In this section, we examine the results of the placement and sizing problem in terms of costs, voltage regulation, number of PV installations, as well as panel area sizes. The first part,

Section 4.1, discusses the results solved by problem (22). The second part,

Section 4.2, provides a comparison between the solutions of problem (22) and (22′).

#### 4.1. Results of Placement and Sizing with Optimal Inverter Design

Voltage regulation is the task of maintaining voltages across the distribution network close to the nominal value of the substation voltage. When there are no PV units present, the feeder voltage profile obtained by running the load-flow using the

`LinDistFlow` is depicted in

Figure 5. It can be observed that, when consumption is at

$150\%$ of the peak load, voltages drop as nodes get farther away from the substation. When a maximum installation of one PV unit per node is allowed, the optimal solution from problem (22) yields the voltage profile in

Figure 6 where every nodal voltage in the system remains within appropriate bounds for every generation scenario. This constraint is enforced via (19), where

$\u03f5=0.03$. This simple experiment motivates the voltage regulation benefits of placing PV generation units in distribution networks. In what follows, we will examine the optimal setups in cases where more homes per node are allowed to install PV units.

The number of installations allowed per node for

${B}_{n}=1,5,10,15,24$ is shown in

Figure 7. By observing

Figure 7, it is clear that most of the installations are allowed at nodes 11, 19, and 20 which are the terminal nodes of the radial network; compare with

Figure 3. The voltage drop at those particular nodes are high, and more installations are required.

Figure 8 depicts the average value of the panel area

${A}_{n,l}$ of all the installations allowed per node for

${B}_{n}=1,5,10,15,24$. For example, if

${B}_{n}=5$, the optimal locations of installations are nodes 11, 19, and 20. Out of the seven homes of node 20, only five homes have installations with an average panel area of 38.7878

${\mathrm{m}}^{2}$ and this value is shown in

Figure 8 for node 20 and

${B}_{n}=5$. The same applies for

${B}_{n}=1,10,15,24$. The average nameplate DC rating of the PV module is depicted in

Figure 9, which is calculated using

${A}_{n,l}$ through (1).

Figure 10 depicts the average value of the inverter capacity

${S}_{{\mathrm{ac}}_{n,\ell}}$ of all the installations allowed per node for

${B}_{n}=1,5,10,15,24$. For example, when

${B}_{n}=5$, the installations occur on nodes 11, 19, and 20. Out of the seven homes of node 20, five homes have installations with average inverter capacity of

$15.35\phantom{\rule{3.33333pt}{0ex}}\mathrm{kVA}$ and this value is displayed for node 20 and

${B}_{n}=5$. The same applies for

${B}_{n}=1,10,15,24$.

Using

Figure 8 and

Figure 10, the maximum of the average panel area and the maximum of the average inverter capacity for each

${B}_{n}$ are listed in

Table 6. For example, it follows from

Figure 8 that, if

${B}_{n}=5$, the installations are allowed at nodes 11, 19, and 20. The average values of the panel area for allowed installations at nodes 11, 19, and 20 are 83.95

${\mathrm{m}}^{2}$, 40.33

${\mathrm{m}}^{2}$, and 37.92

${\mathrm{m}}^{2}$, respectively. The maximum of the three values is 83.9548

${\mathrm{m}}^{2}$, which corresponds to the maximum of the average panel area for

${B}_{n}=5$. The same is applicable for

${B}_{n}=1,10,15,24$. The maximum of the average inverter capacity is given as the maximum value of the average inverter capacity for each of the

${B}_{n}$ bar plots in

Figure 10. For example, it is inferred from

Figure 10 that, when

${B}_{n}=5$, the installations are allowed at nodes 11, 19, and 20. The average values of the inverter capacity for allowed installations at nodes 11, 19, and 20 are, respectively,

$33.25\phantom{\rule{3.33333pt}{0ex}}\mathrm{kVA}$,

$15.98\phantom{\rule{3.33333pt}{0ex}}\mathrm{kVA}$, and

$15.03\phantom{\rule{3.33333pt}{0ex}}\mathrm{kVA}$. The maximum of the aforementioned three values is

$33.25\phantom{\rule{3.33333pt}{0ex}}\mathrm{kVA}$, which corresponds to the maximum of the average inverter capacity for

${B}_{n}=5$.

Table 6 shows that, as the number of installations allowed per node increases, the maxima of the average panel area and the average inverter capacity decrease.

Table 7 lists the optimal value, installation cost, including inverter and DC cost, and thermal loss cost. From

Table 7, it is observed that the optimal value and the installation cost remain the same for

${B}_{n}=15\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\mathrm{and}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}24$. It is also observed that the DC costs remain the same for

${B}_{n}=5$ and

${B}_{n}=10$ although there is a significant difference between the maximum of the average panel area for

${B}_{n}=5$ and

${B}_{n}=10$ in

Table 6. The reason is that the sum of the panel areas of all the allowed installations per node for

${B}_{n}=5$ and

${B}_{n}=10$ is the same and is equal to

$735\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{2}$. A similar observation holds for

${B}_{n}=15$ and

${B}_{n}=24$. In this work, the maximum value of

${A}_{n,\ell}$ is relatively large, that is,

$100\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{2}$ (cf.

Section 3.1). Therefore, the results for

${B}_{n}=1$ give large panel sizes, but, for larger values of

${B}_{n}$, e.g., for

${B}_{n}=15$, the resulting panel area sizes are realistic.