Flexible system of transmission in alternative course

A flexible system of transmission in alternative course, more known under the english acronym of FACTS is equipment of electronics of auxiliary power used to control the burden-sharing in the network by thus improving the capacity of transit and by reducing the losses, to control the tension in a point or to ensure the dynamic stability of the networks of transmission of electricity and groups of productions which are connected there. It can also filter certain harmonics and thus improve quality of electricity.
The liberalization of the market of electricity has as a consequence which the production is now made sometimes very far away from the place of consumption, it is also foreseeable only before, the capacities of transit of the network must thus be increased, the arranged bottlenecks of the network. The lack of acceptance of the public opinion for the construction of electric new lines with high voltage and new powerplants encourages the managers of network to build FACTS in order to increase the capacities of transit of the network.
The FACTS are very numerous. They always form part of the family of the active electric compensations, but this compensation is series, parallel (shunt) or hybrid, thay is to say series and parallel at the same time. They are mainly made up of capacities and inductances to generate the reactive power, as well as electronics of power or circuit breakers to stop and turn on the power through the first elements. A transformer is sometimes used to decrease the tension to which the other components are subjected. The types of the most widespread FACTS are the benches of capacities mechanically commutated series or by thyristors, the SVC, which combine capacity and inductance in parallel with the line, the phase-shifting transformers, the STATCOM and SSSC which are sources of tension placed respectively simultaneously with and series the line.
Control transit of power
Principle of the compensation series
The active power depends mainly on the angle of transport
The reactive power depends mainly on the amplitude of the tensions
In normal weather the electric lines are in charge in a fork going of a third to half of their maximum capacity. In the event of overload, line can to reach its limit, it starts then in some seconds, which involves the overload of the neighbouring lines which start in their turn, one then has releases cascades about it which lead to a cut of generalized electricity. The FACTS try to balance the load between the lines to prevent this situation. If a defect has despite everything place, it will try to rebalance the load between the various lines to stop the cascade of releases.
Moreover one better load balance, between lines reduces the losses. Known as differently, they make it possible to avoid the formation of loop of current. Indeed, the losses take primarily the form of Joule losses and are thus proportional to the square of the current, by decreasing the current by a factor 2, one reduces the losses by 4.
The powers activates P and reactivates Q transported in an electric line in alternative course
Express themselves as follows for a line without loss :
P = V1 * V2 ⁄ X sinΔ
Q = V1 * (V1 - V2cos(Δ)) ⁄ X
Where Δ is the angle of transport. For summary 3 parameters are important: the amplitude of the tensions, the angle of transport and impedance. For the networks in alternative course, control binds the active power to the frequency on the one hand, and the reactive power with the control of the tension of the other.
To regulate the transfer of power, one can thus choose to control
The amplitude of the tensions
The value of the impedance
The angle of transport:
The condensers or reels assembled in series make it possible to change the value of the impedance. The condensers series make it possible to decrease the impedance of the electric connections and to increase the transmissible power by these connections. In a certain manner, one can say that they reduce the length of the line. Contrary the reactances series increase the impedance of an electric connection, in order to better distribute the currents on the various connections.
The condensers or reels assembled in shunt make it possible to change the amplitude of the tension into a point. Indeed the condensers shunts provide reactive power, and increase the tension of the network locally. While the reactances shunts consume reactive power, and decrease the tension of the network.
Dephasings created by the phase-shifting UPFC or transformers influential on the angle of transport. In the case, where more than two lines are in parallels but do not have the same electric length, thay is to say the same angle of transport, an adjustment of this last parameter makes it possible to rebalance the load. It also avoids the current loops.
Known as differently, the condensers and the reels make it possible the FACTS to provide or consume dynamically reactive power on the network. This causes to increase or decrease the amplitude of the tension at its point of connection, and consequently the transmissible maximum active power.
The objective is to increase the capacity of transit of power while approaching the thermal limits of the lines. Bad public acceptance for the electric lines, for environmental reasons mainly, returns this use of the increasingly widespread FACTS. It is good however to recall that the FACTS does not change this thermal limit and cannot thus indefinitely increase the transportable electric output of an electric line with high voltage. Devices FACTS thus do not replace the construction of new lines. They are a means of differing the investments by allowing a more effective use of the existing network.
Control stationary normal voltage
The long lines have tendency has to have overpressures at their ends in the event of weak load, one speaks about Ferranti effect, and on the contrary a weak tension in the event of strong load. To maintain a tension constant, or at least not to exceed the limits imposed by the regulation, a FACTS assembled in parallel can be interesting.
The compensation is localized in stations which are distributed at strategic places on the line. Since the compensation is not distributed in a uniform way, it is impossible to maintain the tension with its face value in any point of the line. II is thus important to select the places well where the compensation shunt is installed to prevent that the tension does not deviate too much from its face value.
If the SVC functions in adjustment of tension, the control system adjusts the current in the SVC so that running and tension follow a characteristic curve.
Dynamic stability of the electrical communications.
In the network, the generation of electricity is ensured by synchronous machines. The defects on the lines, the openings and closings of selector, the breakdowns of certain equipment can make oscillate the active power of the generators, concretely the axes of the generators start to accelerate, others to decelerate. In other words, the angle of transport starts to oscillate. The capacity of the network to find its synchronism, is called dynamic stability.
The FACTS make it possible to regulate the tension and the angle of transport quickly and thus make it possible to quench the oscillations of power activate and increase thus the availability and the reliability of the network. In transitory mode, in the event of defect, the reaction time must be lower than 100ms.
Quality of electricity
The suppliers of electricity must ensure a good quality of the tension, that wants to say a frequency and a level of tension as constant as possible, a form of sinusoidal wave and finally a symmetry between phases.
However the network can undergo the following defects.
Hollows of tension and short cuts. after short-circuit in the network.
Fast variations of tension.
Temporary or transitory overpressures.
Short interruptions or short cut.
An imbalance of tension.
These defects can come from the network itself or the customers, occasional, like the storms, or recurring, like the starting of industrial machines a such light-arc furnace. The FACTS limit the effects of the defects and the failures of the equipment thanks to their control of tension for example.
With the level of the shape of wave of the filters are often associated with the FACTS in order to limit the harmonics surrounding or created by the installation itself.
To limit the currents of short-circuit
Certain FACTS, like the TCSR and the IPC, in certain configurations, can also limit the current of short-circuit.
The first FACTS appeared in the years 1930 in order to increase the capacity of certain lines having a strong impedance. The compensators assembled in series have been used for more than 60 years in the world.
The first SVC appeared in the years 1970 and were developed by the EPRI. First was brought into service at Nebraska in 1974 by General Electric to stabilize the tension become fluctuating because of the rolling mills and the surrounding light-arc furnaces. It is during the same period that the thyristors started to be employed.
The GTO are marketed since the end of the year 1980. The first TCSC was built in 1992 by ABB with Kayenta in the United States, it made it possible to increase the line capacity of the line of 30%. The principle of the STATCOM was invented in 1976 by Laszlo Gyugyi. First was installed in Inuyama in Japan in 1991. It was marketed by Kansai Electric Power Corporation and Mitsubishi Electric Power Corporation. Second was installed in Sullivan in the North-East of Tennessee by Westinghouse Electric Corporation in the United States in 1995. The principle of the UPFC was presented in 1990 by Laszlo Gyugyi. The first UPFC was brought into service in 1998 at the electric station of AEP Inez for the account of American Electric Power.
In the beginning of the year 1990, the United Kingdom and Norway dérégulé their electric market. Thereafter the other Scandinavian countries and Europe in general encased the step. In the United States and in South America the market is also largely liberalized. Whereas the electric network of drive is regarded as a natural monopoly in Europe, Australia, New Zealand and the United States also partially liberalized it. That returns the prediction of the flow of energy much more difficult. It becomes necessary to develop the network and to make it more controllable with FACTS.
Types of FACTS
A construction containing electronics of power to the advantage of being much faster than a mechanical engineering, which makes it possible to react to transitory defects and to adapt instantaneously to the load. Those commutated mechanically are slow, can be commutated only some times per day and are used to correct foreseeable and often cyclic problems.
The FACTS with converters of tension have the advantage of being more compact than those with thyristors. There thus needs less often to increase the electric stations and thus to buy ground what is an advantage.
Converters assembled in power source are theoretically possible, but are less interesting economically and in terms of performance.

FACTS shunt.
The parallel compensation acts mainly on the tension and as limit the oscillations. It can also limit the oscillations of active power, but is less effective in this role than the compensation series. It functions in power source. The choice of their site is determining.
MSC : mechanically commutated condenser, sometimes arranged in the form of filter anti-harmonic. Allows to stabilize the tension in the event of strong load. (50 = MVAr = 500)
MSR : mechanically commutated inductance. Allows to stabilize the tension in the event of weak load. (50 = MVAr = 500)
TSC: condenser commutated by thyristors. The inductance of attenuation present in series, is used to limit the current in the event of abnormal operation and to avoid resonance with the network at particular frequencies. In practice several capacities are assembled in parallel, the connection of one or more capacities in a discrete way makes it possible to control the full value of the capacity connected to the network.
TSR : inductance commutated by thyristors
TCR : inductance controlled by thyristors. Thus it always does not consume the entirety of the possible reactive power.
Equivalent susceptance of a TCR
That is to say the angle of lag of the thyristors of the TCR noted α , the current in the TCR is :
iTCR(t) = 1 ⁄ L ∫αωt √2V sin(ωt)dt = √2V ⁄ ωL (cos(α) - cos(ωt)
for α < = ωt < = 2π - α
and iTCR(t) = 0else
By using a transform of Fourier one finds
iTCR, fundamental = V ⁄ jωLπ * (2(π - α) + sin(2α))
One can then calculate an equivalent susceptance with this value of fundamental of the current
iTCR = -jBV
By identification
B = 1 ⁄ Lω * 2(π - α) + sin(2α) ⁄ π
Beyond 90° the current is not perfectly any more sinusoidal, one has production of harmonics then
For alpha = 90°, the tension is in blue, the current in red
For alpha = 100°, the fundamental one of the current is in green
For alpha = 130°, the fundamental one of the current is in green
For alpha = 150°, the fundamental one of the current is in green

Diagram are equivalent of a SVC

Curve VI of a SVC, when the tension is low the system sends reactive power, in the contrary case absorbs some
SVC Static: compensator of energy reactivates shunt, known as also CSPR, static compensator of reactive power or statocompensator, combine TCR, TSC, fixed benches of capacities and filters of harmonics, whose first model was installed in 1979 in South Africa, is most widespread of the FACTS. They behave like a variable susceptance connected in shunt to the network. They at the same time make it possible to increase and decrease the tension, thus fulfilling the requirements in the event of weak and of strong load (50 = MVAr = 1000).
The SVC have a response time of about 30 to 40ms what is much faster than the mechanical switches whose response time is approximately 100 with 150ms. In addition, the FACTS with mechanical switches are not very flexible and their high costs of maintenance.
TCBR: This type of compensator connected in parallel is used to improve the stability of the network hang the presence of the disturbances.
STATCOM or SVG : it is used to compensate for reactive energy in the line, it is connected in shunt.
It corresponds basiquement in a circuit to D.C. current, made up in its simpler form by a condenser, connected by the electronics of power to the line. This unit behaves like a source of tension, by regulating it one can control the exchange of reactive power between line and STATCOM. Thus if the output voltage of the STATCOM is higher than that of the line, the current runs out in the reactance towards the line bringing of the power reactivates with this one. The source of tension can be on two levels or multilevel. The electronics of power can be realized using GTO, IGCT or IGBT.

STATCOM in detail

Diagram are equivalent of a STATCOM
The STATCOM can regulate its VT tension to provide or absorb reactive power with the line. It is able to provide its rated current, even when the tension is almost null.
One of the advantages of the STATCOM is to be able to provide an significant amount of power reactivates even when the tension of the network is low contrary to a SVC. Its response time is also very weak.
Direction of the exchanges between Source of tension and line
Parameter exchange power
VSTATCOM > Vline Reactive power sent towards the line
VSTATCOM < Vline Reactive power consumed by the STATCOM
Angle of transport of the STATCOM > Angle of transport of the line Active power sent towards the line
Angle of transport of the STATCOM < Angle of transport of the line Active power consumed by the STATCOM

FACTS for compensation series
The compensation series acts mainly on the reactance. It thus reduces the report ⁄ ratio of dependence tension/load, and can influence the burden-sharing between various lines. Its damping capacities of the oscillations of power activates are good. It functions like a source of tension. The choice of their site is not as sensitive as in the case of parallel compensation.
FSC consists of a capacity which can be connected or not to the line. Influence the value of the inductance of the line. It also makes it possible to limit under-synchronous resonances.
TCSR consists of a reel put in series with thyristors, the whole put in parallel with another reel.
TCSC consists of a condenser put in parallel with thyristors and a reel in series. Their control makes it possible to vary the electric length of the line. The reel in series with the thyristors behave like a TCR, but as the impedance of the capacity is lower than that of the line, the harmonics are only propagated very little in the network.
Equivalent inductance of a TCSC is worth : XTCSC = jLω ⁄ (2 ⁄ π ((π - α) + [sin(2α) ⁄ 2]) - LCω²)

Diagram are equivalent of a TCSC

SSSC in detail

Diagram are equivalent of a compensation SSSC
TSSC is a TCSC which can be only connected or disconnected.
TSSR is a TCSR which can function only with one angle of lag of 90° or 180°
TPFC functions like a FSC by influencing the inductance of the line, with the difference that instead of being to protect by lightning protectors, thyristors are used to shunt the current where necessary (QN = 401 MVAr).
SSSC: compensator series. This type of compensator series is the most important device of this family. It consists of a three-phase inverter, in other words a source of tension, coupled in series with the electric line using a transformer, that makes it possible to vary the capacity or the impedance seen by the line. The control of dephasing between this source of tension and the tension of the line also make it possible to influence the transit of active power. Better, if a source of power in D.C. current is present in the SSSC, this one can compensate for the resistance of the line, improving ratio X ⁄ R what increases the power considerably being able to forward on the line. Thus a SSSC can vary at the same time the active power and reactivates independently of the current crossing the electric line contrary to the FSC and TCSC.
Hybrids (parallel series)

Diagram of a UPFC

Diagram are equivalent of a UPFC

Diagram of an IPC

Diagram are equivalent of an IPC
UPFC, It makes it possible to influence independently the tension, the impedance and dephasing. It can also limit the current of short-circuit
The principle of the UPFC consists in deriving part of the current circulating in the line to reinject it with a suitable phase. Converter 1, connected in parallel, has as a function to take the active power and to deliver it with the converter series 2. This last generates a Upq tension controlled in amplitude and phase, which is inserted in the line. It can be compared with a phase-shifting transformer being able moreover exploit the reactive power. The UPFC are the most sophisticated FACTS and also most expensive.
IPFC, It was proposed in 1998. It uses converters DC-DC placed in series with the line to compensate. Concretely, it is used in the case of multiple lines connected to the same station. The IPFC is made of several SSSC, each one of them providing a compensation series to a different line. The IPFC makes it possible to transfer from the active power between the lines compensated to equalize the transits of powers active and reactive on the lines or to discharge the line overloaded towards another charged. The tensions injected have a component in squaring and a component in phase with the respective currents of the lines. The component in squaring allows a compensation independent series in each line, whereas the component in phase defines the level of active power exchanged with the other lines. On the connection continues, the assessment is always null.
IPC, It acts of an inspecting device which is composed of two impedances per phase: one inductive and the other capacitive one, each one being directly related to a unit of dephasing. The values of these impedances are high in order to limit the currents in the event of short-circuit. From its design, the IPC has the following aptitudes :
The control of flows of active power
The limitation of the currents of short-circuit
The decoupling of the tensions between two nodes, even in the event of defect
Explanation of the diagram of the IPC
I1 = U2 ⁄ ZIPC
And I2 = U1 ⁄ ZIPC
Or XIPC = XA ⁄ (2 sin(αA - αB ⁄ 2))
And αIPC = αA + αB ⁄ 2

Equivalent of a TCPST
TCPAR or TCPST or SPS are a phase-shifting transformer whose changer of catches is carried out containing thyristors in order to enable him to answer the transitory modes. The mechanical changers of catches are indeed too slow for this use. It cannot generate or to absorb reactive power but can distribute it between the phases, it is one of the disadvantages of the system. Even if their principle is clear, this system remains for the moment with the state of project.

Diagram of a VTF
VFT is a rotary transformer, it makes it possible to connect two asynchronous networks between them. By doing this, it can regulate the angle of transport. It is a three-phase transformer with two rollings up with a revolving secondary. General electric defends the idea that this system is highly reliable and requires little maintenance because of its low complexity. In the event of defect on one of the two connected networks, the transfer of power is not stopped. It has as a defect to consume reactive power, it is thus necessary to add benches of capacities to it. In addition, it is not dynamic because of the inertia of its axis.
Transformer Sen of the name of its inventors, is a rotary transformer, it creates an adjustable impedance in series on the line, thus emulating either a capacity, or an inductance. It also makes it possible to regulate the angle of transport. Slow device at the base, a changer of catches containing thyristor can allow, the manner of the TCPAR, to answer the transitory phenomena. If not a mechanical changer of catches can be built.
GUPFC Consists of two converters: 1 gone up in shunt, and two assembled in series on two different electric lines in an electric station.
It makes it possible to control the tension, as well as the active power and reactivates forwarding on the two lines. The active power is exchanged between the converters series and parallel using an element in common direct current.

Diagram of a bipolar station HVDC
A HVDC known as: head-spades, whose two ends are on the same place and who thus does not comprise a line of transmission with D.C. current, can be comparable with a FACTS, since its single role is to order the transit of power between two alternative networks. In this case, it can be called GPFC.
They can be of two types: either containing thyristors and thus commutated by the lines, they function then in power source, or containing IGBT and thus autocommutés, they function then in source of tension. These last converters make it possible to control the active power independently and reactivates, which are correlated respectively with the angle of transport and the tension of the station.
Storage systems of energy
The storage systems temporary of energy are not strictly speaking FACTS, but can be to them combined or associated and take part in the quality of the electrical communication.
BESS, storage of energy per battery, Generally, units BESS are relatively small but allow an exchange of high power. Their capacity to adjust the quantity of energy quickly required or to absorb is used for transitory stability.
SMES : storage of energy by superconductive inductance.

Diagram are equivalent of a phase-shifting transformer

Vectorial representation of the tensions in a phase-shifting transformer

Vectorial representation of the tensions in a QBT
The SMES is used mainly with the dynamic check of the transits as power in the electrical communication.
SCES : storage of energy by supercondensator
KESS : storage in the form of energy kinetic
CAES : storage of energy by compressed air
Phase-shifting transformer
Although very different from the FACTS by their technology, the phase-shifting transformers have a role similar to the FACTS : to control the transits of energy in an electrical communication. They are rather traditional transformers, to which the report ⁄ ratio of transformation is close to 1, but whose tensions of entry and exit are out of phase of an angle in general adjustable. The control of this angle makes it possible to modify the transit of power. They are abreviatted PST or PAR. If they are controlled by thyristors TCPST.
Squaring booster rocket transformers, QBT are also phase-shifting transformers, but whose operation is slightly particular. The tension that they bring, UT, is indeed always orthogonal with the tension of entry : U1 = U2 + UT.
The choice of the type of FACTS must be carried out according to the configuration of the network, no general conclusion cannot be made. One can however list the forces of the various systems.
Summary table of the differences between FACTS
  SVC STATCOM CSC TCSC Phase-shifting transformer UPFC IPC SSSC HVDC
Control tension +++ +++ + + + +++ + + ++
Control transit of power (mesh network) 0 0 + ++ +++ +++ +++ ++ +++
Dynamic stability (point-to-point line) + + +++ +++ ++ +++ +++ +++ +++
Damping of the oscillations of power (point-to-point line) + + +++ +++ ++ +++     +++
Damping of the oscillations of power (mesh network) + + + ++ ++ +++     +++
Types of semiconductors used
Symbol of a thyristor
Curve characteristic of a thyristor according to the current of trigger IG
The FACTS of first generation use thyristors which are used to engage and to start the components being used to consume or provide reactive power, concretely the reels and the condensers. The thyristors are engaged with a certain angle of lighting has and lead alternatively on a half-period. One defines the angle of lighting has starting from the passage by zero in the positive direction of the terminal voltage of the thyristor to light. The angle of conduction is the angle during which the thyristors lead. A thyristor starts to lead when a signal of trigger is sent to him and the tension on its terminals is positive. It stops leading when the current which crosses it cancels. One says that they are commutated by the lines, to the frequency of the network.
In order to start the thyristors, the triggering must be transmitted in a simultaneous way to a great component count located at different potentials. The connection must thus be isolated electrically. Two methods are used: optics and magnetic. Optical technology being able to start directly or indirectly thyristors. In the indirect method, the electronics of order located at low tension sends information to electronics under high voltage, which provides the power necessary to the starting of the terminal voltage of the thyristor. The direct method on the other hand uses the energy of the optical impulse to start the thyristor with luminous starting.

illustration of the modulation of width of impulse
On the level of the characteristics the thyristors can have on their terminals a tension going until 8kV and can lead in a continuous way a current going until 4,2kA.
The GTO and the IGBT, contrary to the thyristors, can open and to close itself when good seems to them, they are completely controlled. The elements which can be commutated in general make it possible to obtain better performances, to have a better control of the parameters. They make it possible as private individuals to carry out sources of alternating voltages starting from a source of continuous tension like a capacity.

Converter on two levels used with the dispatcher modulation of impulse to reproduce sinusoidal

Multilevel converter
The GTO make it possible to reach a frequency of commutation of approximately 1khz, the IGBT until 10khz. Another advantage of the IGBT compared to the GTO is that they can regulate the tension and current fluctuation, respectively di ⁄ dt and dv ⁄ dt, that makes superfluous the addition of a reel to limit the rise of the current.
To obtain a high rated current a IGBT consists of several chips assembled in parallel in the same assembly. The free diode of wheel makes it possible to ensure the passage of the current in opposite direction and to avoid the appearance of an opposite tension. If a IGBT is failing, it should be shorted-circuit to allow the operation of IGBTs healthy remainders, for that a switch, typically a thyristor, must be gone up in parallel with the modules.
The IGBT generate losses which one can divide into 2 categories: losses by conduction, losses by commutation. These last are rather important, the IGBT having on its terminals at the same time an high voltage and an important current during its commutation.
Three strategies exist to connect and control the GTO and IGBT, either on two levels with modulation of width of impulse, or on two levels with commutation at the frequency of the network, or a construction into multilevel. The modulation with width of impulse makes it possible to reproduce a tension of fundamental sinusoidal with only two discrete levels of tension, the frequency of commutation is higher than that of the network. In addition to its high level of harmonic, this method has the defect to involve strong loss by commutation. There exists also a technology where the semiconductors are opened or closed only once by cycle, the losses of commutation then are raised, but the transformer must be adapted consequently.
A transformer is often present between the line and the electronics of power in order to lower the tension seen by these last components. Thus the electric insulation for the cooling system and between the high voltage part and the part controls, in low tension, is limited and thus more economic.
Another advantage of the three-phase transformers is to make it possible to filter the third harmonic if their coupling is star-delta. For the TCR logic is thorough further, by installing a transformer star-star-triangle (YNyn0d5), the TCR are connected to the secondary and the tertiary sector. In this case, if the loads are balanced harmonics 5 and 7 are also filtered.
The TCR are characterized by a strong production of harmonics of order 5, 7, 11, 13.
The TSC do not generate distortions. Although the condensers commutated by thyristors do not produce themselves not harmonics, one installed reactances in order to avoid the amplification of the existing harmonics in the network.
The MCR can cause harmonics if the engine has an iron core and that it saturates. The using FACTS of multilevel technologies with IGBT typically cause few harmonics.
To limit these harmonics, or those coming from outside, of the high-pass or low-pass filters are installed to limit them. They are connected to the sets of bars via circuit breakers.
Equations of the powers with FACTS
The inductance of the condenser is noted XC
In the presence of a CSC, the formulas of the power activates and reactivates transmitted become
P1 = V1 * V2 ⁄ X - Xc sinδ
Q1 = V1 * (V1 - V2 cos(δ)) ⁄ X - Xc

Diagram are equivalent of a SVC after transformation of Kennely
The susceptance of the SVC is noted BSVC
P1 = V1 * V2 ⁄ X(1 - X * BSVC ⁄ 4) sinδ
Q1 = V1 * (V1 - V2 cos(δ)) ⁄ X(1 - X * BSVC ⁄ 4)
P1 = V1 * V2 ⁄ X * (1 + ISVG * X ⁄ 2 * √(U12 + U12 + 2U1U2 cos(δ))) sinδ
Q1 = U12 ⁄ X + -V1 * V2 cos(δ) ⁄ X + ISVG ⁄ 2
One can as write, as the power brought by the STATCOM is worth
Phase-shifting transformer
P1 = V1 * V2 ⁄ X sinδ UT * V2 ⁄ X cosδ
Q1 = (U12 + UT2 - U2 * U1 cos(δ) + U2 * UT sin(δ)) ⁄ X
SSSC, with the V1 tension considered equal to V2
P1 = V1 * V2 ⁄ X sinδ + V1 * VSSSC ⁄ X cos(δ ⁄ 2)
Q1 = (V1 * (V1 - V2 cos(δ)) ⁄ X - V1 * (VSSSC sin(δ ⁄ 2)) ⁄ X
The power brought by the SSSC is worth
PSSSC = VSSSC * (V2 sin(δ2 - δSSSC) - V1 sin((δ1 - δSSSC) ⁄ X
QSSSC = -V1 * (V1 - VSSSC cos(δ1 - δSSSC) + V2(V2 + VSSSC2 - δSSSC) - 2V1V2 cos(δ1 - δ2)) ⁄ X
With δ SSSCla phase of the tension produced by the SSSC
δ12 phases respectively in entry and exit of the line
If the SSSC does not exchange any power with the network the first equation activates gives
0 = V2 sin(δ2 - δSSSC) - V1 sin(δ1 - δSSSC)
In other words, the tension of the SSSC, the tension series injected must know always orthogonal about the line to ensure a pure reactive compensation
P1 = U1U2 ⁄ XIPC cos(δ + αIPC)
Q1 = U1U2 ⁄ XIPC sin(δ + αIPC) - U12X ⁄ X2IPC
Q2 = U1U2 ⁄ XIPC sin(δ + αIPC)
UM = √(U1U2 + UTU2 + 2 * U1 * UT cos(ΦT - δ)
α = arctan (UT * sin(ΦT - δ) ⁄ U1 + UT cos(ΦT - δ)
P1 = V1 * UM ⁄ X sinδ + α
Q1 = UM ⁄ X (UM - U2 cos(δ + α) + UT sin(δ + α) + X * Iq)
Installation out of container
The STATCOM and the SSSC are often placed in containers. In detail, at ABB and Siemens converters IGBT, the capacities side D.C. current, the control system of the installation and the cooling system of the converters is inside, while the transformer, the reels and the exchangers of heat are outside. In addition to the advantage of the standardization for the manufacturer, and thus of reduction of the costs, to make a préassemblage possible what limits work on site, and of the limitation of the noise, a container has the advantage of allowing an easier relocalization of the FACTS. That is made possible thanks to the low level of harmonics of the STATCOM and thus to the weak risk of resonance with the preexistent network. It is not necessary of redimensionner the filters to each relocalization.
That is an advantage because these devices are often used to reinforce a weak network, but it is sometimes difficult for the managers of network to envisage the evolution of this one in the long run. In the event of construction of a powerplant in the surroundings, the FACTS becomes useless and could be employed again elsewhere, the container simplifies the operation.

Block diagram of a regulation in simple tension for STATCOM
Le control system is an essential part of the FACTS. The SVC can according to the case being control in tension, in reactive power brought to the line, damping of the oscillations of power, or in active power forwarding in the line. The STATCOM can either be controlled in active power and reactivates, or in tension and angle of transport.
In the event of defect and of risk of oscillations of the power, the loop of damping can take over loop of tension. This loop of damping is also called stabilizing of the systems of power.

Block diagram of the PSS of a STATCOM
The most traditional means of carried out a PS uses the principle of the loop of damping of the oscillations of power of the STATCOM is composed of a block Gpc advance-delay making it possible to obtain the compensation of optimal phase, and of a block of profit KST ensuring desired damping. As for the PS, a high-pass filter, named “English wash-out”, ensures that the loop of damping of the oscillations of power does not react to the normal variations of its entry signal. A filter of the low-pass type possibly accompanied by a limiting device can also be used to attenuate the high frequency profit.
Ustabilisation = K * Filter passes * Filtre low advances and retardn * Filtre passes * signal high entered
With p the variable of Laplace
Low-pass filter = 1 ⁄ 1 + A1p + A2p2
Filter advances ⁄ delay = 1 + pT1 ⁄ 1 + pT2
High-pass filter = pTω ⁄ 1 + pTω
In the case of a UPFC one can allot the control of a parameter to the STATCOM, for example the tension or the reactive power, and another parameter with the SSSC, for example the impedance of the line, the active power or the compensation series. Often the STATCOM controls tension, the SSSC the flow of power, but sometimes a control of the reactive power by the STATCOM and to fix a tension injected fixed for the SSSC proves more suitable.
For several FACTS
When many devices FACTS are connected together in a complex network, inter-connected, if coordination between them is not sufficient, of the undesirable effects can appear for the stability of the network. It is possible to see appearing or developing phenomena of under-synchronous resonance by the presence of capacitive and inductive elements in the networks. These interactions are normally more complex in the networks comprising a compensation series since interactions with several types of modes can appear. Their frequency band is very wide since the frequency can be of some hertz as it can be close to that of synchronism. These interactions can be several types: harmonic interactions, effect of under-synchronous resonances, torsionnelles interactions under-synchronous and interactions of regulation. The electromechanical oscillations in the systems of power occur in a frequency band varying from 0.2 to 2.5 Hz. They correspond to an exchange, between machines, of energy stored in the revolving masses, thay is to say of kinetic energy.
To avoid these problems, of the control methods synchronized are developed. Among the candidates in addition to the traditional straight-line methods one finds algorithms minimax, the decentralized quadratic straight-line method or method LMI. These methods in general use the models small-signals, thay is to say they study the small variations around a point of balance.
Applications particulières
Longues lines
The lines with high voltage big length are confronted with two problems: first is that their angle of transport becomes very important. That can be problematic for the stability of the groups of productions, of broad oscillations of power being able to take place. The FACTS are then useful to quench these oscillations. Those having functions of dephasing also make it possible to limit the angle of transport. The second problem is related to the fact that the electric lines being mainly inductive, they consume reactive power. In order to improve the transit of power, it is useful to place a series condenser to limit this impedance. A TCSC is very effective to limit the oscillations of power as on the line binding north to the south of Brazil, which has a length of 1020km and a tension of 500kV.
Connection of firm wind mills
The firm wind mills are sometimes far away from the powerplants, they can then be connected to a weak network which is likely to have problems of stability in tension and power of short-circuit because of this new connection. To avoid that, the FACTS make it possible to stabilize the network.
The construction of a FACTS making it possible to improve the transfer of power, it consequently allows occasion to further buy more economic power instead of a expensive and close power. It thus makes it possible to reduce the costs of exploitation. The estimate of the value of this type of profit must be done on a case-by-case basis.
One of the advantages of the FACTS on the environmental level is to avoid the construction of line with high voltage, as explained in the Contrôle paragraph transit of power, and thus their harmful effects.
The introduction of a capacity in series with the line, that it is commutated mechanically or by thyristors, can involve resonances between the FACTS and other elements of the network. That being most alarming is that between the impedance of the line and the capacity for frequencies lower than the rated frequency of the network. This resonance low frequency can result in to make enter in resonance machine elements of the generators of the power stations and damage them. One speaks about under-synchronous resonance.
The SSSC do not have this problem because of the parasitic inductance introduced by their transformer.
Like treaty in the filters part, certain FACTS are sources of harmonics, if filtering is not effective that can decrease the quality of electricity.
The converters, transformers, as their respective cooling systems are sources of noise pollutions. Construction out of container, the construction of an anti-noise box for the transformers, of the slower ventilators can make it possible to reduce this factor however.
Electromagnetic pollution
The commutation of the converters in the FACTS is an electromagnetic source of pollution and radio interferences in particular.
So reels are present, a magnetic field is created. To avoid phenomena of induction in metal and thus of heating, the reels cannot be placed in a reinforced concrete building.
Pollution by oil and explosion
The FACTS containing of the transformers also have of them the environmental risks related to oil. This one can flee and contaminate the ground water. In addition mineral oil can lead to the explosion of the transformer in the event of fire.

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