### Three basic compensation strategies

As shown in Fig. 1, DVR system consists of the capacitor (storage device), inverter circuit, LC filter circuit, series transformer. When the voltage sag happens in grid, the dc voltage in energy storage system is converted to ac voltage by the inverter. Then, the ac voltage with a certain amplitude and phase is injected into grid feeder through filter and transformer to restore the sagged load voltage. These operations ensure normal operation of sensitive loads.

As shown in Fig. 2, phase diagrams are applied to analyze characteristics of three basic compensation methods. The Fig. 2(a) shows the in-phase compensation, the Fig. 2(b) and 2(c) shows the minimum energy compensation. The Fig. 2(d) shows the pre-sag compensation. The amplitude and phase angle of DVR compensating voltage are given in the diagrams respectively. In Fig. 2, *U*
_{
G
} and *U*
_{
S
} are the system voltages before and after the sag, *U*
_{
L
} and *U*
_{
A
} are the load voltages before and after the sag, *U*
_{
DVR
} is the DVR output compensating voltage, *P*
_{
DVR
} and *Q*
_{
DVR
} are the DVR active and reactive power injected to system, *δ* is the phase jump of sagged voltage, *θ*
_{
L
} is the angle of load power factor. *I*
_{
L
} is the load current which is set as the reference vector. Δ*U* is set as voltage sag depth, so Δ*U* = (*U*
_{
L
}-*U*
_{
S
})/*U*
_{
L
}.

#### In-phase compensation

For in-phase compensation strategy, the DVR compensating voltage has the same phase with the sagged grid voltage. The amplitude of compensating voltage is equal to the difference between the reference voltage of load and the grid voltage (shown in Fig. 2(a)). This compensation strategy only needs to measure instantaneous voltage in the grid, so it has high speed. Though DVR injects the minimum compensating voltage amplitude, it cannot correct phase jump of voltage sag. In this case, it may lead to interruption of load voltage. The DVR compensating voltage amplitude and phase angle injected to the power system are:

$$ {U}_{DVR}=\sqrt{\frac{2}{3}}{U}_L\varDelta U $$

(1)

$$ \angle {U}_{DVR}={\theta}_L $$

(2)

#### Minimum energy compensation

In reactive power compensation, the injected voltage of DVR is orthogonal to load current. This strategy only provides reactive power to compensate voltage sag without active power consumption. As shown in Fig. 2(b), the DVR injected voltage amplitude and phase in the system are:

$$ {U}_{DVR}=\sqrt{\frac{2}{3}}{U}_L\sqrt{1-2\left(1-\varDelta U\right) \cos \left(\alpha +\delta \right)+{\left(1-\varDelta U\right)}^2} $$

(3)

$$ \angle {U}_{DVR}=\frac{\pi }{2} $$

(4)

where *α* is the phase change caused by DVR reactive compensation. The maximum sag depth in reactive power compensation, Δ*U*
_{
max
}, is closely related to the load power factor:

$$ \varDelta {U}_{\max}\le \left(1- \cos {\theta}_L\right) $$

(5)

When the sagged voltage of grid *U*
_{
S
} is in phase with load current *I*
_{
L
}, the compensating voltage reaches the maximum value, i.e:

$$ {U}_{\max }=\frac{U_S}{1-\varDelta {U}_{\max }} \sin {\theta}_L $$

(6)

When the voltage sag depth is over the boundary of Eq. (5), DVR must inject some active power into the system of grid to maintain compensation. In this situation, the minimum energy method is put forward to enhance the performance of reactive power compensation. As shown in Fig. 2(c), the amplitude and phase of DVR injected voltage are:

$$ {U}_{DVR}=\sqrt{\frac{2}{3}{U}_L}\sqrt{1-2\left(1-\varDelta U\right) \cos {\theta}_L+{\left(1-\varDelta U\right)}^2} $$

(7)

$$ \angle {U}_{DVR}={ \tan}^{-1}\left(\frac{U_L \sin {\theta}_L}{U_L \cos {\theta}_L-{U}_S}\right) $$

(8)

#### Pre-sag compensation

When the voltage sag happens in the grid system, the phase jump often accompanies with the decrease of voltage amplitude. These methods mentioned above can’t compensate the phase jump. Pre-sag compensation ensures that the amplitude and phase of compensating voltage are completely in accord with the pre-sag voltage. However, the storage device has to provide abundant energy in pre-sag compensation. As shown in Fig. 2(d), the amplitude and phase of DVR injected voltage are:

$$ {U}_{DVR}=\sqrt{\frac{2}{3}}{U}_L\sqrt{1-2\left(1-\varDelta U\right) \cos \delta +{\left(1-\varDelta U\right)}^2} $$

(9)

$$ \angle {U}_{DVR}={ \tan}^{-1}\left(\frac{U_L \sin {\theta}_L-{U}_L \sin \left({\theta}_L+\delta \right)}{U_L \cos {\theta}_L-{U}_S \cos \left({\theta}_L+\delta \right)}\right) $$

(10)

### Control requirements

The DVR active power injected to the system is equal to the difference between the active power in the grid and load. From Fig. 2(a), (c) and (d), the active power of pre-sag compensation (*P*
_{
pre
}), in-phase compensation (*P*
_{
in-phase
}) and the minimum energy compensation (*P*
_{
opt
}) can be expressed as:

$$ {P}_{pre}=\sqrt{3{U}_L{I}_L}\left( \cos {\theta}_L-\left(1-\varDelta U\right) \cos \left({\theta}_L-\delta \right)\right) $$

(11)

$$ {P}_{in- phase}=\sqrt{3{U}_L{I}_L\varDelta U \cos {\theta}_L} $$

(12)

$$ {P}_{opt}=\sqrt{3{U}_L{I}_L\left( \cos {\theta}_L-1+\varDelta U\right)} $$

(13)

The relation between the DVR active power and other variables (sag depth, phase jump, the load power factor) can be obtained from the three equations. It can be seen that the pre-sag strategy has the highest power consumption among three compensation methods. The operations of the pre-sag compensation require a plenty of energy in dc energy-storage capacitor. In practice, the energy storage of DVR is limited. When the output power reaches a threshold, the compensating voltage will drop. In order to ensure appropriate operations of DVR, it should be satisfied:

$$ \frac{U_{DVR}}{n_t}\le \frac{m_{i \max }{U}_{dc}}{2} $$

(14)

where *n*
_{
t
} is the turns ratio of series transformer, *m*
_{
imax
} is the maximum modulation index of the DVR inverter, *U*
_{
dc
} is the DVR dc voltage. Once the dc voltage decreases to the minimum threshold value, that is, over the boundary in Eq. (14), DVR will stop compensating process to avoid harmonic pollution of load voltage. The initial energy stored in dc capacitor is:

$$ {W}_{dc}=\frac{1}{2}{C}_{dc}{U}_{dc}^2 $$

(15)

After a period of time Δ*t*, the value of voltage in dc container decreases from *U*
_{
dc
} to *U*
_{
dcf
}. The variation of voltage is Δ*U*
_{
dc
}. The final energy stored in the capacitor at this time is:

$$ {W}_{dcf}=\frac{1}{2}{C}_{dc}{U}_{dcf}^2 $$

(16)

During Δ*t*, the total active power provided by capacitor in steady state is:

$$ {P}_{dc}=\frac{W_{dc}-{W}_{dcf}}{\varDelta t}=\frac{1}{2}{C}_{dc}\frac{d}{dt}{U}_{dc}^2=\frac{1}{2}{C}_{dc}\left(\frac{U_{dc}^2-{U}_{dcf}^2}{\varDelta t}\right) $$

(17)

In an ideal system, the DVR dc power in Eq. (11) is equal to the ac power in Eq. (17), so the capacity of capacitor C can be obtained from the relation. However, the dc voltage will decrease with the flow of power, so as the output compensating voltage. Accordingly, the DVR output active power is also limited by the capacity of dc capacitor and the minimal dc voltage, which is sufficient for a proper restoration of the load voltage. In addition, the dc voltage gradient *dU*
_{
dc
}/*dt* is proportional to the DVR injected active power *P*
_{
DVR
} directly. The less *P*
_{
DVR
} is, the smaller the slope of dc voltage is. Hence, the compensating time is extended. It can be known that improving the rate of dc voltage drop can prolong the compensation time.

According to Eq. (14) and Eq. (17), the maximum compensation time can be expressed as:

$$ {t}_{\max }=\frac{C\ast \left[{U}_{dc}^2-{\left(\frac{2\ast {U}_{DVR}}{m_{i \max}\ast {n}_t}\right)}^2\right]}{2\ast {P}_{DVR}} $$

(18)

The maximum compensation time *t*
_{max} directly reflects the utilization level of stored energy in dc capacitor.