OUE SERIES COMPENSATOR INSERTS REAL AND REACTIVE IMPEDANCE IN ELECTRICAL ENERGY SYSTEMS TO CUSHIZE POWER OSCILLATIONS
FIELD OF THE INVENTION This invention relates to an apparatus for dampening power oscillations in electric power systems. More particularly, it relates to a series compensator which can insert both real and reactive impedance in a transmission line to dampen energy oscillations. BACKGROUND OF THE INVENTION Energy oscillation frequently occurs in electric power systems due to disturbances, such as transmission line failures, line and load commutations, equipment failures and other events "that cause the system to change rapidly. Such energy oscillations have the undesirable effect of avoiding the maximum transmittable power in the system. U.S. Patent No. 5,198,746 discloses a solid state series compensator that injects a controllable voltage of 60 Hz in quadrature with the current of the transmission line. This injected voltage has the same compensating effect as that obtained with a capacitor or variable inductor connected in series with the line. The damping of energy oscillation is carried out by means of the appropriate modulation of the reactive effective impedance of 60 Hz that the compensator injects in series with the line. That is, when the power in the line increases as a result of the acceleration of the power generator at the "sending end" of the line and the consequent increase in the transmission angle, the capacitive impedance of the compensator increases as the magnitude increases. of the inserted 60 Hz voltage that is delayed from the line current by 90 electrical degrees, in order to increase the degree of compensation in series and thus the transmittable power. Conversely, when the power in the line decreases, as a result of the deceleration of the generator "sending end and the consequent decrease in the transmission angle, the inserted voltage causes the line current to lead in order to create, in effect , an inductive output impedance and thereby increase the total impedance of the transmission line .. The series compensator of the patent No. 5,198,746 uses a cd to ac converter to generate the voltage quadrature.The ca terminals of the inverter are connected in series with the transmission line through a coupling transformer A capacitor connected through the terminals cd provides the dc voltage for the converter, since • the converter is theoretically exchanging reactive power with the transmission line, there is no drain However, there are actually losses in the switching circuits of the inverter. The necessary to make up these losses is provided by generating the compensation voltage at a phase angle slightly less than 90 °. U.S. Patent No. 5,343,139 discloses a generalized power flow controller for controlling power flow through the transmission line on a sustained basis. This power flow controller also uses a DC to AC inverter which injects a voltage in series with the voltage of the transmission line. However, the phase angle of the voltage injected with respect to the line current is not limited to substantially 90 ° as in the case of the patent compensator Number 5,198,746, but is controllable at any phase angle between 0 and 360 °. The particular phase angle and the magnitude of the injected voltage can be selected to adjust any or all of the effective impedance of the transmission line, the effective phase angle between voltages at the two ends of the transmission line, and the magnitude of the the transmission line. This results in the adjustment of both real and reactive line impedance. The actual power needed to adjust the actual impedance component is provided to the cd to ac converter by a ac to dc converter having its ac terminals connected in shunt to the transmission line and its dc terminals connected to the dc terminals of the dc. compensating converter through a cd link which includes a capacitor. The response of this flow controller is fast enough "that it can provide dynamic power flow control, and it is also appropriate to dampen oscillations that can develop in the power system. However, this equipment designed primarily for power flow control is more complex and expensive than the series compensator described in U.S. Patent No. 5,198,746. There is a need for an improved compensator to dampen oscillations in the electrical power system. More particularly, there is a need for such an improved compensator that can provide real power modulation as well as reactive power modulation to dampen the oscillations of the power system. SUMMARY OF THE INVENTION These needs and others are met by means of the invention which is directed to the improved apparatus to provide series compensation to dampen the oscillations in an electric power transmission line, which provides exchange of both real and reactive power with the power system. A particular feature of the invention is that it can provide virtual positive real power injection in series with the transmission line to absorb the actual power when the generator is accelerated during power oscillations. During those periods of the oscillation when the real power in the transmission lines decreases, the virtual negative real impedance is inserted in series with the transmission line to supply real power to the system. The virtual positive real impedance can be provided by a resistor which is selectively connected to absorb the real power of the transmission line, and the virtual negative real impedance, which provides real power, can be provided by an energy storage device . The energy storage device, such as a battery or a superconducting magnet, can be used to supply both the positive real impedance when absorbing the real power, and the virtual negative real impedance when supplying real power. The real positive and negative virtual impedances are injected to the transmission line together with the reactive compensation. A power commutating converter generates a compensation voltage having a magnitude and a phase angle with respect to the current of the transmission line that provides the required reactive and virtual real impedances. This voltage is injected in series with the voltage of the transmission line through a coupling transformer. The switching power converter generates both the positive and negative reactive power required internally. The actual power is supplied and absorbed by a power exchange element connected to the cd terminals of the power switching converter. It is not necessary to provide both the positive and negative real impedances, although the more precise control can be made through the injection of both positive and negative real impedance in the transmission line. The fact that it is not necessary according to the invention to inject positive and negative real impedance in the transmission line is advantageous since providing positive impedance is easier and less expensive than providing negative real impedance. More particularly, the invention is directed to: An apparatus for providing line compensation in series with improved capacity to dampen oscillations in electric transmission lines carrying AC current to a transmission line voltage and fundamental frequency, said apparatus comprises: compensation element coupled in series with the transmission line that injects into the transmission line a compensation voltage at said fundamental frequency with controllable magnitude and phase angle with respect to said ac current in the transmission line; a power exchange element connects to the transmission line only through said compensation element; and a control element "which controls the magnitude and phase angle of said compensation voltage to insert reactive and real virtual impedance in the transmission line to dampen the oscillations, and selectively connect the power exchange element with the compensation element. to enable said compensation element to provide the mentioned virtual real impedance. BRIEF DESCRIPTION OF THE DRAWINGS A thorough understanding of the invention can be achieved from the following description of the preferred embodiments when read in conjunction with the accompanying drawings, in which: Figure 1 is a schematic diagram of a simple power system . Figure 2 is a graph of transmitted electrical power versus transmission angle for the power system shown in Figure 1. Figure 3A is a schematic diagram of a part of the power system of Figure 1 incorporating the present invention. Figure 3B is a diagram illustrating the compensating voltage injected according to the invention in the power system shown in Figure 1. Figures 4A-4D illustrate relevant waveforms for understanding the operation of the invention. Figure 5 is a schematic diagram of a series compensator for the electrical power system of Figure 1, according to one embodiment of the invention. Fig. 6 is a schematic diagram of a second embodiment of a series compensator for damping power oscillations in the electric power system of Fig. 1. Fig. 7 is a schematic diagram of a generalized form of the series compensator of the invention. to dampen power oscillations in the electric power system of Figure 1. Figure 8 is a block diagram of the series compensator according to the invention. Figure 9 is a block diagram of an internal control for the series compensator of the present invention. Figure 10 is a block diagram illustrating in greater detail the portions of the internal control system of Figure 9. DESCRIPTION OF THE PREFERRED MODALITY The subject invention is directed to a novel series compensator for damping oscillations in electric power systems. To explain the proposed operation and control of this novel series compensator, reference is made to the simple power system in Figure 1. This system 1 consists of a terminal generator 3, a transmission line 5, represented by its inductive impedance in series, X, and a receiver end power system 7 that can be considered as an infinite capacity power channel. If the voltage phasor at the sending end of the transmission line 5 is s and that at the receiving end is r, then the transmitted electrical power, Pe, can be expressed by the following equation: Pe =. { Vs. Vr / X } sen d Ec. (1)
where Vs and Vr are the magnitudes of Vs and Vr, respectively, and d is the angle between Vs and Vr. The power Pe is shown plotted in Vs, Vr, and X against the angle d in Figure 2. It can be seen «that the maximum transmissible power is Pemax 0 Va.Vr / X Eq. (2) obtained at d = 90 degrees. In the present power systems the power generator 3 is an electromagnetic machine that converts the mechanical input power Pm, usually provided by a steam or gas turbine 9, into the electric power Pe that feeds the transmission line 5. Under steady-state conditions, the mechanical input power Pm is equal to the electric output power Pe (neglecting losses), and the turbine-generator equipment 3, 9 runs at a constant speed in such a way that a transmission angle d0 is maintained / necessary to satisfy the equilibrium Pm = Pe (see figure 2). The occurrence of power oscillation is due to the fact that, while the electric power Pe can change almost instantaneously in the transmission line (often due to unforeseen or accidental events such as transmission line failures and equipment failures), the Mechanical input power Pm can change only very slowly due to the nature of the mechanical controls and the large inertia involved in the rotation of the system. Therefore, in the case of a fault in the transmission line or other alterations in the electric power system l, the generator 3 receives more mechanical input power than the electric power transmitted by the defective transmission system. As a result, the generator 3 starts to accelerate and the transmission angles become larger than the steady state angle d0. The acceleration process of course means that the excess mechanical energy is being mechanically stored by the rotational inertia of the turbine-generator equipment. In order to restore steady-state equilibrium after eliminating the fault, this stored energy must be absorbed from the rotational mechanical system. Depending on the total factor Q (quality) of the electro-mechanical system, the absorption of this energy can take a number of oscillatory cycles during which the electrical power in the transmission system can fluctuate widely. In some cases, when the total system has negative damping, the magnitude of the oscillation keeps increasing until the protection of the system turns off the altered generator. It is clear that, mainly, the oscillatory energy can be eliminated from the affected system in two ways. One is increasing and decreasing the power transmitted in resonance with the acceleration and deceleration of the generator and thus, in resonance with the changes when it becomes positive and when the electric power becomes negative. The other is to directly subtract the surplus energy from the transmission system when the generator approaches and dissipates it or returns it to the system, from temporary storage, when the generator is in the interval of the declaration. U.S. Patent No. 5,198,746 proposes a novel method for the implementation of only the first technique. That is, the series compensator is controlled to decrease the impedance of the transmission line, and thereby increase the power transmitted during the acceleration periods of the machine, and to increase the impedance of the transmission line, and thereby decrease the power transmitted during periods of deceleration. This is done by injecting a voltage at the fundamental frequency (60 Hz) into the transmission line. The injected voltage is delayed from the current by 90 ° by the capacitive impedance and carries it by 90 ° by the inductive impedance. It is evident that the power oscillation method described in patent number 5,198,746 is based on a dimensional modulation of the output of the series compensator. This is, the injected voltage is maintained essentially in quadrature with respect to the line current so that the power exchange between the series compensator and the ac system is substantially reactive. Our invention is based on a two-dimensional modulation of the output of a series compensator. That is, the phase angle between the fundamental voltage (60 Hz) and the current of the transmission line is chosen to force the exchange of reactive and real power between the series compensator and the ac system. This is illustrated in Figures 3A and 3B. As shown in Figure 3A, the series compensator 11 injects a compensation voltage Vc into the transmission line 5 in series with the line voltage Vx through a coupling transformer 12. Figure 3B is a phasor diagram that shows the injected voltage Vc, the line current IL, and the voltage phasors of the Vsis system. It can be seen that the injected phasor voltage Vc has a component Vci which is in quadrature with the line current and another component Vcr which is in phase with the line current IL. The quadrature voltage component Voi represents the virtual reactive impedance that the series compensator provides for the compensation of the reactive impedance in the manner proposed by patent number 5,198,746. The actual voltage component Vcr represents a virtual positive real impedance (resistor) or a virtual negative real impedance (power source) in series with the line. The variation either of the reactive impedance injected or of the actual impedance injected in resonance with the variation of the transmission angle is effective to achieve the power oscillation damping. The mechanism of damping with the injection and reactive impedance is explained in detail in the patent number 5,198,746 which is incorporated herein for reference. It can be seen that, in addition to the increased power transmission provided by the reactive compensation in series by injecting a real positive impedance (resistor) into the line at the time when the sending end generator 3 accelerates due to excess mechanical power, part The excess real power is dissipated, helping to establish a balance between the mechanical input power and the electrical output power for the stable operation of the system. Similarly, the injection of a negative real impedance (in effect, a real power source) into the line, when the sending end generator is unable to momentarily satisfy the demand for electrical power and therefore is By decelerating, it could clearly remove part of the generator load and thus could help stabilize the system. Generally, the general technique proposed by this invention is to provide highly effective damping of power oscillations by simultaneously injecting reactive and real impedance in series with the line. Both the reactive and real impedance are controlled in resonance with the variation of the transmission angle (and the corresponding power). The reactive impedance is controlled to increase the electrical power transmitted when the generator decelerates. The real impedance is controlled so that it is positive and thus consumes real power when the generator accelerates and is negative and thus generates real power when the generator decelerates. The proposed method for damping power oscillations in the systems is illustrated in Figures 4A-D, wherein the transmission angle d, the transmitted electrical power 9, and the outputs of reactive Xc and real Rc, respectively, of the series compensator controlled between its maximum values of virtual and inductive capacitive impedance, * and respectively, the maximum positive virtual and real negative impedance values are plotted against time. The dashed lines in the graph of the transmission angle and the transmitted power represent an undamped oscillation, while the solid lines illustrate the damping effect according to the invention. It should be noted that the variation of the real (or reactive) output between the maximum positive and maximum negative values is not a necessary requirement to effect the damping. For example, effective damping can be achieved even if the real or reactive impedance inserted is varied between zero and a positive or negative maximum. As will be discussed, this is an important practical consideration because the implementation of a series compensator capable of injecting a positive real impedance, in addition to a capacitive or inductive impedance, is relatively simple and economical, while the implementation of one that also can inject a virtual negative real impedance is relatively complex and expensive. The last case in which only virtual positive real impedance is provided is illustrated in Figure 4D where the virtual negative real impedance is shown with dashed line. As explained above, the serial compensator of the invention is capable of concurrently injecting a virtual and real virtual reactive impedance in series with the line. The magnitude of these virtual impedances is controlled independently with the VA limitation ratio of the series compensator. This flexibility allows the selection of different modes of operation of the compensator in series under conditions of differing systems. Thus, several control strategies can be implemented to meet particular system requirements. In the preferred operative arrangement, the proposed series compensator is configured to provide reactive line compensation as the primary function to achieve steady-state line impedance compensation and, in the case of parallel lines, equalization of line current. However, whenever alterations of dynamic systems are found, the series compensator can also execute the injection of a real virtual impedance in series with the line. The reactive virtual and real virtual impedances are controlled to maximize the electric charge in the generator during its first and subsequent acceleration periods, and to maximize the electric charge during the deceleration periods. This strategy provides the maximum possible improvement in transient stability (first oscillation) and in dynamic stability (oscillation damping) in addition to achieving the steady state line compensation requirements. A series compensator capable of generating reactive and actual impedance outputs according to the invention is shown in Figure 5. The series compensator 11 includes a switching power converter in the form of a DC-AC voltage source inverter. 13 which employs firing-to-shutdown thyristors 15 (or other appropriate semiconductor power switches) to generate the voltage Vc required for real and reactive series compensation. The terminals 17 of the inverter 13 are connected to the coupling transformer 12, which in turn is connected in series in the transmission line. The transformer 19 and the transmission line 5 are shown in a single line for clarity of representation; however, it is understood that they represent the three-phase implementation. A control 21 detects the currents and voltages of the transmission line through the current transformers 23 and the potential transformers 25, respectively (again, multiphase for three-phase system). The control 21 uses these currents and voltages together with sets of parameters and control inputs to generate fire signals for the switches 15 in the inverter 13 to generate the appropriate instantaneous values of real and reactive impedance represented by the voltage output Vc by the investor. As discussed previously, the inverter 13 is capable of generating the positive virtual and negative reactive impedances internally. The actual power to effect the actual virtual impedance injection in the transmission line is provided by a power exchange device in the form of a real impedance source 27 connected to the terminals 29 of the inverter 13. In the embodiment of the figure 5, the actual impedance source 27 comprises an energy storage device 31 and a cd to cd converter which provides charge and discharge control of the energy storage device. The energy storage device 31 can be, for example, a battery bank or a superconducting magnet. In this case, the energy storage device absorbs the actual power of the transmission line 5 through the inverter 13 to insert virtual positive real impedance in the transmission line, and return real power through the inverter 13 to the line of transmission to provide virtual negative real impedance. A capacitor 35 connected through the terminals cd 29 stabilizes the dc voltage applied to the inverter 13. Another embodiment of the invention is illustrated in figure 6, wherein the actual impedance source 27 'comprises a resistive load 37 selectively connected through of the cd terminals 29 of the inverter 13 by means of a switch 39 operated by a fly-back diode 41. This series compensator mode can supply both positive and negative reactive impedance, but only realpositive impedance. However, it is considerably less expensive than the embodiment of Figure 5 and still provides substantially improved performance over the compensator of the patent number 5,198,746 which only provides reactive impedance compensation. A generalized embodiment of the invention is shown in Figure 7, where the power exchange device in the form of the actual power source 27"comprises a current source 43 which can be selectively connected by the switch 45 to supply real power. (negative real impedance) to the inverter 13, and a current reservoir 47 which is selectively connected by a switch 49 to absorb real power (provides positive real impedance) of the inverter 13. For the purpose of analysis to be explained, the transformer of coupling 12 is represented as voltage sources 51 a, b and c connected in each phase to the transmission line 5. The functional operation of the series compensator 11 is carried out by appropriate control of the power inverter 13 which produces the voltage injected in Thus, the complete control system has two major functions: one, referred to as the function of internal control, is the synchronization and proper operation of the solid state inverter 13 to generate the required voltage, with the appropriate magnitude and phase angle in relation to the current line, in series with the line to achieve the degree of impedance compensation reactive and real impedance insertion; the other referred to as the external control function, is the determination of what reactive and real series compensation is needed (ie, what magnitude and phase angle the inserted voltage must have) in order to provide the maximum improvement in transient stability and power oscillation damping. External control measures the appropriate system variables, such as line voltages, currents and frequency or rotational speed, derives reference signals for the internal control of the inverter. The operation and implementations of practical external control circuits related to the reactive impedance control are described in patent number 5,198,746. The operation of external circuits can be extended to the control of the virtual real impedance, to achieve the damping of power oscillation, by means of the following simple rule. Whenever a series compensator will inject a capacitive reactive impedance to increase the transmitted power, the compensator also injects a positive real impedance in series with the line to absorb the real power of the ac system. (This means that the absorption of generator power accelerating is maximum - the line transmits the increased power due to capacitive compensation in series and the series compensator absorbs the additional real power). Conversely, whenever the compensator will inject an inductive (reactive) impedance to decrease the transmitted power, the compensator simultaneously injects a negative real impedance (if it has energy storage capacity as shown in figure 5) or real impedance. zero (if equipped with only a switched-energy absorber device (resistor)) as shown in figure 6. The internal control of the inverter is responsible for regulating the magnitude of the ac output voltage of the inverter and its phase angle in response to the requirements of the system as determined by external control. Figure 7 defines a polarity convention of each of the electrical variables that will be referred to in the following description of the internal control. In this simplified representation of Figure 7, the losses of the power system are neglected, and the inverter 13 is viewed in a general form as a three-phase device whose terminal voltages appear in series with the transmission line. The leakage inductance of the power transformers 12 coupling the inverter in the transmission line is shown as an additional nominal inductance 53a, b and c in series with the line. In order to describe the internal control scheme it is necessary to first define the mathematical model of the series compensator on which it is based. The mathematical model uses the concept of instantaneous vectors to represent the three-phase voltage and current sets on the AC side of the inverter. Using this concept, the three line currents are instantaneously described by a single vector with orthogonal, ida, and iqs components. The definition of ids, and iqs in terms of current line currents is as follows:
In these equations 3-5,? is the angle of the current vector with respect to the ds axis of the coordinate system, and i is the instantaneous magnitude of the current vector. Similarly, the voltage injected in series is represented vectorically in coordinates (ds, qs) and then further transformed to a reference frame (d, q) in which the x-axis is always coincident with the line current vector . This voltage transformation is defined as follows:
In this frame of reference, the components of the series voltage vector, ed and eq, count for the real and reactive instantaneous power of the line respectively. The expressions for these components are as follows: Actual power (P) = _3__ied Eq. (7) 2 Reactive power (q) = _3_ieq Eq. (8) 2 In Figure 8, these definitions are used to establish a block diagram of the series compensator (that is, the system that is to be controlled). In this diagram, the harmonic voltages generated by the inverter 13 are disregarded and an instantaneous balance is assumed between the power on the ac side and on the cd side of the inverter. A constant, KIf defines the ratio between the capacitor voltage Vc and the peak phase voltage ed, eq on the inverter's AC side as shown in 53, 55 and 59. A control angle, jß, is defined as the angle whereby the voltage vector of the inverter (ie, the voltage vector injected in series) drives the line current vector. This angle is established by the control system and can be changed quickly and arbitrarily. As shown in Figure 8, it sets the instantaneous value of the voltage injected at 53 and 57. The ratio of the magnitude of the line current vector, i, constitutes the instantaneous "reactance" represented by the line compensator, and the The ratio of ed to i constitutes instantaneous "resistance." It is further assumed that the line current is determined substantially by extraneous factors and the currents is therefore viewed as an input independent of the system. control ß, is seen to influence the inverter ac terminal, Pca, determined at 61 and 63 from ed and the line current i directly through its effect on ed as shown at 57 and 59. The power in the ac side of the inverter is combined with an equal power of cd which defines at 65 an instantaneous charge current of the capacitor, i ^ An additional load current, i2, is provided by the actual power source / reservoir cd 27. sum of these charge currents is integrated by the capacitor 35 as shown at 67, obtaining approximately a change in the capacitor CD voltage and hence a corresponding change in ed and eq. Figure 9 is a block diagram of the proposed internal control scheme. The external control 69 generates reactance demands, X *, and resistance, R *, based on its observation of the transmission line conditions (determined from current and line voltage measurements) and power system considerations. These are combined into 71 to form a demand of magnitude of impedance, Z *. The reactance demand, X *, and an adjusted resistance demand, R ± *, are used to calculate the control angle, ß. The impedance magnitude demand is multiplied by 73 by the magnitude of the line current vector, i, to obtain the magnitude of the desired series voltage, e *. Because the magnitude of the capacitor voltage, Vc, directly determines the magnitude of the AC voltage in series, a feedback control loop 75 is provided to regulate the value of Vc. An error signal is calculated at 77 as the difference between e * and Vc multiplied by a constant KS at 79 and passed to a controller 81 which either activates the power source cd 43 or the power tank 47 as appropriate to In order to correct any «deviation of the desired capacitor cd voltage. In Figure 9, this controller 81 is shown as a simple type of hysteresis (bang-bang) although linear drivers must be used in practice. In addition to this main control action, additional action is provided to cover the possibility that only one power tank 47 or only one power source is included. When the cd voltage vector signal exceeds a given magnitude limit, an amplified signal is produced at 83 which is converted to an impedance when dividing by the current at 85. The quotient is scaled at 87 and added at 89 in demand of resistance, R *, to form the adjusted resistance demand R ± *. This ensures that the real power can always be negotiated to and from the transmission line to regulate the DC voltage whenever the regulation can not be achieved by the source and / or cd deposit. In effect, this additional action does not exceed the resistance demand of external controls to the point that it does not require more real power than that which the investor can provide or reserve. The angle, ß, is added to the angle 0 of the line current vector at 91 to obtain a total angle, f, for the series voltage vector, which is used at 93 to determine the state of the switches. in the inverter 13. The selection of the switch state is carried out by means of a state check table which is sequentially stored and accessed only as a function of f, the desired angle of the series voltage vector. The content of the verification table is naturally different for the different possible inverter topologies and harmonic reduction schemes, but in all cases the input is an angle (f) and the output is a set of switch states that is fed to the gate controller circuits for controlling the switch devices 15. The feedback signals, i, and 0, are generated from actual current measurements. A locked phase loop of the vector 95 calculates the angle 0 from the orthogonal components ids and iqs of the single vector representing the three-phase current which is generated by a resolver vector 97 from the phase currents. The orthogonal components odB and iqs are also used by a vector magnitude calculator 99 to calculate the magnitude of current i. The block 101 is a limiter that imposes a positive lower limit on the magnitude i before it passes to the divider block 85. This prevents the output of the block 85 from becoming very large. Figure 10 illustrates in greater detail, the elements of the resolver vector 97, the locked phase loop of vector 95 and the calculator of the magnitude vector 99. The resolver vector 97 generates the real and imaginary components idB and iqs of the current vector 97 in the rotational reference system from the measured phase currents ia, ib and ic. This is achieved by implementing equation 3, above. The magnitude calculator of vector 99 generates the magnitude i of the single current vector through the implementation of equation 5, above. The angle 0 of the line current is not obtained in the preferred embodiment of the invention by the arctangent calculation of equation 4, but on the contrary by means of the locked phase loop of vector 95 which closely follows the angular rotation of the current vector represented by the ida and iqs components provided by the resolver vector 97. The real component idB is multiplied by 103 by the sine of the angle 0 derived at 105. This product is subtracted from the difference junction 107 of the product calculated in 109 of the imaginary component iqs of the current and cosine of the angle 0 is derived at 111. Additional integral control proportional to the difference is applied at 113 and integrated into 115 to generate the angle 0. The subject invention provides reactive compensation in series of line and a real virtual impedance insertion which results in the dramatic improvement in transient stability and damping of oscillations in a system of t transmission of electric power. While «that the specific embodiments of the invention have been described in detail, it will be appreciated by those skilled in the art that various modifications and alternatives to those details can be developed in light of all the teachings of the disclosure. Accordingly, the particular arrangements disclosed are only intended to be illustrative and not limiting of the scope of the invention to which all the amplitude of the appended claims and any of their equivalents are given.