In this section, the compound transformer model has been tested in the 5-node benchmark system in [12] in order to precisely determine its potential influence on deviating the optimum operating point in the OPF algorithm. The compound transformer in this system acts as a series compensator regulating active power across the line connecting between nodes “Lake” and “Main”.
According to the system data presented in Appendix II, the active power demand in “Main” is 40 megawatts, which, in normal operation, where no active power regulation is carried out, is provided by the system through the lines connected to “Main”. The optimum active power flow solution as obtained by the OPF algorithm
at “Main” in the 5-node benchmark system has been presented in figure (4.15) below.
Figure 4.15 - Partial OPF Solution for the normal 5-node Benchmark system; Notice the active power flow between “Lake” and “Main”
However, by connecting the compound transformer between “Lake” and “Main” the active power transferred between these nodes can be regulated seamlessly using the phase shifter angle variations in the compound model.
Figure 4.16 - Partial OPF Solution for the 5-node Benchmark system with the compound transformer regulating power between "Lake" and "Main"
Within the OPF algorithm, setting the power control constraint requires activation of equation (4.16) via adding its Lagranigan in equation (4.24) to the system Lagrangian. The compound transformer model is added to the system configuration using the dummy bus “Comp1” and the OPF is run for operating mode 3.2 (power control on receiving end). The OPF converges in 6 iterations with the results shown in figure (4.16).
As it can be seen from the results obtained by the OPF shown in figure (4.16), the compound transformer has successfully regulated the amount of active power between “Lake” and “Main” to 40 MWs, with the active power at “Main” arriving at 39.82 MWs to support the active power demand there. The compound !
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transformer model has regulated the pre-determined amount of active power by varying its phase shifter angle to -5.114 degrees, which means that the overall incurred phase angle difference between voltage phase angles in “Lake” and “Main” has been shifted by approximately 5 degrees resulting in increasing the power flow to 40 MWs from previously 14.87 MWs in the first test run in figure (4.15). A significant shift in the bulk of active power flow between the transmission lines connected to “Main” is noticeable by comparing the OPF results shown in figures (4.15) and (4.16). The bulk of active power now flows in the line connecting “Lake” and “Main”.
On the other hand, comparing the results in figures (4.15) and (4.16) shows that in the second test run the generators active power dispatch particularly in “South” has been increased, which indicates that using the compound transformer model to regulate the power while adding more flexibility to the system operation comes at the price of increasing the objective function final value by approximately 1%, from 748 $/hr to 757 $/hr. However, this increase in the objective function value (generators’ increase in fuel consumption) is justified by the fact that the compound transformer model improves system stability margins by improving its controllability in active power flow.
The main reason as to why the cost function increases in the second test run, in other words why the optimal power flow does not converge towards the same point, is that by using the compound transformer to regulate active power between “Lake” and “Main”, a new functional equality constraint is introduced, which is suitable for controlling the active power along the transmission line between these two nodes. Therefore, by simply adding the new functional constraints, the boundaries of the solution region will change from that of the first test run where no power regulating devices are present. It is precisely due to the change of boundaries of the solution region that there is less degree of freedom for the cost function variations during the course of the solution process that ultimately yields a different optimum operating point. It is imperative to note that FACTS controllers are tasked with improving system stability by actively controlling the system parameters. However, applying the OPF in each system depends on the conditions of that system alone, which are essentially the constraints, which the optimisation problem’s objective functions are subjected to.
On the other hand, if the compound transformer had been used in mode 1 the results would have been similar to those in figure (4.15), since the OPF boundary regions would have remained the same as before.
4.7 Conclusion
In this chapter an advanced model for the voltage source converter in optimal power flow has been introduced. The model is based on the fact that the operation of a PWM-controlled VSC can be modelled more precisely as a compound transformer device with controllable complex tap phasor to include the characteristics of the PWM control in the OPF mathematical formulation.
Based on the operational principles of a PWM-controlled VSC, the complex tap ratio variations accurately model the voltage and phase angle control of the converter’s output voltage phasor giving rise to independent active and reactive power control. Furthermore, by adding a shunt branch to the equivalent circuit of the compound transformer, the converter’s internal switching losses are modelled as a shunt resistive branch, whereas the DC bus capacitor is modelled as a shunt susceptance.
The new model is essentially different from previous modelling approaches [5, 11, 14, 16], which often regard the VSC as a controllable voltage source much like a synchronous condenser behind coupling impedance (or reactance) [4].
The active power in the compound transformer model is controlled via the variations in the amount of variable phase shifter angle in the compound transformer, which is capable of achieving control targets by way of phase angle control much like a real PWM-controlled voltage source converter. The voltage relations between the sending and receiving side of the compound transformer is controlled via the variable tap changer ratio, which in case of a PWM-controlled VSC, corresponds to the amplitude modulation index [13, 14] of the converter. This gives rise to indirect reactive power control by way of direct nodal voltage magnitude control.
The bi-directional power flow control is also maintained by selecting the appropriate modes of operation in the compound transformer. The active power
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control entails activating the active power functional equality constraint, which remains active throughout the solution process. However, in certain modes of operation where no active power regulation is desired, penalising its corresponding multiplier will deactivate the Lagrangian function associated with the active power control constraint.
The behaviour characteristics of the model developed in this chapter within the OPF algorithm has been tested thoroughly in a variety of simulations aimed at portraying different circumstances under which the compound transformer may be chosen to operate. In the AC stand-alone tests, the compound model has been tested as a stand-alone device with the purpose to verify whether all the control modes work properly. In the DC tests, the compound transformer models the operation of a voltage source converter used to feed a DC load. Using the phase shifter angle compensation in the compound transformer model, it has been observed that the new VSC model is capable of providing the required amount of active power to the DC node regardless of the changes in system conditions at the AC side, which means that the new VSC compound transformer model is capable of isolating the DC loads from the AC network. This particular characteristic of the compound transformer model plays a vital role in performing OPF calculations in systems with VSC-HVDC links. In the next chapter the details of modelling the VSC-HVDC systems using the new compound transformer model are explained.