**3. Underlying principle of the proposed algorithm**

Regardless of the modulation technique applied, either SPWM or SVPWM (Bueno et al., 2002), a set of states defines transistors in each phase. These states imply that each phase is connected to 1, 0 or -1, where 1 is P (Udc+), 0 is NP (middle point) and -1 is N (Udc-). Because the topology described in this document is triphasic, it is necessary to work with a three element vector, where each value is the state of each phase, typical of a three-phase converter.

The aim of a converter is to fix a specific phase-to-phase voltage in order to force a current flow through the grid filter. The system will work correctly as long as the appropriate voltage is applied, whichever phase it was connected to 1, 0 or -1. Figure 5 shows the schematic that works with two independent MPPTs, where the original vector is generated. There has been a lot of work done on Maximum Power Points (MPPT) for photovoltaic panels (Weidong et al., 2007), (Salas et al., 2006) and (Jain & Agarwal 2007), and in (Esram et al., 2007) research focuses on the review of maximum power point tracking. In (Patel & Agarwal 2008), the influence of shadows on MPPTs is studied.

Fig. 5. Schematics of the control system.

The algorithm we propose explains how to calculate a fixed or dynamic voltage band around the reference voltage in NP, which can be Vdc/2 or something else. This voltage band, which will determine whether the measured NP voltage is above, below or within the range, will force the algorithm to increase, reduce or apply no offset to the NP voltage.

Once a reference shift is drawn, NP voltage must be changed, which means that one capacitor will charge as the other discharges with the same amount of energy. So, globally, energy and DC-bus voltage remain unchanged.

The instant power is distributed in the system in the following way:

$$P\_G = P\_{PV1} + P\_{PV2} \tag{1}$$

where *PG* is the total available or generated power,

$$p\_{C1} + p\_{C2} = P\_{DC} \approx P\_G - P\_{Load} \tag{2}$$

and stored power in the capacitors is approximated, ignoring power losses in wires and the inverter. Nevertheless, this power can be considered more or less constant for transients, where *pC1≠pC2* is not only different, but not constant either; the variation being a consequence of the ripple in the NP. Hence, these powers can be separated into a continuous component and a variable component, as follows:

$$p\_{C\_1} = P\_{C\_1} + \frac{1}{2}C\_1 \frac{du\_{C\_1}}{dt}^2\tag{3}$$

$$p\_{\text{C}\_2} = P\_{\text{C}\_2} + \frac{1}{2}C\_2 \frac{d u\_{\text{C}\_2}}{dt}^2\tag{4}$$

considering *PDC* more or less constant, and existing for a short-period of time , we can say that:

$$\frac{1}{2}\mathbf{C}\_1 \frac{d\boldsymbol{u}\_{\mathbb{C}\_1}}{dt} \approx -\frac{1}{2}\mathbf{C}\_2 \frac{d\boldsymbol{u}\_{\mathbb{C}\_2}}{dt} \approx \boldsymbol{\Delta}\_p \tag{5}$$

and if (5) is substituted in (3) and (4), the equations become:

$$p\_{\mathbb{C}\_1} \approx P\_{\mathbb{C}\_1} + \Delta\_p \tag{6}$$

$$p\_{\mathbb{C}\_2} \approx P\_{\mathbb{C}\_2} - \Delta\_p \tag{7}$$

There are certain moments in which all phases are connected to the same capacitor, forcing that capacitor to get charged while the other one gets discharged. That is precisely the right time to change the normal behaviour.

The object is to modify the state vector in order to overrule the standard sequence and charge the appropriate capacitor.

A list with all shifted vectors is shown in Table 2, but some conditions have to be met: the appropriate state combination and a true need of shift.


Table 2. Vector-Shift implementation.

330 Solar Radiation

The algorithm we propose explains how to calculate a fixed or dynamic voltage band around the reference voltage in NP, which can be Vdc/2 or something else. This voltage band, which will determine whether the measured NP voltage is above, below or within the range, will force the algorithm to increase, reduce or apply no offset to the NP voltage. Once a reference shift is drawn, NP voltage must be changed, which means that one capacitor will charge as the other discharges with the same amount of energy. So, globally,

*C C DC G Load* 1 2 *p p P PP* (2)

and stored power in the capacitors is approximated, ignoring power losses in wires and the inverter. Nevertheless, this power can be considered more or less constant for transients, where *pC1≠pC2* is not only different, but not constant either; the variation being a consequence of the ripple in the NP. Hence, these powers can be separated into a continuous

2

2

*p*

*C*

*du*

*C*

*du*

1 1 2

2 1 2

1 1

2 2

1 2 1 1 2 2

*C C*

*pP C dt*

*pP C dt*

considering *PDC* more or less constant, and existing for a short-period of time , we can say

*C C*

*C C* 1 1 *<sup>p</sup> p P* (6)

*C C* 2 2 *<sup>p</sup> p P* (7)

There are certain moments in which all phases are connected to the same capacitor, forcing that capacitor to get charged while the other one gets discharged. That is precisely the right

The object is to modify the state vector in order to overrule the standard sequence and

*du du C C*

2 2

*C C*

*PP P G PV PV* 1 2 (1)

(3)

(4)

*dt dt* (5)

energy and DC-bus voltage remain unchanged.

where *PG* is the total available or generated power,

component and a variable component, as follows:

<sup>1</sup>

<sup>2</sup>

1 2

time to change the normal behaviour.

charge the appropriate capacitor.

and if (5) is substituted in (3) and (4), the equations become:

that:

The instant power is distributed in the system in the following way:

The results found when a shifted vector is applied are that signs in (6) and (7) get changed. This forces a modified offset in the NP with twice the value and opposite sign. This is different from the original offset when shift is not done, giving:

$$
\boldsymbol{p\_{C\_1}} \approx \boldsymbol{p\_{C\_1}} - 2\boldsymbol{\Lambda}\_p \tag{8}
$$

$$
\vec{p\_{C\_2}} \approx p\_{C\_2} + 2\Lambda\_p \tag{9}
$$

and keeping the relationship in (2):

$$
\vec{p\_{C\_1}} + \vec{p\_{C\_2}} = \vec{p\_{C\_1}} + \vec{p\_{C\_2}} = P\_{\rm DC} \tag{10}
$$

Therefore, both aims of not modifying DC-bus voltage and quickly applying an appropriate change to the NP voltage are achieved.

Figure 6 shows where the change of the original vector for the shifted vector is put.

Fig. 6. Schematic with vector-shift implementation.
