*2.3. PV Inverter Impact on Distribution Feeder Voltage Profile*

To provide a better insight into the voltage problem in DN caused by PVs, a case study is carried out. Two PV inverter control modes are chosen to clarify its capabilities for voltage optimization. The DN model, presented in Figure 4 [40], consists of three radial feeders supplied by a 10/0.4 kV substation. Each feeder supplies 20 residential consumers. More information about the network model can be found in [40]. It is assumed that half of the residential consumers have PV systems on their rooftops. The nominal power of each PV plant is 5 kW. The case studied in the simulations corresponds to maximum production and consumption of 0.2 kW with an inductive power factor of 0.9.

DIgSILENT PowerFactory [41] software is used for case study implementation and the conventional power flow is analyzed. Two modes of PV inverter operation are used in simulations: fixed power factor mode and volt–VAR control mode. The results for the fixed power factor mode are presented in Figure 5. The power factor range is taken from a real-life example of an inverter [42]. The voltage profiles at different power factor values are compared. There is an increase in the voltage profile at the unit power factor. The voltage profile is corrected by changing the power factor.

The comparison of voltage profiles at the unit power factor and the applied volt–VAR control mode is shown in Figure 6. In the case without voltage control (unit power factor), there is a voltage rise in the distribution feeder caused by PVs. The voltage values are in the range of 1.005 p.u. to 1.05 p.u. In the volt–VAR control mode, the voltage values are lower than the unit power factor and are in the narrower range of 0.992 p.u. to 1.005 p.u. These values are more acceptable for the operation of DN, i.e., voltage deviation is smaller.

**Figure 5.** Voltage profile in a distribution feeder obtained using the fixed power factor control mode.

**Figure 6.** Voltage profile in the distribution feeder obtained using the volt–VAR control mode.

The obtained results show an improvement in the voltage profile of DN compared to the case without voltage control (unit power factor of the PV inverter). Voltage control in PV prosumer-rich DNs has a positive impact. The case study shows the possibilities of PV inverters regarding voltage control and the situation when there are lots of inverters placed at different positions in the DN. Determining the optimal operating point of the PV inverters imposes using optimization algorithms from which the OPF are imposed as a logical solution.

#### **3. Voltage Optimization in PV-Rich Distribution Networks—Objectives and Variables**

The OPF concept was proposed in the early 1960s [43] as an enhancement of economic dispatch to find the optimal solution for controlling variable settings under different constraints. The OPF is used as a universal term for problems associated with network optimization [44–47]. The OPF is ordinarily modeled to the appliance on transmission level considering large generating units. Besides the fundamental variables, the OPF model may contain ancillary generation units and variables representing the other segments of the power system used for optimal operation.

The transmission network (TN) diverges from DN in topology, nature, electrical parameters, power flow values, and a number of control devices. Unlike TNs, DNs are inherently unbalanced and more complex [48]. The reason for the imbalance is that the DN supplies unequal single-phase loads and contains unequal conductor interspace of threephase segments [49,50]. The *R/X* ratio is high in DNs and contributes to the complexity of

control and optimization. In contrast, the *R/X* ratio is low for TNs. Compared to DNs, TNs have a few direct consumers. The simple control and a well-built communication system of TNs are the main reasons why OPF has applied only at the transmission level. The integration of DGs and flexible loads such as EVs makes OPF feasible in DN optimization. To incorporate unpredictable DG and to exploit the potential of flexible loads, OPF became imminent for DNs [51]. Although there is no official record in the literature of the beginning of the application of OPF in DNs, it can be said it started with the integration of different types of prosumers in DNs [48].

#### *3.1. General Formulation—Objectives and Variables*

The OPF problem can be described as minimizing the objective function while taking equality and inequality constraints into account [48]:

$$
min F(\mathbf{x}\_t \mathbf{u}) = 0 \tag{2}
$$

$$
\xi(\mathbf{x}\mathfrak{u}) = 0 \tag{3}
$$

$$h(\mathbf{x}; \mathbf{u}) \le 0 \tag{4}$$

where *F*(**x,u**) represents the objective function and *g*(**x,u**) represents nonlinear equality constraints i.e., power flow equations, *h*(**x,u**) represents nonlinear inequality constraints. The vectors **x** and **u** present state variables, and control variables, respectively.

In [48], the generally used objectives for OPF formulation are given. It should be noted that the objectives and constraints must be modeled accurately to obtain a satisfactory solution.

Scientific papers are included in this review if at least one of the objectives is voltage optimization and if one of the optimization variables is PV inverter reactive power injection. Furthermore, the voltage optimization problem is mostly described as the objective of voltage deviation (VD) minimization, i.e., maintaining voltages within boundaries determined by grid codes. The general mathematical expression for VD is:

$$V\_{d\sigma\upsilon} = \sum\_{i \in \mathcal{N}} \left( V\_i - V^{nom} \right)^2 \tag{5}$$

where:

*Vdev*—voltage deviation;

*Vi*—voltage at bus *i*;

*Vnom*—nominal voltage.

Another objective that appears is related to the voltage unbalance, commonly presented as the voltage unbalance factor (VUF). The definition of VUF is given in [52] as the ratio of negative *V*− *sequence* and positive *V*<sup>+</sup> *sequence* voltage sequences and is most often expressed in percentages:

$$VUF = \frac{V\_{scquence}^{-}}{V\_{scquence}^{+}}.\tag{6}$$

In addition to voltage optimization, the following objectives also appear: (i) power loss minimization [53–55], (ii) on load tap changer (OLTC) switching operation minimization [56], (iii) PV cost minimization [38], (iv) reactive power injection/absorption minimization [57], (v) active power curtailment (APC) minimization [58], (vi) cost of purchased energy minimization [59], (vii) peak shaving minimization [59], and (viii) security margin index (SMI) minimization [59]. The mathematical expressions of the commonly used objectives are given in Table 2.


**Table 2.** Mathematical expressions of the commonly used objectives in voltage optimization problems.

In power systems, the conventional power flow is both nonlinear and nonconvex and commonly solved by the Newton–Raphson iterative method. In constrained OPF applications, equality constraints incorporate conventional power flow equations and other constraints to ensure balance. A detailed version of the power flow is named AC power flow [60]. AC power flow as a constraint in OPF is most often formulated in the polar form [60]:

$$P\_l = \sum\_{k=i}^{N} |V\_i||V\_k||Y\_{ik}|\cos(\delta\_i - \delta\_k - \theta\_{ik})\tag{7}$$

$$Q\_i = \sum\_{k=i}^{N} |V\_i||V\_k||Y\_{ik}|\sin(\delta\_i - \delta\_k - \theta\_{ik})\tag{8}$$

where:

*Pi*—active power at bus *i*;

*Qi*—reactive power at bus *i*;


Besides AC power flow, the authors use two other formulation approaches: decoupled AC power flow [49] and DC power flow [50]. In decoupled AC power flow, active and reactive powers are decoupled as a function of voltage angle and voltage magnitude, respectively. Assumptions made for the DC power flow formulation include purely imaginary elements of Y and a small difference between two voltage angles of two adjacent busses.

Various inequality constraints are given in [48,61]:


$$V\_i^{\min} \le V\_i \le V\_i^{\max} \tag{9}$$

where *Vmin <sup>i</sup>* and *<sup>V</sup>max <sup>i</sup>* are the lower and upper voltage limits.

3. PV active and reactive power constraint

$$0 \le P\_{PV,i} \le P\_{PV,av,i} \tag{10}$$

$$-\sqrt{S\_{PV,i}^2 - P\_{PV,i}^2} \le Q\_{PV,i} \le \sqrt{S\_{PV,i}^2 - P\_{PV,i}^2} \tag{11}$$

where *PPV*,*i*, , *QPVi* , and *SPV*,*<sup>i</sup>* are active, reactive, and apparent powers at bus *i*. *PPV*,*av*,*<sup>i</sup>* is available active power at bus *i*.

4. Line current (thermal) constraint

$$I\_{ik}^{min} \le I\_{ik} \le I\_{ik}^{max} \tag{12}$$

where *Imin ik* and *<sup>I</sup>max ik* are the lower and upper limits of the line current between buses *i* and *k*.

5. OLTC tap position constraint (if it is included)

$$T\_i^{\min} \le T\_i \le T\_i^{\max} \tag{13}$$

where *Tmin <sup>i</sup>* and *<sup>T</sup>max <sup>i</sup>* are the lower and upper positions of OLTC tap at bus *i*.

6. Capacitor constraint (if it is included)

$$\mathbf{Q}\_{\rm Ci}^{\rm min} \le \mathbf{Q}\_{\rm Ci} \le \mathbf{Q}\_{\rm Ci}^{\rm max} \tag{14}$$

where *Qmin Ci* and *<sup>Q</sup>max Ci* are the lower and upper limits of capacitor reactive power at bus *i*.

7. Energy storage constraint (if it is included)

$$So\mathbb{C}\_{i,t}^{\min} \le SoC\_{i,t} \le SoC\_{\prime}t^{\max} \tag{15}$$

where *SoCmin <sup>i</sup>*,*<sup>t</sup>* and *SoCmax <sup>i</sup>*,*<sup>t</sup>* are the lower and upper limits of the charge state of the storage system at time *t*.

The voltage optimization problem can be single-objective or multi-objective. OPF objectives and variables used in the review papers are categorized and summarized in Table 3.


#### **Table 3.** *Cont.*


The abbreviations are as follows: CVR: conservation voltage reduction; VSI: voltage stability index; SC: shunt capacitor; AVR/VR: automatic voltage regulator; CB: capacitor bank.

According to the literature review, the multi-objective problem prevails.

Besides PV inverter reactive power, other variables include: (i) PV active power, (ii) OLTC, (iii) CB, (iv) static compensator, (v) reactive power from the substation, (vi) VRs, (vii) charge/discharge rate of ESS, (viii) EV active power, and (ix) SC.
