Учебники / 0841558_16EA1_federico_milano_power_system_modelling_and_scripting

VSC models can be connected as shunt or series devices to the ac network. Equations (17.2) and (18.9)-(18.11) can be used as base for both connections, as follows.

VSC Shunt Connection: Assuming that the VSC is connected to the ac bus h, the shunt connection is obtained imposing:

VSC Series Connection: Assuming that the VSC is connected to the ac buses h and k, the series connection is defined by:

0 = qh + qk − qac

0= p2h + qh2 − (vhiac)2

0= ph sin(θh − φac) − qh cos(θh − φac)

0= p2k + qk2 − (vk iac)2

0= pk sin(θh − φac) − qk cos(θk − φac)

The VSC model is completed by the controls that regulates the modulating amplitude am and the firing angle α. These controls depend on the application and should be defined separately to maintain the VSC model as general as possible (similarly to the case of the inverter and rectifier models used in the HVDC link).

A special care has to be devoted to the operating limits of the VSC device. Both the firing angle and the modulating amplitude are limited:

amaxm ≤ am ≤ aminm

These limits are used for bounding the controls of the VSC device. Then the ac and dc current limits indicated in Table 18.4 have also to be checked during VSC operation. If a current limit is reached, then some of the VSC controls have to be locked. In other words, current limits impose indirect limits on the firing angle α and the modulating amplitude am (e.g., αmax(idc)).

The following examples describe two typical applications of the VSC device for connecting distributed dc resources to the ac network, namely the fuel cell and the photovoltaic cell.

Example 18.1 Solid Oxide Fuel Cell Control

The SOFC model described in Subsection 17.6.1 is linked to ac networks through a shunt-connected VSC device. The two controllers that have to be included to complete the system composed of the SOFC and the VSC are shown in Figure 18.6. The ac voltage magnitude vh is regulated by means of the VSC inverter modulating amplitude am:

The amplitude control undergoes an anti-windup limiter. The fuel cell dc current set point irefdc is defined based on power reference pref :

The set point irefdc is limited by the dynamic limits proportional to the hydrogen flow:

where qH2 is the hydrogen flow and U max are U min are the maximum or the minimum hydrogen utilization, respectively. Then the current idc is regulated through the VSC firing angle α by means of a PI controller:

Fig. 18.6 Power and ac voltage controls for the solid oxide fuel cell

Example 18.2 Solar Photovoltaic Cell Control

Similarly to the fuel cell, photovoltaic panels are linked to the ac network through a shunt-connected VSC device. The voltage control on the ac side can

be obtained using the VSC inverter modulating amplitude am. The voltage control equation is, for example, (18.15).

The other control is aimed to maximize the power production of the photovoltaic cell:

This control is called Maximum Power Point Tracking (MPPT). As discussed in Section 17.6.2, the output current idc is a function of the voltage vdc, of the temperature Θ and of the solar irradiance G. The exponential does not allow writing an explicit expression of the current idc with respect of the other variables. Thus the static pv characteristic of the photovoltaic cell can be found only numerically.

Figure 18.7 shows the pv characteristic of a typical photovoltaic cell. The curve has always a maximum, which depends on the temperature and the solar irradiance values. The power produced by the cell increases as the solar irradiance increases and the temperature decreases.

The maximum power point satisfies the condition:

= idc(vdc, Θ, G) + vdc

∂idc(vdc, Θ, G)

∂vdc

which can be solved numerically. The solution of (18.20) provides the optimal value of the dc voltage reference. Then a PI controller can be used for regulating the VSC firing angle. The resulting equations are:

Fig. 18.7 E ect of solar irradiance and temperature on the pv characteristic of the photovoltaic cell

In most practical applications, the first equation of (18.21) is often substituted by empirical systems based on measures. Most MPPT tracking systems vary periodically the dc voltage of the photovoltaic cell. The MPPT firstly tries increasing the dc voltage. If the output power increases, then the voltage is further increased, otherwise, the MPPT decreases the dc voltage. This kind of control is easy to implement since it requires only the measures of the electrical dc power (no need of measuring the temperature and the solar irradiance and of knowing cell parameters).

The ac voltage and the power controls of the photovoltaic cell are depicted in Figure 18.8 and all cell regulator parameters are defined in Table 18.6.

Example 18.3 Superconducting Magnetic Energy Storage

The Superconducting Magnetic Energy Storage (SMES) is a shunt-connected VSC-based device where in the dc side a superconducting coil is connected in parallel with the VSC capacitor through a dc/dc system (chopper ) [123]. The SMES scheme is shown in Figure 18.9. The coil is able to store magnetic energy and to release it to the network depending on the duty cycle of the chopper. Since the resistance of the superconducting coil and of the dc system is very small with respect to the inductance, the time constant of the RL circuit is relatively high with respect to conventional electro-magnetic time scales.

Fig. 18.8 Maximum power point tracking for the photovoltaic cell

Table 18.6 Photovoltaic cell regulator parameters

The equations for the SMES can be divided into three parts: (i) dc network model, (ii) VSC model, and (iii) controllers.

Dc network : The dc network is composed of the parallel the VSC capacitor and the superconducting coil behind the chopper. The capacitor can be considered a parallel RC element described by equations (17.10) and is interfaced to the dc network through (17.2). The coil-chopper block is modelled as:

where all voltages and currents are average values, L is the superconducting coil inductance and dc is the chopper duty cycle. The duty cycle can be used a control variable of the SMES, in fact for dc = 0.5 the average coil voltage vs and the average dc current idc are zero. For dc > 0.5 the coil is charged, while for dc < 0.5 the coil is discharged.

Finally, the dc network must contain two nodes, one of which is connected to the ground. The VSC and the RL element are connected in parallel to the dc nodes.

Shunt-connected VSC : Equations are (17.2) and (18.9)-(18.11) and (18.12). Regulators: The regulator are aimed to provide correct handling of the coil charge and discharge. As indicated in (18.22), the coil charge/discharge is defined by the value of the duty cycle. Thus, the active power ph at the ac side is regulated by means of the duty cycle. Then the VSC firing angle and modulation amplitude can be used for regulating the dc voltage vdc and the ac voltage vh, respectively. PI or lag controllers similar to the

ones defined for the SOFC can be used (see Figure 18.6).

To avoid overand under-charging, it is necessary to monitor the energy stored in the superconducting coil, for example measuring and integrating the dc voltage and current:

Then, based on the maximum and minimum coil energy limits, one can define the maximum and minimum duty cycle value and disable the active power control. As discussed in [11], properly handling VSC firing angle and modulating amplitude control limits can also improve the transient behavior of the SMES.

18.2.1Simplified Dynamic VSC Model

Due to the fast response of the power electronic switches and of the capacitor, in most transient stability applications, the VSC can be modelled taking into account only the power balance and simplified control equations. If the power flow is from the dc side to the ac one, the power balance is:

where ploss is due to commutation and conduction losses of switches and diodes and capacitor losses, and can be evaluated as a function of the rms value of the current circulating in the electronic switches (approximated by the dc current idc) and of the dc voltage vdc:

where the coe cients a, b, c and d are provided by the VSC manufacturer. The simplified control equations do not explicitly include the firing angle α and the modulating amplitude am but only consider input and output variables. Hence, to regulate the active and reactive powers on the ac side, the control di erential equations can be written as:

Similar equations can be written for controlling the ac voltage, the power factor, the dc current or the dc voltage. VSC current limits can be taken into account by replacing one of the previous equations (18.26) with the current limit that is violated (e.g., idc − imaxdc = 0).

18.2.2Power Flow VSC Model

In power flow analysis, it can be convenient to simplify the VSC model using only static equations. A very simple model is depicted in Figure 18.10, in both versions, i.e., shunt and series connections [2].

vh θh

Fig. 18.10 Power flow VSC equivalent circuit: (a) shunt connection and (b) series connection

Shunt-connected VSC : The equations for the shunt-connected VSC are:

ph = vh2gsh − vhvsh(gsh cos(θh − αsh) + bsh sin(θh − αsh)) (18.27) qh = −vh2bsh − vhvsh(gsh sin(θh − αsh) − bsh cos(θh − αsh))

psh = vsh2 gsh − vhvsh(gsh cos(θh − αsh) − bsh sin(θh − αsh)) qsh = −vsh2 bsh + vhvsh(gsh sin(θh − αsh) + bsh cos(θh − αsh))

where gsh + jbsh = 1/(rsh + jxsh) = 1/z¯sh. These equations approximate the VSC through an independent generator and a connection that models the embedded VSC transformer and internal VSC losses. The variables are psh, qsh, vsh, θsh for the VSC internal bus and vh, θh at the point of connection with the ac network. Since there are six quantities and four equations, two additional equations are needed. These equations strictly depend on the application. In general, one has:

Section 19.3 of Chapter 19 describes an example of shunt-connected VSC configuration, namely the STATCOM device.

Series-connected VSC : The equations for the series-connected VSC are:

ph = vh2gse − vhvk (gse cos(θh − θk ) + bse sin(θh − θk)) (18.29) −vhvse(gse cos(θh − αse) + bse sin(θh − αse))

qh = −vh2bse − vhvk (gse sin(θh − θk ) − bse cos(θh − θk )) −vhvse(gse sin(θh − αse) − bse cos(θh − αse))

pk = vk2gse − vhvk (gse cos(θh − θk ) − bse sin(θh − θk )) −vkvse(gse cos(θk − αse) − bse sin(θk − αse))

qk = −vk2bse + vhvk (gse sin(θh − θk ) + bse cos(θh − θk )) +vkvse(gse sin(θk − αse) + bse cos(θk − αse))

pse = vse2 gse − vhvse(gse cos(θh − αse) − bse sin(θh − αse))

−vkvse(gse cos(θk − αse) + bse sin(θk − αse))

qse = −vse2 bse + vhvse(gse sin(θh − θse) + bse cos(θh − θse))

−vkvse(gse sin(θh − αse) − bse cos(θh − αse))

where gse + jbse = 1/(rse + jxse) = 1/z¯se. These equations approximate the VSC through an independent generator and a series connection that models the embedded VSC transformer and internal VSC losses. The variables are pse, qse, vse, θse for the VSC internal bus and vh, θh, vk and θk at the points of connection with the network. Since there are eight quantities and six equations, two additional equations are needed. These equations strictly depend on the application. In general, one has:

Section 19.4 of Chapter 19 describes an example of series-connected VSC configuration, namely the SSSC device.

system. In fact, the power flow model (18.27) does not explicitly model the dc systems and, thus the power balance is the only way to link the two VSC devices that form the HVDC system. The power balance can be expressed as (see Figure 18.12):

It is interesting to note that the detailed hybrid model (17.2), (17.10) and (18.9)-(18.11) takes implicitly into account the power balance since dc node equations impose the Kirchho ’s current law, while ac bus equations impose the active and reactive power balances. This is another advantage of the detailed model over the simplified ones.