-
Notifications
You must be signed in to change notification settings - Fork 4
Expand file tree
/
Copy pathline-losses.html
More file actions
369 lines (333 loc) · 31 KB
/
line-losses.html
File metadata and controls
369 lines (333 loc) · 31 KB
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
<!DOCTYPE html>
<html lang="en-US">
<head>
<title>I2R Losses in Single- and Three-Phase Distribution Networks</title>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<!-- SEO Description -->
<meta name="author" content="Nicholas S. Selby">
<meta name="description" content="How unbalanced do loads on a three-phase distribution network have to be to warrant reconfiguring the network to be single-phase?">
<link rel="canonical" href="https://rupumped.github.io/line-losses.html">
<script type="application/ld+json">
{
"@context": "https://schema.org",
"@type": "BlogPosting",
"headline": "I2R Losses in Single- and Three-Phase Distribution Networks",
"image": [
"https://rupumped.github.io/blog/line-losses.jpg",
"https://rupumped.github.io/blog-posts/losses-cover.jpg"
],
"datePublished": "2023-03-04T00:00:00+00:00",
"author": [{
"@type": "Person",
"name": "Nicholas S. Selby",
"url": "https://rupumped.github.io/"
}]
}
</script>
<!-- Open Graph Tags -->
<meta property="og:title" content="I2R Losses in Single- and Three-Phase Distribution Networks">
<meta property="og:description" content="How unbalanced do loads on a three-phase distribution network have to be to warrant reconfiguring the network to be single-phase?">
<meta property="og:image" content="https://rupumped.github.io/blog/line-losses.jpg">
<meta property="og:image:width" content="3000">
<meta property="og:image:height" content="3000">
<meta property="og:image:alt" content="A power line pole">
<meta property="og:url" content="https://rupumped.github.io/line-losses.html">
<meta property="og:type" content="article">
<!-- Favicon -->
<link rel="icon" href="./favicon.ico" type="image/x-icon">
<!-- Fonts -->
<link rel="preconnect" href="https://fonts.googleapis.com">
<link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
<link href="https://fonts.googleapis.com/css2?family=Open+Sans:wght@400;700&family=Raleway:wght@300;400;700&display=swap" rel="stylesheet">
<!-- Custom CSS -->
<link rel="stylesheet" type="text/css" href="main.css">
<link rel="stylesheet" type="text/css" href="secondary.css">
<link rel="stylesheet" type="text/css" href="blog-posts/blog-post.css">
<!-- KaTeX for Math -->
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.16.11/dist/katex.min.css" integrity="sha384-nB0miv6/jRmo5UMMR1wu3Gz6NLsoTkbqJghGIsx//Rlm+ZU03BU6SQNC66uf4l5+" crossorigin="anonymous">
<script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.11/dist/katex.min.js" integrity="sha384-7zkQWkzuo3B5mTepMUcHkMB5jZaolc2xDwL6VFqjFALcbeS9Ggm/Yr2r3Dy4lfFg" crossorigin="anonymous"></script>
<script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.11/dist/contrib/auto-render.min.js" integrity="sha384-43gviWU0YVjaDtb/GhzOouOXtZMP/7XUzwPTstBeZFe/+rCMvRwr4yROQP43s0Xk" crossorigin="anonymous" onload="renderMathInElement(document.body);"></script>
<script src="https://rupumped.github.io/NicksAPPS/JavaScript/katex-support.js"></script>
</head>
<body>
<header>
<div class="header-content">
<a href="index.html" id="name">NICHOLAS S SELBY</a>
<nav>
<input class="menu-btn" type="checkbox" id="menu-btn">
<label class="menu-icon" for="menu-btn" tabindex="0"><span class="navicon"></span></label>
<ul class="menu">
<li><a href="index.html">HOME</a></li>
<li><a href="about.html">ABOUT</a></li>
<li><a href="selected-work.html">PROJECTS</a></li>
<li><a href="blog.html">BLOG</a></li>
<li><a href="service.html">SERVICE</a></li>
</ul>
</nav>
</div>
</header>
<main>
<h1>I2R Losses in Single- and Three-Phase Distribution Networks</h1>
<p class="date">Posted on March 4, 2023 • 15-minute read</p>
<figure class="cover">
<img src="blog-posts/losses-cover.jpg" alt="Photo CC0: Kistler, Ken. 'Electric Power Lines At Sunset.'">
<figcaption>Photo CC0: Kistler, Ken. "Electric Power Lines At Sunset."</figcaption>
</figure>
<section class="toc">
<h2>Contents</h2>
<div>
<ul>
<li><a href="#Introduction">Introduction</a></li>
<li><a href="#Three-Phase">I2R Losses in a Three-Phase Distribution Network</a></li>
<li><a href="#Single-Phase">Reconfiguring for Single-Phase Distribution</a></li>
<li><a href="#Balance">Defining Balance</a></li>
<li><a href="#Opt-Single">Optimizing the Single-Phase Distribution Network</a></li>
<li><a href="#Energy">Extension to Energy</a></li>
<li><a href="#Intuition">Developing an Intuition</a></li>
<li><a href="#Additional">Additional Considerations</a></li>
</ul>
</div>
</section>
<section id="Introduction">
<h2>Introduction</h2>
<p>Power line losses are a significant concern in mini-grid systems. These losses not only impact the efficiency and reliability of the system, but also increase the overall cost of electricity for end-users. One of the key factors that contributes to power losses in distribution networks is load unbalance. When loads are unevenly distributed across the phases of a three-phase network, the resulting current imbalance can cause additional losses in the system.</p>
<p>In this essay, we will explore the impact of load unbalance on power losses in both three-phase and single-phase distribution networks for mini-grids. Specifically, we will answer the question: "How unbalanced do loads on a three-phase distribution network have to be to warrant reconfiguring the network to be single-phase?" By analyzing the theoretical and practical implications of load unbalance, we will gain insights on the most optimal configuration for mini-grids in terms of minimizing power losses and improving energy efficiency. This analysis will help mini-grid operators and designers make informed decisions about the configuration of distribution networks, ultimately improving the quality and affordability of electricity for end-users.</p>
</section>
<section id="Three-Phase">
<h2>I2R Losses in a Three-Phase Distribution Network</h2>
<img class="img-center" style="width: 12cm;" src="blog-posts/losses-3p.jpg" alt="Diagram of a three-phase distribution network">
<p>Consider a three-phase AC distribution network in a wye configuration with four cables: Line 1, Line 2, Line 3, and Neutral. Assume each conductor has the same resistance, \(\mathrm{R} > 0\), and each of the three lines, \(k\), is supplying a load with current \(i_k \geq 0\). The current flowing back to the source along the neutral is:</p>
<div class="equation">
\begin{equation}
i_{\mathrm{N},3\mathrm{p}}=\sqrt{i_1^2+i_2^2+i_3^2-i_1i_2-i_1i_3-i_2i_3}
\end{equation}
</div>
<p>The power lost along a conductor due to the current, \(i\), being impeded by resistance \(\mathrm{R}\) is given by</p>
<div class="equation">
\begin{equation}
\mathrm{P}=i^2\mathrm{R}
\end{equation}
<!-- (2) \label{eq:i2r} -->
</div>
<p>Therefore, the power loss in the above three-phase network is given by the sum of the power losses along each line:</p>
<div class="equation">
\begin{equation}
\begin{array}{rl}
\mathrm{P}_{3\mathrm{p}} & =i_1^2\mathrm{R}+i_2^2\mathrm{R}+i_3^2\mathrm{R}+i_{\mathrm{N},3\mathrm{p}}^2\mathrm{R} \\
& =\left(2\left(i_1^2+i_2^2+i_3^2\right)-i_1i_2-i_1i_3-i_2i_3\right)\mathrm{R}
\end{array}
\end{equation}
<!-- (3) \label{eq:p3} -->
</div>
</section>
<section id="Single-Phase">
<h2>Reconfiguring for Single-Phase Distribution</h2>
<img class="img-center" style="width: 12cm;" src="blog-posts/losses-1p.jpg" alt="Diagram of a three-phase distribution network converted to a single-phase network with three line conductors">
<p>Now consider a single-phase distribution network made by connecting Lines 1, 2, and 3 of the above three-phase network. Lines 1, 2, and 3 now act as a single conductor with resistance \(\mathrm{R}/3\) carrying current \(\sum_{k=1}^3 i_k\). Assuming the AC power source was reconfigured to output single-phase AC along the three conductors, Kirchhoff's Current Law dictates the current along the neutral is equal to the current along the new, combined line:</p>
<div class="equation">
\begin{equation}
i_{\mathrm{N},1\mathrm{p}}=\sum_{k=1}^3 i_k
\end{equation}
</div>
<p>Applying Eq. (3), the power loss in this single-phase network can be expressed:</p>
<div class="equation">
\begin{equation}
\begin{array}{rl}
\mathrm{P}_{1\mathrm{p}} & = \left(\sum_{k=1}^3 i_k\right)^2\frac{\mathrm{R}}{3}+\left(\sum_{k=1}^3 i_k\right)^2\mathrm{R}\\
& =\frac{4}{3}\left(i_1+i_2+i_3\right)^2\mathrm{R}
\end{array}
\end{equation}
<!-- (5) \label{eq:p1} -->
</div>
<p>If \(\mathrm{P}_{1\mathrm{p}}< \mathrm{P}_{3\mathrm{p}}\), then total power loss would be reduced by reconfiguring the original three-phase network to be single-phase using the method described above. This occurs when:</p>
<div class="equation">
\begin{equation}
\begin{array}{rcl}
\mathrm{P}_{1\mathrm{p}} & < & \mathrm{P}_{3\mathrm{p}} \\
\frac{4}{3}\left(i_1+i_2+i_3\right)^2\mathrm{R} & < & \left(2\left(i_1^2+i_2^2+i_3^2\right)-i_1i_2-i_1i_3-i_2i_3\right)\mathrm{R} \\
11\left(i_1i_2+i_1i_3+i_2i_3\right) & < & 2\left(i_1^2+i_2^2+i_3^2\right)
\end{array}
\end{equation}
<!-- (6) \label{eq:p1ltp3} -->
</div>
</section>
<section id="Balance">
<h2>Defining Balance</h2>
<p>Note that the above inequality has no dependence on resistance \(\mathrm{R}\) if \(\mathrm{R}\neq 0\). We can define a unitless measure, \(\beta\), to quantify how well the load is balanced in the three-phase network:</p>
<div class="equation">
\begin{equation}
\beta \triangleq \begin{cases}
\frac{i_1i_2+i_1i_3+i_2i_3}{i_1^2+i_2^2+i_3^2} & \text{if max}\left(i_1, i_2, i_3\right)\neq 0 \\
1 & \text{otherwise}
\end{cases}
\end{equation}
</div>
<p>When the loads are perfectly balanced, i.e. \(i_1=i_2=i_3\), \(\beta=1\), and as the currents become further from equal, \(\beta\) decreases to \(0\). Note that \(\beta\in [0,1]\ \forall i_1,i_2,i_3\geq 0\).</p>
<p>Rewriting Eq. (6) assuming that at least one of the currents is nonzero yields \(\beta < 2/11\). Therefore, it can be said that, if the loads are unbalanced such that \(\beta < 2/11\), the line losses in the above single-phase distribution network would be less than those in the original three-phase network.</p>
</section>
<section id="Opt-Single">
<h2>Optimizing the Single-Phase Distribution Network</h2>
<img class="img-center" style="width: 12cm;" src="blog-posts/losses-1ps.jpg" alt="Diagram of a three-phase distribution network converted into a single-phase network with two line conductors">
<p>Now consider a single-phase distribution network that has been optimized by moving the load originally on Line 3 to Line 1 or 2, then connecting Line 1 to Line 2 and Line 3 to Neutral. Thus, we have a single-phase network with current \(i_{1\mathrm{p}^*}=\sum_{k=1}^3 i_k\) and total resistance \(\mathrm{R}\). Applying Eq. (2), we can compute the power line loss for this new network:</p>
<div class="equation">
\begin{equation}
\begin{array}{rl}
\mathrm{P}_{1\mathrm{p}^*} & =\left(\sum_{k=1}^3 i_k\right)^2\mathrm{R} \\
& =\left(i_1+i_2+i_3\right)^2\mathrm{R}
\end{array}
\end{equation}
<!-- (8) \label{eq:p1s} -->
</div>
<p>If \(\mathrm{P}_{1\mathrm{p}^*} < \mathrm{P}_{3\mathrm{p}}\), then total power loss would be reduced by reconfiguring the original three-phase network into the optimized single-phase network. This occurs when:</p>
<div class="equation">
\begin{equation}
\begin{array}{rcl}
\mathrm{P}_{1\mathrm{p}^*} & < & \mathrm{P}_{3\mathrm{p}} \\
\left(i_1+i_2+i_3\right)^2\mathrm{R} & < & \left(2\left(i_1^2+i_2^2+i_3^2\right)-i_1i_2-i_1i_3-i_2i_3\right)\mathrm{R} \\
3\left(i_1i_2+i_1i_3+i_2i_3\right) & < & i_1^2+i_2^2+i_3^2
\end{array}
\end{equation}
<!-- (9) \label{eq:beta-gt-2}-->
</div>
<p>or \(\beta < 1/3\).</p>
</section>
<section id="Energy">
<h2>Extension to Energy</h2>
<p>Redefine current \(i\) to be a function of time, \(t\): \(i(t)\). Thus, the currents \(i_1\), \(i_2\), and \(i_3\) are now functions of time: \(i_1(t)\), \(i_2(t)\), and \(i_3(t)\). Substituting \(i(t)\) into Eq. (2) similarly redefines power loss to be a function of time as well:</p>
<div class="equation">
\begin{equation}
\mathrm{P}(t)=i^2(t)\mathrm{R}
\end{equation}
</div>
<p>The energy lost due to power line losses \(\mathrm{P}(t)\) between times \(0\) and \(\mathrm{T}\) is:</p>
<div class="equation">
\begin{equation}
\mathrm{E}=\int_0^\mathrm{T} \mathrm{P}(t)dt
\end{equation}
<!-- (11) \label{eq:energy} -->
</div>
<p>Applying Eq. (11) to Eqs. (3), (5), and (8), we can compute the energy lost due to power line losses between times \(0\) and \(\mathrm{T}\) for the three-phase, single-phase, and optimized single-phase networks, respectively, as:</p>
<div class="equation">
\begin{equation}
\begin{cases}
\mathrm{E}_{3\mathrm{p}} &= \int_0^\mathrm{T} \left(\left(2\left(i_1^2(t)+i_2^2(t)+i_3^2(t)\right) -i_1(t)i_2(t)-i_1(t)i_3(t)-i_2(t)i_3(t)\right)\mathrm{R}\right)dt\\
\mathrm{E}_{1\mathrm{p}} &= \int_0^\mathrm{T} \left(\frac{4}{3}\left(i_1(t)+i_2(t)+i_3(t)\right)^2\mathrm{R}\right)dt\\
\mathrm{E}_{1\mathrm{p}^*}&= \int_0^\mathrm{T} \left(\left(i_1(t)+i_2(t)+i_3(t)\right)^2\mathrm{R}\right)dt
\end{cases}
\end{equation}
</div>
<p>Just as with the power inequalities, if \(\mathrm{E}_{1\mathrm{p}} < \mathrm{E}_{3\mathrm{p}}\) or \(\mathrm{E}_{1\mathrm{p}^*} < \mathrm{E}_{3\mathrm{p}}\), reconfiguring the network from three-phase to single-phase or optimized single-phase, respectively, would reduce energy loss during the period from time \(0\) to \(\mathrm{T}\). As an example, consider the comparison of the three-phase system to the optimized single-phase system:</p>
<div class="equation">
\begin{equation}
\begin{array}{rcl}
\mathrm{E}_{1\mathrm{p}^*} & < & \mathrm{E}_{3\mathrm{p}} \\
\int_0^\mathrm{T} 3\left(i_1(t)i_2(t)+i_1(t)i_3(t)+i_2(t)i_3(t)\right)dt & < & \int_0^\mathrm{T} \left(i_1^2(t)+i_2^2(t)+i_3^2(t)\right)dt
\end{array}
\end{equation}
</div>
<p>Note that, as before, the inequality does not depend on resistance \(\mathrm{R}\) and can be summarized using a single, unitless measure:</p>
<div class="equation">
\begin{equation}
\Beta \triangleq \begin{cases}
\frac{\int_0^\mathrm{T}\left(i_1(t)i_2(t)+i_1(t)i_3(t)+i_2(t)i_3(t)\right)dt}{\int_0^\mathrm{T}\left(i_1^2(t)+i_2^2(t)+i_3^2(t)\right)dt} & \text{if } \exists_t \left(\mathrm{max}\left(i_1(t), i_2(t), i_3(t)\right)\neq 0\right) \\
1 & \text{otherwise}
\end{cases}
\end{equation}
<!-- (14) \label{eq:phi_e} -->
</div>
<p>Rewriting the energy inequalities using \(\Beta\) yields the same numeric cut-offs for \(\Beta\) as \(\beta\):</p>
<ul>
<li>If \(\Beta < 2/11\), the energy lost due to power line losses between times \(0\) and \(\mathrm{T}\) would be reduced by reconfiguring the three-phase system as a single-phase system.</li>
<li>If \(\Beta < 1/3\), the energy lost due to power line losses between times \(0\) and \(\mathrm{T}\) would be reduced by reconfiguring the three-phase system as an optimized single-phase system.</li>
</ul>
<p>The discrete-time approximation of \(\Beta\) can be computed with Riemann sums:</p>
<div class="equation">
\begin{equation}
\Beta \approx \begin{cases}
\frac{\sum_{\tau=1}^\mathrm{M}\left(i_1(t_\tau)i_2(t_\tau)+i_1(t_\tau)i_3(t_\tau)+i_2(t_\tau)i_3(t_\tau)\right)\Delta t(t_\tau)}{\sum_{\tau=1}^\mathrm{M}\left(i_1^2(t_\tau)+i_2^2(t_\tau)+i_3^2(t_\tau)\right)\Delta t(t_\tau)} & \text{if } \exists_\tau \left(\mathrm{max}\left(i_1(t_\tau), i_2(t_\tau), i_3(t_\tau)\right)\neq 0\right) \\
1 & \text{otherwise}
\end{cases}
\end{equation}
</div>
<p>where \(\Delta t(t_\tau)\) is the sampling period at time \(t_\tau\), the time at sample \(\tau\in \{1, 2, ..., \mathrm{M}\}\), such that \(t_1=0\) and \(t_\mathrm{M}=\mathrm{T}\), where \(\mathrm{M}\) is the number of samples. Note that \(\Delta t\) need not be constant over time. The cut-off values of \(2/11\) and \(1/3\) do not change.</p>
</section>
<section id="Intuition">
<h2>Developing an Intuition</h2>
<figure>
<img class="img-center" style="width: 20cm;" src="blog-posts/losses-intuition.jpg" alt="Plots show how much load imbalance is required before conversion to a single-phase network would reduce line losses">
<figcaption>These plots show how much load imbalance is required before conversion to a single-phase network would reduce line losses. Relative load sizes on Lines 2 and 3 that fall within the green domain minimize line losses with a three-phase network, and those that fall within the red domain minimize line-losses with a single-phase network. The left plot shows this relationship for a single-phase network with three line conductors and one neutral conductor, and the right plot for two line conductors and two neutral conductors.</figcaption>
</figure>
<p>Without loss of generality, let Line 1 carry the largest load, i.e. \(i_1\geq i_2\) and \(i_1\geq i_3\). Scale the loads on Lines 2 and 3 to be expressed as a fraction of the load on Line 1, i.e. \(0\leq i_2/i_1\leq 1\) and \(0\leq i_3/i_1\leq 1\). We can use these two scaled currents, \(i_2/i_1\) and \(i_3/i_1\), to visualize the domain of relative loads for which a single- or three-phase network would minimize line losses. The above figure illustrates this.</p>
<p>The amount of load imbalance required for the line losses of a single-phase distribution network to be less than those of a corresponding three-phase network is significant. For example, if the load on Line 2 is 40% of the size of the load on Line 1, even a system in which there was no load on Line 3 would still have less line losses with a three-phase distribution network than a corresponding single-phase network.</p>
<p>For most distribution networks, <b>three-phase networks will usually be more efficient at reducing I2R losses than single-phase networks.</b> In three-phase networks, the neutral conductor carries only the unbalanced current, which is typically smaller than the currents in the lines. In a single-phase distribution network, there is no balancing of loads, so the current flowing along the return path must be equal to the current flowing along the line. The only time line losses are reduced by reconfiguring a three-phase network into a single-phase network is when the benefit gained from the neutral conductor in a three-phase network carrying only the unbalanced current is outweighed by the additional resistance on the lines from not combining them into a single conductor. From Eq. (9), this only happens when \(\beta < 1/3\). Even for Renewvia's smaller networks that are harder to consistently balance, \(\beta\) tends to be greater than 0.5, and \(\Beta\) tends to be greater than 0.7, significantly greater than the cut-off value of 1/3.</p>
</section>
<section id="Additional">
<h2>Additional Considerations</h2>
<p>This essay has only addressed the question of when the line losses of a three-phase distribution network are greater than those of a single-phase network with the same four conductors. However, several additional considerations exist when considering which system will lead to a more efficient mini-grid. For example, in the power house, batteries are often bussed to enable equal current draw from each module. Battery inverters and charge controllers are connected to the DC bus rather than directly to individual batteries. For most calculations, bus bars are considered to have negligible resistance. However, in three-phase systems, different battery inverters are responsible for different phases, so the resistances between one battery and different battery inverters can be different.</p>
<img class="img-center" style="width: 12cm;" src="blog-posts/losses-busbar.jpg" alt="An electrical diagram of two batteries connected to an inverter">
<p>Consider the above diagram of two batteries connected to one of the battery inverters in a three-phase system. For simplicity, assume this system has only three battery inverters, one for each phase, and only one of the battery inverters is currently outputting power to a load. The ratio of the current output of the two batteries is inversely proportional to the ratio of the resistances between the batteries and the inverter:</p>
<div class="equation">
\begin{equation}
\frac{i_1}{i_2}=\frac{\mathrm{R}_2}{\mathrm{R}_1}
\end{equation}
</div>
<p>Note that, even if the bus bar resistance is small, the ratio of the currents depends only on the <i>ratio</i> of the resistances. Thus, if there is twice as much conductor between the battery inverter and the first battery as between the battery inverter and the second battery, then the second battery will drain twice as fast as the first. If this condition persists, because the batteries are bussed, current will eventually flow from the first battery to the second due to the drop in battery voltage as state-of-charge decreases. This flow will likely overload its battery fuse due to the low absolute resistance between the batteries on the bus bar. Therefore, even though the I2R losses of a three-phase network may be less than those of a single-phase network, the additional OPEX required to replace the battery fuses in an unbalanced three-phase system should be considered.</p>
<p>From a design perspective, most single-phase distribution networks have only two conductors: line and neutral. If the mini-grid distribution network is not already built, the distribution engineer must consider the additional CAPEX involved with running two additional conductors. However, a single-phase network with only two conductors will always have line losses greater than or equal to a three-phase network. This can be shown by computing the power loss of said single-phase system to be \(\mathrm{P}_{1\mathrm{p}2}=2\left(i_1+i_2+i_3\right)^2\mathrm{R}\) and noting that \(\mathrm{P}_{1\mathrm{p}2} < \mathrm{P}_{3\mathrm{p}}\) if and only if \(i_1i_2+i_1i_3+i_2i_3 < 0\) or \(\beta < 0\), which is impossible for nonnegative currents. It is perhaps easier to inspect that, in the worst case of only one line in a three-phase network being loaded, the three-phase network is acting as a single-phase network.</p>
<p>Other considerations include:</p>
<ul>
<li>The CAPEX of any additional battery inverters necessary for a three-phase network</li>
<li>The superior efficiency of three-phase diesel backup generators compared to single-phase ones</li>
<li>The ability to use battery inverters of single-phase systems to collaborate to fulfill the total load</li>
<li>The existence of three-phase loads, which may require the purchase of additional transformers if the grid is single-phase</li>
<li>Three-phase systems with only one battery inverter per phase have no backup mechanisms in the event that a battery inverter fails</li>
</ul>
</section>
</main>
<footer>
<div id="top-footer">
<div id="footer-socials">
<a aria-label="My LinkedIn profile" rel="external" href="https://www.linkedin.com/in/nicholas-selby-5278b334/" target="_blank">
<svg viewBox="0 0 448 512" xmlns="http://www.w3.org/2000/svg" aria-labelledby="LinkedIn-title LinkedIn-desc">
<title id="LinkedIn-title">LinkedIn</title>
<desc id="LinkedIn-desc">LinkedIn icon</desc>
<path d="m416 32h-384.1c-17.6 0-31.9 14.5-31.9 32.3v383.4c0 17.8 14.3 32.3 31.9 32.3h384.1c17.6 0 32-14.5 32-32.3v-383.4c0-17.8-14.4-32.3-32-32.3zm-280.6 384h-66.4v-213.8h66.5v213.8zm-33.2-243c-21.3 0-38.5-17.3-38.5-38.5s17.2-38.5 38.5-38.5c21.2 0 38.5 17.3 38.5 38.5 0 21.3-17.2 38.5-38.5 38.5zm282.1 243h-66.4v-104c0-24.8-.5-56.7-34.5-56.7-34.6 0-39.9 27-39.9 54.9v105.8h-66.4v-213.8h63.7v29.2h.9c8.9-16.8 30.6-34.5 62.9-34.5 67.2 0 79.7 44.3 79.7 101.9z"/>
</svg>
</a>
<a aria-label="My Google Scholar Profile" rel="external" href="https://scholar.google.com/citations?user=SKcs1pEAAAAJ" target="_blank">
<svg viewBox="0 0 448 512" xmlns="http://www.w3.org/2000/svg" aria-labelledby="Google-Scholar-title Google-Scholar-desc">
<title id="Google-Scholar-title">Google Scholar</title>
<desc id="Google-Scholar-desc">Google Scholar icon</desc>
<path d="m48 32c-26.5 0-48 21.5-48 48v352c0 26.5 21.5 48 48 48h352c26.5 0 48-21.5 48-48v-352c0-26.5-21.5-48-48-48zm140.69531 64h178.24024l-16.47071 12.85938v22.69335c6.1159.78532 5.46875 4.46019 5.46875 8.85743v107.22265c0 4.96563-4.06205 9.0293-9.02734 9.0293h-3.32422c-4.96563 0-9.0293-4.06403-9.0293-9.0293v-107.22265c0-4.40781-.63995-8.08659 5.52149-8.86133v-14.26563l-47.42383 38.89258c.54795 1.01573 1.06775 1.6675 1.5625 2.51953 4.16684 7.37735 6.28906 16.54919 6.28906 27.75977 0 8.59325-1.43204 16.31497-4.33593 23.13086-2.89082 6.81736-6.4066 12.38451-10.50782 16.67383-4.10119 4.30429-8.21545 8.23614-12.33008 11.77734-4.11326 3.54813-7.62906 7.24664-10.50781 11.08789-2.90139 3.82814-4.34765 7.78747-4.34765 11.88867 0 4.10814 1.87594 8.28085 5.61328 12.48633 3.72386 4.2186 8.30529 8.30635 13.72265 12.34375 5.42938 4.01043 10.84634 8.46278 16.26368 13.30664 5.42904 4.83584 9.98706 11.06107 13.71093 18.62109 3.75047 7.58518 5.625 15.93732 5.625 25.11719 0 12.10928-3.08756 23.04599-9.24609 32.7793-6.17204 9.69317-14.21877 17.42887-24.10156 23.09961-9.90904 5.70939-20.50722 10.00631-31.8086 12.91015-11.32797 2.87733-22.56501 4.32227-33.78906 4.32227-7.08423 0-14.23265-.5475-21.42187-1.66602-7.21219-1.12-14.43823-3.09828-21.7168-5.88476-7.29167-2.80655-13.75106-6.25079-19.34961-10.39063-5.61205-4.09467-10.12984-9.38173-13.59375-15.82031-3.46392-6.43852-5.18359-13.67857-5.18359-21.71875 0-9.53863 2.65694-18.38532 7.98242-26.63476 5.32512-8.18928 12.3823-15.02552 21.1582-20.44141 15.31325-9.52549 39.33645-15.4113 72.03125-17.63672-7.47326-9.34283-11.22266-18.13975-11.22266-26.36914 0-4.68271 1.22186-9.69984 3.64454-15.11719-3.90652.54687-7.9286.85352-12.03125.85352-17.56273 0-32.39541-5.70952-44.43946-17.20703-12.04402-11.47856-18.05859-25.8462-18.05859-43.2168 0-1.81599.05219-3.41866.18164-5.18945h-71.378907zm31.37891 38.33398c-11.19892 0-19.79186 4.02906-25.78125 12.07032-5.9879 8.02012-8.98438 17.73465-8.98438 29.14648 0 9.71981 1.64061 19.61644 4.92188 29.70703 3.2681 10.08548 8.62123 19.08948 16.12109 27.02539 7.47324 7.95594 16.17192 11.92578 26.05469 11.92578 11.01591 0 19.62286-3.68944 25.79492-11.07226 6.14505-7.37043 9.23047-16.67343 9.23047-27.87891 0-9.54517-1.6264-19.537-4.89453-29.98633-3.25608-10.46247-8.68505-19.90262-16.26367-28.30664-7.55234-8.4321-16.30333-12.63086-26.19922-12.63086zm18.51758 172.08008c-8.03469 0-15.93905.71228-23.69922 2.13086-7.76017 1.37996-15.41675 3.70986-22.96875 6.99805-7.57861 3.26958-13.69819 8.07499-18.35938 14.42969-4.68784 6.34667-7.01953 13.82169-7.01953 22.42187 0 8.19582 2.05913 15.50628 6.17383 21.87305 4.10265 6.32768 9.51795 11.28394 16.25 14.83203 6.73201 3.55471 13.78885 6.23613 21.1582 8 7.3828 1.75184 14.89682 2.66406 22.55274 2.66406 15.14262 0 28.17578-3.41115 39.11328-10.22851 10.91054-6.8159 16.38086-17.34206 16.38086-31.54883 0-2.98812-.41689-5.93201-1.23828-8.80274-.85861-2.90389-1.70365-5.3892-2.53711-7.45312-.83345-2.02525-2.42173-4.4673-4.76563-7.2793-2.33081-2.81861-4.11569-4.90255-5.31445-6.28906-1.2239-1.43208-3.51215-3.47703-6.88672-6.17969-3.34426-2.71375-5.48032-4.40548-6.43164-5.03125h-.00195c-.93771-.65762-3.3717-2.4355-7.29102-5.32031-3.9189-2.91084-6.06651-4.45407-6.44531-4.63672-2.05736-.37077-4.94607-.58008-8.66992-.58008z"/>
</svg>
</a>
<a aria-label="My GitHub profile" rel="external" href="https://github.com/rupumped" target="_blank">
<svg viewBox="0 0 448 512" xmlns="http://www.w3.org/2000/svg" aria-labelledby="GitHub-title GitHub-desc">
<title id="GitHub-title">GitHub</title>
<desc id="GitHub-desc">GitHub icon</desc>
<path d="M400 32H48C21.5 32 0 53.5 0 80v352c0 26.5 21.5 48 48 48h352c26.5 0 48-21.5 48-48V80c0-26.5-21.5-48-48-48zM277.3 415.7c-8.4 1.5-11.5-3.7-11.5-8 0-5.4.2-33 .2-55.3 0-15.6-5.2-25.5-11.3-30.7 37-4.1 76-9.2 76-73.1 0-18.2-6.5-27.3-17.1-39 1.7-4.3 7.4-22-1.7-45-13.9-4.3-45.7 17.9-45.7 17.9-13.2-3.7-27.5-5.6-41.6-5.6-14.1 0-28.4 1.9-41.6 5.6 0 0-31.8-22.2-45.7-17.9-9.1 22.9-3.5 40.6-1.7 45-10.6 11.7-15.6 20.8-15.6 39 0 63.6 37.3 69 74.3 73.1-4.8 4.3-9.1 11.7-10.6 22.3-9.5 4.3-33.8 11.7-48.3-13.9-9.1-15.8-25.5-17.1-25.5-17.1-16.2-.2-1.1 10.2-1.1 10.2 10.8 5 18.4 24.2 18.4 24.2 9.7 29.7 56.1 19.7 56.1 19.7 0 13.9.2 36.5.2 40.6 0 4.3-3 9.5-11.5 8-66-22.1-112.2-84.9-112.2-158.3 0-91.8 70.2-161.5 162-161.5S388 165.6 388 257.4c.1 73.4-44.7 136.3-110.7 158.3zm-98.1-61.1c-1.9.4-3.7-.4-3.9-1.7-.2-1.5 1.1-2.8 3-3.2 1.9-.2 3.7.6 3.9 1.9.3 1.3-1 2.6-3 3zm-9.5-.9c0 1.3-1.5 2.4-3.5 2.4-2.2.2-3.7-.9-3.7-2.4 0-1.3 1.5-2.4 3.5-2.4 1.9-.2 3.7.9 3.7 2.4zm-13.7-1.1c-.4 1.3-2.4 1.9-4.1 1.3-1.9-.4-3.2-1.9-2.8-3.2.4-1.3 2.4-1.9 4.1-1.5 2 .6 3.3 2.1 2.8 3.4zm-12.3-5.4c-.9 1.1-2.8.9-4.3-.6-1.5-1.3-1.9-3.2-.9-4.1.9-1.1 2.8-.9 4.3.6 1.3 1.3 1.8 3.3.9 4.1zm-9.1-9.1c-.9.6-2.6 0-3.7-1.5s-1.1-3.2 0-3.9c1.1-.9 2.8-.2 3.7 1.3 1.1 1.5 1.1 3.3 0 4.1zm-6.5-9.7c-.9.9-2.4.4-3.5-.6-1.1-1.3-1.3-2.8-.4-3.5.9-.9 2.4-.4 3.5.6 1.1 1.3 1.3 2.8.4 3.5zm-6.7-7.4c-.4.9-1.7 1.1-2.8.4-1.3-.6-1.9-1.7-1.5-2.6.4-.6 1.5-.9 2.8-.4 1.3.7 1.9 1.8 1.5 2.6z"/>
</svg>
</a>
<a aria-label="My RSS feed" href="rss.xml">
<svg fill="#000000" height="800px" width="800px" version="1.1" id="Layer_1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" viewBox="-143 145 512 512" xml:space="preserve" aria-labelledby="RSS-title RSS-desc">
<title id="RSS-title">RSS</title>
<desc id="RSS-desc">RSS icon</desc>
<path d="M329,145h-432c-22.1,0-40,17.9-40,40v432c0,22.1,17.9,40,40,40h432c22.1,0,40-17.9,40-40V185C369,162.9,351.1,145,329,145z M43.1,518.7c-6.2,6.2-14.7,9.9-24.1,9.9c-9.4,0-17.8-3.8-24-9.9c-6.2-6.2-10-14.6-10-23.9c0-9.4,3.8-17.8,10-24s14.6-10,24-10 c9.4,0,17.9,3.8,24,10c6.2,6.2,10,14.6,10,24C53,504.2,49.2,512.6,43.1,518.7z M104.8,529c-0.1-32.1-12.5-62.3-35.1-84.9 c-22.6-22.6-52.8-35.2-84.7-35.2V360c46.6,0,88.7,19,119.3,49.6c30.6,30.6,49.5,72.8,49.6,119.4H104.8z M192,529 c-0.1-114.2-92.8-207.1-206.9-207.1V273c70.6,0,134.5,28.7,180.8,75.1c46.3,46.4,75,110.3,75.1,180.9H192z"/>
</svg>
</a>
</div>
<div id="footer-initials">
NSS
</div>
<div id="footer-nav">
<a href="mailto:nicholas.selby@renewvia.com">CONTACT</a>
<a href="index.html">HOME</a>
<a href="sitemap.xml">SITEMAP</a>
</div>
</div>
<p id="copyright">
© 2018–2025 This work by <a href="https://rupumped.github.io" property="cc:attributionName" rel="cc:attributionURL">Nicholas S. Selby</a> is licensed under a <a rel="license" target="_blank" href="http://creativecommons.org/licenses/by/4.0/">Creative Commons Attribution 4.0 International License</a>. Feel free to fork the <a rel="external" target="_blank" href="https://github.com/rupumped/rupumped.github.io">source code</a> from GitHub and create your own website using this template.
</p>
</footer>
</body>
</html>