink model of the system is shown in Fig. It is clear from the results that the proper load reference control reduces the speed error signals to zero and also enhances the overall dynamic performance of the system.

This chapter presents the fundamentals of power system dynamics. Simplified models and system topology i. The following chapters include further analysis of larger power systems. In addition, the dynamics of recent energy production technologies will also be presented. The system frequency is then the average of the frequency of individual generator in the system. Just after a disturbance, the system frequency dynamics Fig.

In the previous chapter, the linearized modeling and simulation of various components and controllers in the SMIB system Fig. The model of Fig. In that model, the synchronous machine is represented by the third- order linearized model i. The PSS model is represented by Fig. The AGC supplementary control for restoring the system frequency is represented in Fig. The stated models represented the overall structure of the SMIB system with machine controls included. This overall model is shown in Fig. In this section, this model is simulated considering various levels of modeling details.

The main objective is to investigate the impact of the dynamic response of various components in the SMIB system on the frequency dynamics of the system. The changes in the frequency gradient and frequency deviations are the main focus. This is because the values of these quantities will be shown to have a major impact on the measurement-based equivalence that will be presented in this chapter.

A secondary objective of the considered simulations includes a more clear understanding of power system dynamics and their governing issues. In this example, one lead-lag compensator is considered while the transfer functions of the frequency transducer and the high frequency filter are neglected. Therefore, the transfer function of the PSS is shown in Fig. The transfer function of the turbine is stated in Table 4.

The transmission network connecting the generator terminals to the infinite bus is constructed of a step-up transformer and two parallel lines. The considered disturbance is a momentary outage of one of the lines; a situation that can be theoretically represented — as explained in the previous chapter - by a sudden change in the power angle. This change in the angle is assumed to be 5 degrees. The system model is simulated considering the cases listed in Table 5. A single disturbance is used for comparative analysis purposes.

The period of the TD simulations are set to 5. Table 5. The most important conclusion that can be easily depicted from the results is that the characteristics of the initial frequency dynamics are the same regardless of the modeling details as well as the available controllers. In all cases, the initial frequency gradient, deviation, and nadir are the same.

This can be easily explained based on the fundamentals given in the previous chapter. At the beginning of the transient process, the system dynamics are dependent mainly on the system inertia. This is because the changes in the electrical power response are instantaneous Fig. These time delays depend on the parameters of the machine and thecontrollers as well as the initial conditions, and the severity of the disturbance.

The mechanical power transients associated with the prime mover system and their controllers are shown in the previous chapter to be characterized by very large response time in comparison with the electromagnetic transient. Therefore, their effects will not act on the considered TD duration. The basic characteristics of the SMIB system and impact of various controllers are demonstrated in the previous chapter and the cases shown in Fig.

Although the derived conclusions were based on a simple system and linearized models, they are also applicable to larger systems with more complex details as will be shown later. In this section, an efficient simple method for estimating the equivalent system inertia constant at a specific is presented, performed, and evaluated. Based on the swing equation 4. PMUs can measure the AC power frequency i. Stability-based minimization of load shedding in weakly interconnected systems for real-time applications.

Frequency sensitivity and electromechanical propagation simulation study in large power systems. Demonstration of an inertia constant estimation method through simulation. Estimation of WECC system inertia using observed frequency transients. This is due to the relatively slow mechanical system dynamics in comparison with electromechanical and electromagnetic dynamics.

Just after a disturbance, typical inertial and frequency responses of power systems Fig. Therefore, the measurements used for estimating the inertia of the system should be just after a disturbance i. Coherency grouping of generators is also required for the construction of the equivalence; however, the grouping is not necessary to be performed online. As demonstrated in the previous volume chapters 2 and 3 , a coherent group of generators remains coherent regardless of the disturbance type, location, severity; however, large changes in the system topology may cause upset of the coherency grouping.

This issue can be solved by identification of the coherency grouping from the performance of the generators under various disturbed conditions raised from natural changes in the system. With this coherency identification approach, the coherency grouping remains up-to- date without the need of performing a mathematical simulation for that purpose.

Again, the identification step is only needed in situations of large topological changes in the power system. For simplifying the presentation of the approach, the system shown in Fig. The considered interconnected power system is assumed to be composed of two radially connected areas. The configuration of Fig. The buses at which the tie-link is connected are called the interface buses. These buses are required to be equipped with WAM devices for online measurement of the voltage phasors at the interface buses via PMUs, frequency, and tie-line active and reactive power values.

Either area 1 or area 2 may be considered as the study area SA while the other one is considered as the external area EA.

## Dynamic Security of Interconnected Electric Power Systems - Volume 1

In the following modeling and analysis, the equivalency of both areas will be considered. The objective is the minimization of the dynamic order of the overall interconnected systems for dynamic security studies that will be considered in later in this book. It is assumed that the generators of each area are coherent.

Therefore, their electromechanical aggregation is feasible. In this chapter, only conventional sources are considered while various renewable resources will be considered later in this book. The equivalency approach is based on two stages both of them is based on online synchronized measurements of the frequency, voltage phasors and tie- line power flow.

The first stage as shown in Fig. Classical generator models are considered for the equivalent generator. This simplification can be justified based on the application of the equivalent simplified models. These models are to be used for fast prediction of the transient stability of power systems, for example, using the energy functions or the equal area criterion EAC. As shown in the previous chapter as well as this chapter, during the initial stages of a transient process Fig.

This is based on the use of the measurement of active power and frequency measurements on that bus. These measurements are performed under disturbed conditions. The measurements in a very short window after a disturbance are used for. Since the inertia estimation is based solely on measurements, the internal structure of the considered area is not required and the area can be considered as a black-box. Therefore, the method is general and can be applied to any mix of generating technologies as well as large areas.

This kind of flexibility cannot be provided with traditional methods of equivalency which are mainly based on system data sets. For the purpose of the estimation of the inertia at a specific interface bus, equation 5. Based on the standard model of the steady-state and approximate transient characteristics of synchronous generators chapter 4 and Appendix 1 , the following equations can be easily proved for the equivalent generator shown in Fig.

The interface quantities are assumed to be known from measurements through WAMs. Equations 5. Since these equations are nonlinear transcendental equations, then the iterative approach is a good choice for their simultaneous solution. For this purpose, with the index i represents an iteration number, the following equations are derived,. With is selected, equations 5. The initial value of the equivalent generator emf can be set to reasonable values such as 1.

This method is simple and straight forward in comparison to previous method such as that presented by Chow et. With the presented method in this section, the weakly interconnected power system in Fig. Due to the complexity and inherent nonlinearity of power systems, the online dynamic security assessment and control is considered one of the major challenges in recent power systems.

Chapter 1 and 2 in Volume 1 discuss in details the speed and accuracy requirements for handling the stability or the dynamic security of power systems online. One of the most efficient and fast enough approaches for online security studies is the online based equivalence such as that presented in the previous section. With the possibility of reducing a multi-machine system to a single machine equivalence SME , not only the speed of dynamic security assessment is significantly enhanced, but the fast stability analysis methods - such as the Equal Area Criterion EAC - can be implemented on the SME or.

Power system coherency and model reduction. London: Springer. Estimation of radial power system transfer path dynamic parameters using synchronized phasor data. The SME approach is a hybrid direct-temporal method for fast handling of the dynamic security or transient stability. The transient stability of power systems includes two main aspects; analysis and control. The speed of solution, its accuracy, and the nature as well as the strength of the disturbances define the appropriate modeling and solution techniques. For example, the EEA provides a very fast way for assessing the transient stability of the power system; however, in its original form, the size of the system is limited to one or two machines.

Therefore, an interconnected system such as that shown in Fig. High-order nonlinear modeling of individual components forming a power system results in highly accurate results. The solution, in this case, is in the time domain; however, neither the execution time nor the volume of the results to be analyzed is acceptable especially for online applications. Other methods such as linearized analysis methods are significantly faster in comparison with the solution of nonlinear models; however, linearized models are only applicable for simulating small disturbances that result in small deviations in the operating conditions.

Due to its high economical pressure and constraints, the deregulation of power systems causes near security limit operation of power systems. In addition, the deregulation leads to larger interconnections for energy security, and large power transactions for economical profit maximization. Therefore, the time needed for system monitoring and security analysis becomes too small for the current computational technologies and method to handle power. The SMIB equivalence of power system is shown to be a direct approach to enhancing the real-time handling of the stability and security of power systems The derivation of the SMIB equivalence of a power system will be presented and evaluated in the following.

For simplicity, all resistances are neglected in this analysis too. This is assumption is also usual in the EAC based analysis. Based on the equivalent circuit shown in Fig. Therefore, the swing equation at each interface bus and the equation of the relative motion can be represented by,. The equivalent transient reactance of the SME is simply the equivalent reactance of the two machines. Therefore, the SMIB equivalence of the system is presented by eqs. Although, this model abstracts the original system Fig.

For example, eq. Therefore, changes of Pg2 and Pd2 have opposite impact on the value of Peq but its sensitivity to both of them is the same. This is also applicable to the impact of the changes in Pg1 and Pd1 on Peq; however, a drop in Pg1 reduces Peq and a drop in Pd1 increases Peq. As a direct application of the SMIB equivalence in security analysis is the estimation of the maximum sudden drop in the power generation Pg2. In addition, the presented model can be used to determine corrective actions such as the minimum amount of load shedding for ensuring the stability of the system if the drop in the generation is higher than the maximum limit.

A dynamic system can be described mathematically differential or difference equations. This definition indicates that there are three conditions of system stability; stable, unstable, and critically stable. Based on section 4. The shape of the responses depends mainly on the system damping. Regarding to the previous challenges, the stability and protection coordination issues have become interested and must be highlighted.

The frequency control and protection of the electrical systems are the two main sides to investigate the dynamic security of the MG system. There are several studies have dealt this problem from the control side such as conventional controllers with different algorithms and optimization techniques [ 10 , 11 ], intelligent control, i. Tang et al. While, Sedghi and Fakharian [ 17 ] used the coordination of robust control and fuzzy technique to address the frequency control issue in [ 16 ]. Furthermore, Wang et al. On the other side, the protection systems have changed significantly from the bygone decade and will change continuously as a result of the advancement of technology.

Therefore, power systems designers are seeking to apply digital devices to handle the increasing complexity of power system, which improve the cost and usability. Subsequently, the digital technology has appeared in the protection system of microprocessor relays since and developing to those with communications interfaces in the as [ 20 ]. Today, digital relays have featured with high speed communication, which helped in replacing wires for safety interlocking, control and also circuit breakers tripping action.

Furthermore, there are many applications of digital relays in transmission and generation system protection due to their flexibility, high performance level, and capability of operating under different temperatures compared to the classical electromechanical relays. Therefore, this study focuses on the digital protection device, e. There are several studies have dealt this problem from the short circuit fault side only such as, the optimized time-based coordination of conventional over-current relays; which is the earliest protection technique for utility grids including micro-grids [ 21 ].

This method has a limit in its ability of multi-relay protection because of its high sensitivity to components parameters in high fault levels. Sheng et al. However, this method has been developed to island the MG for any fault in the utility grid and also disconnecting most of distributed generations DGs for faults within the MG.

Furthermore, some studies handled the frequency protection problems such as; Laghariet et al. Moreover, Komsan and Naowarat [ 24 ] discussed the same issue by using the rate of change under frequency relay to improve the load shedding scheme in MGs. Further, Freitas et al. However, they faced a very hard task in relays coordination as their design may not detect the islanded conditions within the required time.

Teimourzadeh et al. However, the proposed approach is an efficient index for providing a quick detection of MG security status. Jose Vieira et al. However, this presented coordination has a drawback, which it did not compensate the frequency fluctuations within the allowable frequency limit due to the action of the relay is energized when the system frequency become out of the allowable limit.

To prove the effectiveness of the proposed coordination in protecting the MG against frequency variations, it has been tested under different scenarios of disturbances such as, high penetration level of RESs, reducing system inertia, and sudden load variations. The remaining of this paper is arranged as follows, Section 2 discusses the problem description.

The structure of the studied MG system with the state equations are presented in Section 3.

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The coordination of control and protection methodology is described in Section 4. Section 5. Finally, the last section concludes the results and advantages of the proposed method. The dynamic security issue is one of the most critical issues, which face the power system designers. Dynamic security refers to the ability of the electrical power system to maintain the synchronism when subjected to a sever trainset disturbance [ 2 ]. Therefore, the dynamic security deals with disturbances that impose momentous changes into the system variables.

Among these are short-circuit faults, loss of a dominant generation source, and loss of a large load. The system response to these disturbances includes large deviations in the system variables such as voltage magnitudes and angles, generator speed, and system frequency [ 12 ].

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Hence, the balance between the input mechanical power and the output electrical power is disturbed. And then, the mismatch makes the synchronous generators SGs either accelerate or decelerate. On the other hand, preserving dynamic security is different between the bulk power systems and MGs. In the case of the bulk power systems, the conventional synchronous generators are considered the source of the dynamics.

Moreover, most of the available methods for preserving the dynamic security of the bulk power systems are considered inefficient for MGs due to these methods are devised based on the features of the bulk power system, which are significant inertia constant and rather slow dynamics. Therefore, this research studies the dynamic security issue in the microgrids. The power electronic interface-based RESs are static devices without any rotating mass so that the associated inertia constant is roughly zero.

On the other hand, synchronous generators-based RESs are small-scale generators with noticeably low inertia constants [ 28 ]. Such a low inertia constant renders the MGs more vulnerable to the transients than the bulk power systems. Furthermore, the power generation from RESs are unpredictable and variable, results in more fluctuations in power flow and frequency in the MG, which significantly affects the power system operation.

Also, the randomly changes in load power demand caused a bad response to the PCC voltage, active, and reactive powers transfer. To solve the dynamic stability problems, it must be determined the effective factors, which steer the MGs toward the insecurity. These factors include a low inertia constant, frequent fault occurrence, and inadequacy of existing protection schemes. Therefore, the stability and protection coordination issues have become a center of interest especially for power system researchers.

Hence, this research proposes an efficient coordination of secondary frequency control i. The MG is distributed through low voltage distribution systems and the electric power is mainly generated by DGs such as photovoltaic PV , wind turbines WT , hydropower plant, fuel cells, etc. The simplified model of an islanded MG with influence of the proposed coordination, the coordination of LFC and digital protection is shown in Fig. In this study, some physical constraints effects are taken into consideration for modelling the actual islanded MG such as; the speed governor dead band i.

Whereas, backlash is defined as the total magnitude of sustained speed change. All speed governors have a backlash, which is important for primary frequency control in the presence of disturbances. The GRC limits the generation rate of the output power, which is given as 0. The dynamic model of the studied MG system with the proposed coordination strategy as shown in Fig. This section describes the state-space equations of the proposed islanded MG considering high penetration of RESs. The dynamic equations of the studied hybrid power system can be derived and written in the state variable form as follows:.

These variations are considered as the MG disturbance signals. While, the damping D and the inertia H are the uncertainty parameters. The complete state-space model of the presented MG considering high RESs penetration level can be obtained through the state variables and definitions from 1 to 7. The linearized state-space model of the MG from Fig.

The power system frequency may have high variations if there is no longer balance between the generation and load demand. The normal frequency deviations can affect the power systems efficiency and reliability, while large deviations can destroy the equipment, overload transmission lines and cause interference with the system protection. Therefore, the frequency control is divided into three main operations based on the size of the frequency deviations.

LFC must recover the system frequency to its steady-state condition within the limits of standard time deviations. In that situation, the protection devices i. This action will interrupt power system supply. Hence, there must be an accurate coordination of LFC or emergency control and protection scheme. The frequency relay is a member of the protection devices group. It is used to protect the power system from a blackout in case of load loss, generation loss, or N-1 emergency.

Furthermore, it is used in the MG network to detect the islanding operation, which occurs in case of DGs because of losing of mains [ 30 ]. Moreover, the main threat occurs when a DG reconnected to the rest of the system without synchronizing operation at first. In the past, DGs are directly disconnected from the system due to over or under frequency problems. Recently, the continuous operation of DGs to supply domestic loads in islanded condition become necessary.

Therefore, the use of digital relays has spread and become more widely used in the MGs as the digital relays can change their settings according to the abnormality conditions. Furthermore, recently, there are many applications of digital relays in transmission and generation system protection due to their advantages such as; flexibility, high performance level, and capability of operating under different temperatures compared to the classical electromechanical relays [ 4 , 31 ].

The digital relay is a basic component in the digital protection system, which includes optical instrument transformer and a digital communication bus as shown in Fig. When an abnormal condition is detected, the relay trips a circuit breaker and make triggering for an alarm. Considering the islanded MG presented in Fig. When a disturbance occurs causing power imbalance, the system frequency starts to deviate due to the transients of DG.

The digital OUFR can be adjusted with the integrator time-delay settings. In this condition, the deviations of system frequency must persist during a pre-defined time interval for energizing the relay. Hence, the delay time setting can present as:. If the frequency is over or under the limit. If the value of the integrator output exceeds the set value, a trip action will occur by the digital relay sent to the circuit breaker to disconnect the variable load or disconnect DG.

The limits of the digital frequency relay are depended on the European codes and could be set to other values based on country standards. The dynamic response of the studied MG with the proposed coordination strategy is evaluated and tested under variation in loading patterns, loading conditions, system parameters i. The islanded MG is tested in the presence of high fluctuated wind power and low fluctuated solar power as shown in Fig.

Whereas, the wind power i. Then, the wind power, solar irradiation power, and load demand are assigned as the disturbance sources to the islanded MG To verify its dynamic security, which is the main target of this research. In this scenario, the robustness of the proposed coordination for the dynamic security of the islanded MG is evaluated by implementing the random domestic load variations as shown in Fig.

Figure 9 c shows the frequency response of the studied islanded MG. This is because the integrator output value does not exceed the set value. Therefore, the LFC succeeded to readjust the frequency to its normal value. This scenario proves the effectiveness of the LFC as it can adjust the frequency to its normal value in all five stages of this scenario without needs to protection action.

In this scenario, the islanded MG is subjected to the power change under different load disturbance profile as shown in Fig. Hence, there is no need for relay action. Furthermore, the integrator output exceeds the integrator set time K. Therefore, the digital relay is energized and sending a trip signal to the generator circuit breaker. In this scenario, the dynamic security of the islanded MG with the proposed coordination strategy is evaluated under variation in wind power penetration levels i.

Moreover, the load disturbance profile of scenario B is applied to the studied MG as in Fig. The secondary control i. Therefore, the digital OUFR sent a trip signal to the generator circuit breaker at that time as shown in Fig. Hence, the effectiveness of the proposed coordination is approved for the MG dynamic security. The effect of half microgrid inertia through the proposed coordination of LFC and digital OUFR is investigated for the microgrid dynamic security. Hence, the digital protection device quickly trips the generation units in this scenario to maintain the equipments from damage as shown in Fig.

This coordination strategy is proposed for supporting the frequency stability and protecting the islanded MG against high-frequency deviations, which increased recently due to high penetration of RESs , random load variations, and system uncertainty. The simulations results proved that the proposed coordination has been achieved an effective performance for maintaining the MG dynamic security. Whereas, the LFC has been succeeded to readjust the frequency deviations to its allowable limits under different conditions of transients, load disturbances, and RESs penetration levels.

However, in some cases of large disturbances and high RESs penetration, the LFC cannot maintain the frequency stability as the frequency fluctuates beyond the normal limits. In that case, the digital frequency relay trips the generation units. Furthermore, the results confirmed that the digital OUFR has superiority in terms of accuracy, sensitivity and wide range controlling. LFC : Load frequency control. Belwin, J. Brearley and R. Journal of Renewable and Sustainable Energy Reviews, 67 , — Dong, Y. An emergency-demand-response based under speed load shedding scheme to improve short-term voltage stability.

Aristidou, P. Contribution of distribution network control to voltage stability: A case study. Bevrani, H. Power system monitoring and control. New Jersey: Wiley. Rakhshani, E. Analysis of derivative control based virtual inertia in multi-area high-voltage direct current interconnected power systems.