1.2. Metal-Organic Frameworks
1.2.1. Métodos de síntesis de los MOFs
In this section, models developed for SVC and STATCOM controllers, suitable for OPF algorithm using Newton’s method for Lagrangian functions, are elaborately depicted. The difference between SVC and STATCOM is in the nature of their operation [1]. Even though both provide reactive power compensation, SVC operation is based on injecting a reactive current (either capacitive or inductive) in the system by switching capacitors. However STATCOM regulates voltage directly by the operation of its VSC. The converter will control its output voltage and inject or consume reactive power to and from the system, more like a synchronous condenser [1]. Therefore SVC model is based on a shunt variable susceptance
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whereas STATCOM model is based on controllable voltage source. Furthermore, as an advantage to SVC, the STATCOM can be configured to provide active power to its point of connection through an energy storage device such as a large capacitor (not the DC voltage source capacitor in VSC) or battery or a group of fuel cells [1, 4, 16]. The active power control capability of the STATCOM is presented for the first time within the OPF algorithm in the model developed in this chapter and its effectiveness is tested.
Shunt compensation is a necessity in modern power systems, which feed several kinds of loads in order to keep a constant voltage profile and ensure good power quality for the consumer. There are two possible cases in shunt compensation [1, 2, 5, 9-11, 16]:
1. The voltage is sagging due to excessive reactive consumption caused by heavy industrial loads; in this case the compensator will inject reactive power to its point of control to compensate the needed reactive power and keep the voltage magnitude constant
2. In some unlikely cases the voltage starts increasing dramatically which may have been caused due to load rejection, Ferranti effects in open end lines are also a main cause of voltage increase; in such peculiar situations the compensator starts consuming the excess reactive power in order to preclude the voltage from increasing any more
SVC comprises two sets of legs, one set includes capacitor banks, which can be switched on or off, and the other contains TCR valves (or Thyristor Controlled Reactor valves). Whenever there is a need for reactive power the TCR’s switching angle will increase making SVC’s current more capacitive and vice versa [8, 10]. STATCOM comprises a VSC behind connecting shunt transformer impedance (or reactance) [16] typically between 0.1 to 0.15 per unit [1] (In empirical examples presented in this chapter the STATCOM model’s coupling impedance has a value of 0.05 per unit). VSC may also be connected to energy storage to provide active power control or account for small switching losses that occur in realistic converters [1]. STATCOM will then provide the required reactive power dispatch to the system through operation of the VSC. The converter voltage is controlled to provide reactive power compensation at the point of connection; the STATCOM model can
also be configured to provide direct reactive power rather than voltage regulation to the point of connection. The STATCOM’s main advantage towards the SVC is that it is capable of providing capacitive compensation even in very low voltage levels and therefore provides a more promising platform on which the system can remain stable [1].
It should be noted that SVC and STATCOM operation principles are discussed in detail in several publications [1, 2, 5, 6, 10, 16, 25] and will not be discussed here any further. In this section the principles outlining the modelling criteria for SVC and STATCOM is however presented. It is stressed here that because of the vital application of shunt compensators in improving system stability by regulating the voltage developing OPF models for such devices is of paramount importance when studying FACTS based systems.
3.3.1 STATCOM OPF Formulation (Controllable Voltage Source Model)
The mathematical approach developed in previous chapter for formulating the OPF problem by forming an augmented Lagrangian function is used to model the STATCOM operation here.
The STATCOM regulates reactive power by controlling its converter voltage magnitude much like a synchronous condenser [1, 16], it is therefore modelled as a controllable voltage source with close to zero active power exchange (converter voltage is in phase with the system nodal voltage) with the system (neglecting the ohmic losses as well as presence of any sort of energy storage units) [4, 16, 22]. The power balance equations given in previous chapter (Equations 2.15 and 2.16) also apply in a system with STATCOM taking into account the effects of STATCOM reactive power output (which is voltage dependent) in the reactive power balance equation as explained thoroughly in the parametric example of chapter two. The STATCOM controllable voltage source model is the first model with an additional functional equality constraint that is presented in this research. The functional equality constraint is on STATCOM’s output reactive power, which is a function of its output converter voltage. Equation (3.2) shows the exclusive Lagrangian formed out of STATCOM’s reactive power constraint. The state variables vector associated with the STATCOM are the voltage magnitude and
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phase angle of the controllable voltage source representing STATCOM’s voltage source converter as such:
!
z
statcom= ["
conv,V
conv,#
qconv]
T 3.1
The vector of variables in equation (3.1) is used to form the STATCOM Lagrangian function in equation (3.2).
!
L
conv(x,") = "
qconv(Q
conv# Q
specified)
3.2This function should account for the STATCOM’s control constraints, which is on converter’s output reactive power as shown in equation (3.3):
!
Q
conv" Q
specified= 0
3.3In addition to the functional constraint in equation (3.3), STATCOM can be configured for direct voltage regulation, in which case it behaves exactly like a synchronous condenser with zero active power output.
The direct voltage regulation shows itself as a variable equality constraint on STATCOM’s output voltage as shown in equation (3.4).
!
V
i" V
specified= 0
3.4As explained in chapter two the variable equality constraints are enforced throughout the OPF solution process using exclusive quadratic penalty functions in forms of equation (3.5).
!
"(V
i) =1
2#(V
i$ V
specified)
This function is defined in such a way that penalises the system Lagrangian for points outside the pre-specified nodal voltage for node
!
i
. It should be stressed herethat STATCOM’s converter voltage should not be penalised as it is assumed that STATCOM regulates voltage with the free operation of the converter, therefore penalising the converter’s voltage along with the nodal voltage at which point the STATCOM is connected will produce inaccurate results. The controllable voltage source model shown in figure (3.1) represents the operation of the VSC in the STATCOM device (designated here as
!
k
) as an adjustable (or controllable) voltagephasor, namely
!
V
conv(k )"#
conv(k ), which should not be confused with the system’svector of voltage phasors.
Figure 3.1 - STATCOM Controllable Voltage Source Model
According to figure (3.1) the STATCOM regulates the voltage magnitude by injecting currents to its point of compensation. The converter output reactive power thus takes the form of equation (3.6):
!
Q
conv= Im{V
k.I
c*} = V
k. V
j.[G
kjsin"
kj# B
kjcos"
kj]
j =k i$
3.6Substituting for the vector of state variables pertaining the STATCOM’s converter shown in equation (3.1), equation (3.6) is re-written as such: