# Zero-Shot Metric Depth with a Field-of-View Conditioned Diffusion Model

Saurabh Saxena†, Junhwa Hur‡, Charles Herrmann‡, Deqing Sun‡, David J. Fleet†
†Google DeepMind ‡Google Research
<EMAIL_ADDRESS>DF is also
affiliated with the University of Toronto and the Vector Institute.

###### Abstract

While methods for monocular depth estimation have made significant strides on
standard benchmarks, zero-shot metric depth estimation remains unsolved.
Challenges include the joint modeling of indoor and outdoor scenes, which
often exhibit significantly different distributions of RGB and depth, and the
depth-scale ambiguity due to unknown camera intrinsics. Recent work [5]
proposed a specialized multi-head architecture for jointly modeling indoor and
outdoor scenes. In contrast, we advocate a generic, task-agnostic diffusion
model, with several advancements such as log-scale depth parameterization to
enable joint modeling of indoor and outdoor scenes, conditioning on the field-
of-view (FOV) to handle scale ambiguity and synthetically augmenting FOV
during training to generalize beyond the limited camera intrinsics in training
datasets. Furthermore, by employing a more diverse training mixture than is
common, and an efficient diffusion parameterization, our method, Diffusion for
Metric Depth (DMD) achieves a 25% reduction in relative error (REL) on zero-
shot indoor and 33% reduction on zero-shot outdoor datasets over the current
SOTA [5] using only a small number of denoising steps. For an overview see
diffusion-vision.github.io/dmd

##  1 Introduction

Monocular estimation of metric depth in general environments, while useful for
applications such as mobile robotics and autonomous driving, has proven
elusive. The two main barriers stem from (1) the large differences in RGB and
depth distributions one finds in indoor and outdoor datasets, and (2)
intrinsic scale ambiguity in images when one lacks knowledge of camera
intrinsics. Not surprisingly, most current models for monocular depth are
either specific to indoor or outdoor scenes, or, if trained for both, they
estimate scale-invariant depth.

Figure 1: Relative depth error for DMD (ours) compared to ZoeDepth (SOTA) on
eight zero-shot and two in-distribution ($\ast$) benchmarks. DMD outperforms
ZoeDepth by a substantial margin on all benchmarks.

Current models for metric depth are often trained solely on indoor or outdoor
scenes, primarily on a single dataset captured with fixed camera intrinsics
(e.g., with an RGBD camera, or, with RGB+LIDAR for outdoor scenes). Such
models avoid challenges caused by different indoor and outdoor depth
distributions, but at the cost of generality. They also overfit to the camera
intrinsics of the training dataset and do not generalize well to out of
distribution data.

The predominant approach to jointly modeling indoor and outdoor data is to
estimate scale- and shift-invariant depth, rather than metric depth (e.g.,
MiDaS [34]). By normalizing the depth distributions, one brings indoor and
outdoor depth distributions closer, and also avoids the problems of scale
ambiguities in the presence of variable camera intrinsics. Recently there has
been growing interest in bridging these different approaches, training joint
indoor-outdoor models that estimate metric depth. To cope with both indoor and
outdoor domains, ZoeDepth [5] adds two heads to MiDaS [34], one for each
domain, to convert from scale-invariant depth to metric depth.

In this paper we advocate denoising diffusion models for zero-shot metric
depth estimation, with the aid of several key innovations to obtain state of
the art performance. In particular, field-of-view (FOV) augmentation is used
during training to improve generalization to different camera intrinsics, and
FOV conditioning in training and inference helps to resolve intrinsic scale
ambiguities, providing a further boost in performance. Representing depth in
the log domain allocates model capacity in a more balanced way for indoor and
outdoor scenes, improving indoor performance. Finally, we find that
$v$-parameterization in neural network denoising greatly increases inference
speed. The resulting model, dubbed DMD (Diffusion for Metric Depth),
outperforms recently proposed metric depth model, ZoeDepth [5]. In particular,
DMD yields much lower relative depth error than ZoeDepth on all eight out-of-
distribution datasets in [5] when fine-tuned on the same data. Expanding the
training dataset leads to further improvement in performance (Fig. 1).

To summarize, we make the following contributions:

  * •

We introduce DMD, a simple yet effective method for zero-shot metric depth
estimation on general scenes.

  * •

We propose synthetically augmenting the FOV for improved generalization, FOV
conditioning to resolve depth-scale ambiguity and representing depth in log-
scale to better utilize the model’s representation capacity.

  * •

DMD establishes a SOTA on zero-shot metric depth, achieving $25\%$ and $33\%$
lower relative error than ZoeDepth on indoor and outdoor datasets,
respectively, while being efficient, owing to the use of $v$-parameterization
for diffusion.

##  2 Related Work

Monocular depth (fine-tuned and evaluated in-domain). Given the challenges of
learning a joint indoor-outdoor model, most approaches have restricted models
to target a single dataset (either indoor or outdoor) with fixed intrinsics.
In this setting, great progress has been made with advancements in specialized
architectures [16, 15] such as the use of binning [8, 18, 1, 3, 30] or loss
functions [28, 15] that are suited for this task. [2] proposed combining
multiple training datasets with variable intrinsics by normalizing the images
to the same camera intrinsic.

Joint indoor-outdoor models. To train joint indoor-outdoor models, one can
mitigate the difficulty of learning diverse scene statistics by estimating
scale- and shift-invariant depth instead. MiDaS [34] trains their model on
diverse indoor-outdoor datasets and demonstrates good generalization to
various unseen datasets. However, they do not provide metric depth. DPT [35]
leverages this for pre-training and further fine-tunes separately for metric
depth on NYU and KITTI. ZoeDepth [5] proposes adding a mixture-of-experts
head, supervised by scene type, on top of a similarly pre-trained model,
thereby handling indoor and outdoor scenes. In contrast, our model, DMD, uses
a relatively generic framework, without domain-specific architectural
components.

Intrinsics-conditioned monocular depth. Incorporating camera intrinsics for
depth estimation has been briefly explored in previous work [17, 22]. They
argue that intrinsic-conditioning allows one to train on multiple datasets
with varying intrinsics, but this is only demonstrated with small-scale
experiments. Similar to our method, ZeroDepth [21] introduces an intrinsic-
conditioned metric-scale depth estimator that is trained on large-scale
training datasets.

While conditioning on the input field-of-view, we introduce a novel field-of-
view augmentation scheme that augments training data by cropping or uncropping
to simulate diverse FOVs and provide a large scale study on zero-shot
generalization to both indoor and outdoor domains as well as robustness to
diverse camera intrinsics.

Diffusion for vision. Denoising diffusion models [43, 23] have recently
emerged as a powerful class of generative models. Although initially proposed
for natural image generation [38, 24, 32, 13], they have recently been shown
to be effective for several computer vision tasks such as semantic
segmentation [26], panoptic segmentation [9], optical flow [40] and monocular
depth estimation [26, 14, 40]. Ours is the first demonstration that diffusion
models can also support state-of-the-art zero-shot metric depth estimation for
general indoor or outdoor scenes.

##  3 Diffusion for Metric Depth (DMD)

In what follows we describe DMD (Diffusion for Metric Depth) and the design
decisions to solve these issues. In particular, we cast monocular depth
estimation as a generative RGB-to-depth translation task using denoising
diffusion. To this end we introduce several technical innovations to
conventional diffusion models and training procedures to accommodate zero-
shot, metric depth.

###  3.1 Diffusion models

Diffusion models are probabilistic models that assume a forward process that
gradually transforms a target distribution into a tractable noise
distribution. A learned neural denoiser is trained to reverse this process,
iteratively converting a noise sample to a sample from the target
distribution. They have been shown to be remarkably effective with images and
video, and they have recently begun to see use for dense vision tasks like
segmentation, tracking, optical flow, and depth estimation. They are
attractive as they exhibit strong performance on regression tasks, capturing
posterior uncertainty, without task specific architectures, loss functions and
training procedures.

For DMD we build on the task-agnostic Efficient U-Net architecture from DDVM
[40]. While DDVM used the $\epsilon$-parameterization for training the neural
depth denoiser, here instead we use the ${\bm{v}}$-parameterization [39]. We
find that the ${\bm{v}}$-parameterization yields remarkably efficient
inference, using as few as one or two refinement steps, without requiring
progressive distillation [39].

Under the ${\bm{v}}$-parameterization, the denoising network is given a noisy
target image (in our case a depth map),
$\mathbf{z}_{t}=\alpha_{t}\mathbf{x}+\sigma_{t}{\bm{\epsilon}}$, where
$\mathbf{x}$ is the noiseless target input (depth map),
${\bm{\epsilon}}\sim\mathcal{N}(0,I)$, $t\sim\mathcal{U}(0,1)$,
$\sigma_{t}^{2}=1-\alpha_{t}^{2}$, and $\alpha_{t}>0$ is computed with a pre-
determined noise schedule, and the denoising network predicts
${\bm{v}}\equiv\alpha_{t}{\bm{\epsilon}}-\sigma_{t}\mathbf{x}$. From the
output of the denoising network, i.e.,
${\bm{v}}_{\theta}(\mathbf{z}_{t},{\bm{y}},t)$, where ${\bm{y}}$ is an
optional conditioning signal (RGB image in this case), one obtains an estimate
of $\mathbf{x}$ at step $t$, i.e.,
$\hat{\mathbf{x}}_{t}=\alpha_{t}\mathbf{z}_{t}-\sigma_{t}{\bm{v}}_{\theta}(\mathbf{z}_{t},{\bm{y}},t)$,
and the corresponding estimate of the noise, denoted
$\hat{{\bm{\epsilon}}}_{t}$. Under this parameterization, with a conventional
L2 norm, the training objective is based on the expected ‘truncated SNR
weighting’ loss, i.e.,
$\text{max}(\lVert\mathbf{x}-\hat{\mathbf{x}}_{t}\rVert_{2}^{2},\lVert{\bm{\epsilon}}-\hat{{\bm{\epsilon}}}_{t}\rVert_{2}^{2})$
[39]. Motivated by the superior performance of the L1 loss in training DDVM
[40] compared to the L2, we similarly employ a L1 loss for DMD as well,
yielding the objective

$\mathbb{E}_{\mathbf{x},{\bm{y}},t,{\bm{\epsilon}}}\left[\text{max}(\lVert\mathbf{x}-\hat{\mathbf{x}}_{t}\rVert_{1},\lVert{\bm{\epsilon}}-\hat{{\bm{\epsilon}}}_{t}\rVert_{1})\right]~{}.$
(1)

| Image | Ground Truth |  ZoeDepth [5] | DMD (ours)
---|---|---|---|---
DIODE Indoor |  |  |  |
Hypersim |  |  |  |
ibims-1 |  |  |  |
NYU |  |  |  |
SunRGBD |  |  |  |
Figure 2: Qualitative comparison between our method and ZoeDepth [5] on indoor scenes. Unlike ZoeDepth, our method estimates depths at more accurate scale over diverse datasets. | SUN RGB-D | iBims-1 Benchmark | DIODE Indoor | HyperSim
---|---|---|---|---
Method |  $\delta_{1}$ $\uparrow$ |  REL $\downarrow$ |  RMSE $\downarrow$ |  $\delta_{1}$ $\uparrow$ |  REL $\downarrow$ |  RMSE $\downarrow$ |  $\delta_{1}$ $\uparrow$ |  REL $\downarrow$ |  RMSE $\downarrow$ |  $\delta_{1}$ $\uparrow$ |  REL $\downarrow$ |  RMSE $\downarrow$
BTS [29] | 0.740 | 0.172 | 0.515 | 0.538 | 0.231 | 0.919 | 0.210 | 0.418 | 1.905 | 0.225 | 0.476 | 6.404
AdaBins [3] | 0.771 | 0.159 | 0.476 | 0.555 | 0.212 | 0.901 | 0.174 | 0.443 | 1.963 | 0.221 | 0.483 | 6.546
LocalBins [4] | 0.777 | 0.156 | 0.470 | 0.558 | 0.211 | 0.880 | 0.229 | 0.412 | 1.853 | 0.234 | 0.468 | 6.362
NeWCRFs [50] | 0.798 | 0.151 | 0.424 | 0.548 | 0.206 | 0.861 | 0.187 | 0.404 | 1.867 | 0.255 | 0.442 | 6.017
ZoeD-M12-NK [5] | 0.856 | 0.123 | 0.356 | 0.615 | 0.186 | 0.777 | 0.386 | 0.331 | 1.598 | 0.274 | 0.419 | 5.830
DMD-NK | 0.914 | 0.109 | 0.306 | 0.801 | 0.130 | 0.563 | 0.402 | 0.298 | 1.407 | 0.356 | 0.382 | 5.527
DMD-MIX | 0.930 | 0.091 | 0.275 | 0.859 | 0.118 | 0.447 | 0.380 | 0.291 | 1.292 | 0.497 | 0.318 | 4.394
Table 1: Zero-shot results on unseen indoor datasets (evaluated at pixels with GT depth less 8m for SUN RGB-D, 10m for iBims-1 and DIODE Indoor, and 80m for Hypersim). Best and second-best results are highlighted. Trained on the same data (NYU and KITTI), DMD-NK outperforms ZoeD-M12-NK. With more data (Taskonomy and nuScenes), DMD-MIX outperforms by a much larger margin.  | Image | Ground Truth |  ZoeDepth [5] | DMD (ours)
---|---|---|---|---
DDAD |  |  |  |
DIML Outdoor |  |  |  |
DIODE Outdoor |  |  |  |
KITTI |  |  |  |
Virtual KITTI 2 |  |  |  |
Figure 3: Qualitative comparison between DMD and ZoeDepth [5] on outdoor scenes. Compared with ZoeDepth [5], our method is able to estimate a more accurate depth scale. | Virtual KITTI 2 | DDAD | DIML Outdoor | DIODE Outdoor
---|---|---|---|---
Method |  $\delta_{1}$ $\uparrow$ |  REL $\downarrow$ |  RMSE $\downarrow$ |  $\delta_{1}$ $\uparrow$ |  REL $\downarrow$ |  RMSE $\downarrow$ |  $\delta_{1}$ $\uparrow$ |  REL $\downarrow$ |  RMSE $\downarrow$ |  $\delta_{1}$ $\uparrow$ |  REL $\downarrow$ |  RMSE $\downarrow$
BTS [29] | 0.831 | 0.115 | 5.368 | 0.805 | 0.147 | 7.550 | 0.016 | 1.785 | 5.908 | 0.171 | 0.837 | 10.48
AdaBins [3] | 0.826 | 0.122 | 5.420 | 0.766 | 0.154 | 8.560 | 0.013 | 1.941 | 6.272 | 0.161 | 0.863 | 10.35
LocalBins [4] | 0.810 | 0.127 | 5.981 | 0.777 | 0.151 | 8.139 | 0.016 | 1.820 | 6.706 | 0.170 | 0.821 | 10.27
NeWCRFs [50] | 0.829 | 0.117 | 5.691 | 0.874 | 0.119 | 6.183 | 0.010 | 1.918 | 6.283 | 0.176 | 0.854 | 9.228
ZoeD-M12-NK [5] | 0.850 | 0.105 | 5.095 | 0.824 | 0.138 | 7.225 | 0.292 | 0.641 | 3.610 | 0.208 | 0.757 | 7.569
DMD-NK | 0.872 | 0.093 | 4.828 | 0.842 | 0.122 | 6.740 | 0.544 | 0.300 | 2.522 | 0.162 | 0.627 | 9.577
DMD-MIX | 0.890 | 0.092 | 4.387 | 0.907 | 0.108 | 5.365 | 0.602 | 0.190 | 2.089 | 0.187 | 0.553 | 8.943
Table 2: Zero-shot results on four unseen outdoor datasets. Best and second-
best results are highlighted. Following pre-training, DMD-NK is trained on
NYUv2 and KITTI, while DMD-MIX adds Taxonomy and NuScenes for training. Using
the same fine-tuning data, DMD-NK outperforms ZoeD-M12-NK on all benchmarks
except DIODE Outdoor where it outperforms on REL but is behind on other
metrics. DMD-MIX shows further performance improvements over DMD-NK with the
expanded training mixture.

###  3.2 Joint indoor-outdoor modelling

Training a joint indoor-outdoor model can be difficult because of the large
differences in depth distributions one finds in indoor and outdoor scenes.
Much of the available indoor training data have depths up to $10m$, while
outdoor scenes include ground truth depths up to $80m$. Further, training data
is often lacking the variation in camera intrinsics needed for robustness to
images from different cameras. Rather, many datasets are captured with a fixed
camera. To mitigate these issues we propose three innovations, namely, the use
of log depth, field of view augmentation, and field of view conditioning.

Log depth. Diffusion models usually model data distribution in [-1, 1]. One
might convert metric depth to this range with linear scaling, i.e.,

$d_{\mathrm{lin}}=\mathrm{normalize}(d_{\mathrm{r}}/d_{\mathrm{max}})~{},$ (2)

where $d_{\mathrm{r}}$ is the raw depth in meters, $d_{\mathrm{max}}$ is often
taken to be 80m, to accommodate the usual outdoor depth range, and
$\mathrm{normalize}(d)=\mathrm{clip}(2*d-1,-1,1)$. This, however, allocates
little representation capacity to indoor scenes (where depths are usually less
than $10m$).

Instead, we can allocate more representation capacity to indoor scenes with
log-scaled depth ($d_{\mathrm{log}}$) as the target for inference, i.e.,

$d_{\mathrm{log}}=\mathrm{normalize}\left(\frac{\log(d_{\mathrm{r}}/d_{\mathrm{min}})}{\log(d_{\mathrm{max}}/d_{\mathrm{min}})}\right)~{},$
(3)

where $d_{\mathrm{min}}$ and $d_{\mathrm{max}}$ denote the minimum and maximum
supported depths (e.g., $0.5m$ and $80m$). Empirically, we find log scaling to
be remarkably beneficial.

|  |  |
---|---|---|---
Image | Ground Truth | Prediction with linear-scaling | Prediction with log-scaling
Figure 4: Linearly scaling depth leads to noisy predictions for images with shallow depth. See Section 3.2 for more details. Predicting depth in a log-scale fixes this. Note that here we use a max depth of 5 meters for better visualization.  |  | Indoors | Outdoors
---|---|---|---
| Experiment | NYU | SunRGBD | DIODE Indoor | ibims-1 | Hypersim | KITTI | DIML Outdoor | DIODE Outdoor | Virtual KITTI 2 | DDAD
REL | Linear scaling | 0.082 | 0.108 | 0.324 | 0.146 | 0.398 | 0.056 | 0.467 | 0.630 | 0.092 | 0.122
Log scaling | 0.076 | 0.109 | 0.298 | 0.130 | 0.382 | 0.055 | 0.300 | 0.628 | 0.093 | 0.122
RMS | Linear scaling | 0.340 | 0.314 | 1.526 | 0.612 | 5.693 | 2.516 | 3.126 | 10.129 | 4.788 | 6.288
Log scaling | 0.313 | 0.306 | 1.407 | 0.563 | 5.527 | 2.527 | 2.522 | 9.577 | 4.828 | 6.740
Table 3: Ablation showing that log depth improves quantitative performance on
indoor datasets which is understandable since log-scaling increases the share
of representation capacity allocated to shallow depths.

Field-of-view augmentation. Because datasets for depth estimation often have
little or no variation in the field of view, it is easy for models to over-fit
and thus generalize poorly to images with different camera intrinsics. To
encourage models to generalize well to different fields of view, we propose to
augment training data by cropping or uncropping to simulate diverse FOVs.
While cropping is straightforward, for uncropping it is unclear how best to
pad the enlarged image. Our preliminary experiments used generative uncropping
with Palette [37], however, we found that padding the RGB image with Gaussian
noise (mean-zero, variance 1) works as well, is simpler and more efficient.

For missing ground truth depth with uncropping augmentation, we adopt the
approach in [40], using a combination of near-neighbor in-filling and step-
unrolled denoising during training. It is shown in [40] that this technique is
effective in coping with the inherent distribution shift between training and
testing when ground truth data are noisy or incomplete.

Field-of-view conditioning. Metric depth estimation from a single image is
ill-posed in the presence of unknown camera intrinsics, as depth scales
inversely with the field-of-view. While one might hope that diversifying
camera intrinsics through FOV augmentation will help generalization to
different cameras, we and others [48] observe that it is not sufficient in
itself. FOV augmentations do help simulate some variation in camera
intrinsics, but variations in other factors, like focal length, are hard to
simulate.

As a conditioning signal, to help disambiguate depth scale, we use
$\tan(\theta/2)$, where $\theta$ is the vertical FOV. We explored conditioning
on the horizontal FOV as well, but that did not improve results substantially.

| Image | Ground Truth | DMD-NK | DMD-MIX
---|---|---|---|---
ibims-1 |  |  |  |
DDAD |  |  |  |
Figure 5: Qualitative comparison between DMD-NK (fine-tuned on NYU and KITTI)
and DMD-MIX (fine-tuned on KITTI, NYU, nuScenes, and Taskonomy). DMD-MIX
further improves depth scale estimation as well as fine details on depth
boundaries.

##  4 Experiments

Method | NYU | KITTI
---|---|---
$\delta_{1}$$\uparrow$ |  $\delta_{2}$$\uparrow$ |  $\delta_{3}$ $\uparrow$ |  REL $\downarrow$ |  RMS $\downarrow$ |  $\log_{10}$ $\downarrow$ |  $\delta_{1}$$\uparrow$ |  $\delta_{2}$$\uparrow$ |  $\delta_{3}$$\uparrow$ |  REL $\downarrow$ |  Sq-rel $\downarrow$ |  RMS $\downarrow$ |  RMS log $\downarrow$
Domain-specific models:
BTS [29] | 0.885 | 0.978 | 0.994 | 0.110 | 0.392 | 0.047 | 0.956 | 0.993 | 0.998 | 0.059 | 0.245 | 2.756 | 0.096
DPT [35] | 0.904 | 0.988 | 0.998 | 0.110 | 0.357 | 0.045 | 0.959 | 0.995 | 0.999 | 0.062 | – | 2.573 | 0.092
AdaBins [3] | 0.903 | 0.984 | 0.997 | 0.103 | 0.364 | 0.044 | 0.964 | 0.995 | 0.999 | 0.058 | 0.190 | 2.360 | 0.088
NeWCRFs [50] | 0.922 | 0.992 | 0.998 | 0.095 | 0.334 | 0.041 | 0.974 | 0.997 | 0.999 | 0.052 | 0.155 | 2.129 | 0.079
BinsFormer [30] | 0.925 | 0.989 | 0.997 | 0.094 | 0.330 | 0.040 | 0.974 | 0.997 | 0.999 | 0.052 | 0.151 | 2.098 | 0.079
PixelFormer [1] | 0.929 | 0.991 | 0.998 | 0.090 | 0.322 | 0.039 | 0.976 | 0.997 | 0.999 | 0.051 | 0.149 | 2.081 | 0.077
IEBins [41] | 0.936 | 0.992 | 0.998 | 0.087 | 0.314 | 0.038 | 0.978 | 0.998 | 0.999 | 0.050 | 0.142 | 2.011 | 0.075
MIM [49] | 0.949 | 0.994 | 0.999 | 0.083 | 0.287 | 0.035 | 0.977 | 0.998 | 1.000 | 0.050 | 0.139 | 1.966 | 0.075
DDVM [40] | 0.946 | 0.987 | 0.996 | 0.074 | 0.315 | 0.032 | 0.965 | 0.994 | 0.998 | 0.055 | 0.292 | 2.613 | 0.089
Joint indoor-outdoor models:
ZoeD-M12-NK | 0.953 | 0.995 | 0.999 | 0.077 | 0.277 | 0.033 | 0.966 | 0.993 | 0.996 | 0.057 | 0.204 | 2.362 | 0.087
DMD-NK | 0.944 | 0.986 | 0.996 | 0.076 | 0.313 | 0.033 | 0.964 | 0.994 | 0.999 | 0.055 | 0.219 | 2.527 | 0.087
DMD-MIX | 0.953 | 0.989 | 0.996 | 0.072 | 0.296 | 0.031 | 0.967 | 0.995 | 0.999 | 0.053 | 0.203 | 2.411 | 0.084
Table 4: In-domain results showing better relative error than ZoeDepth for our
models on both the NYU and KITTI datasets. Best results (amongst indoor-
outdoor models only) are bolded. For reference, we provide results for models
trained separately for the indoor and outdoor domains showing that our results
are competitive despite being a more general model.

###  4.1 Training data

DDVM [40] showed that using large amounts of diverse training data is
important for generic models with no task-specific inductive biases. Here we
follow the training strategy proposed in [40], initializing the model with
unsupervised pre-training on ImageNet [12] and Places365 [52], with tasks
proposed in [37]. This is followed by supervised pre-training on ScanNet [11],
SceneNet-RGBD [31], and Waymo [45]. Unlike [40], we also include the DIML
Indoor [10] dataset for more diversity. No FOV augmentation or conditioning is
used for this pre-training stage.

For the final training stage we train on a mixture of NYU [42], Taskonomy
[51], KITTI [19] and nuScenes [7]. At this stage we apply FOV augmentation to
NYU, KITTI and nuScenes, but not Taskonomy as it is large and has substantial
FOV diversity and also add the FOV conditioning. Finally, to enable fair
comparisons with the current SOTA ZoeDepth [5] model, we also train a model
solely from NYU and KITTI, like ZoeDepth.

|  | Indoors | Outdoors
---|---|---|---
Metric | Experiment | NYU | SunRGBD | DIODE Indoor | ibims-1 | Hypersim | KITTI | DIML Outdoor | DIODE Outdoor | Virtual KITTI 2 | DDAD
REL | No FOV cond | 0.081 | 0.116 | 0.316 | 0.18 | 0.400 | 0.057 | 1.257 | 0.613 | 0.100 | 0.121
With FOV cond | 0.076 | 0.109 | 0.298 | 0.130 | 0.382 | 0.055 | 0.300 | 0.628 | 0.093 | 0.122
RMS | No FOV cond | 0.319 | 0.325 | 1.474 | 0.712 | 5.196 | 2.574 | 5.382 | 8.582 | 5.021 | 6.826
With FOV cond | 0.313 | 0.306 | 1.407 | 0.563 | 5.527 | 2.527 | 2.522 | 9.577 | 4.828 | 6.740
Table 5: Depth errors for models trained with and without field-of-view conditioning. Results shows that FOV conditioning provides a substantial boost in performance. DIML Outdoor benefits the most, which is understandable given its large FOV, for which generalization is a major challenge for simple FOV augmentation. |  Indoors |  Outdoors
---|---|---
In-distribution |  |
OOD |  |
Figure 6: Plots showing the effect of perturbing the FOV during inference.
Optimal performance is at or near the true FOV. Performance degrades with
larger perturbation.

###  4.2 Design choices

Denoiser Architecture. We adopt the modifications of the Efficient U-Net [38]
proposed in DDVM [40], with one further modification to support FOV
conditioning. The FOV embedding, like the timestep embedding, is constructed
by first building a sin-cos positional embedding [47] followed by linear
projection. The sum of these two embeddings is used to modulate different
layers of the denoiser backbone using FiLM [33] layers. Following [5] we train
at a resolution of 384$\times$512\. The predicted depth maps are resized to
the ground-truth resolution for evaluation, following prior work [5]. Other
training hyper-parameters such as the batch size and optimizer details are
like those in [40].

Augmentations. In addition to the FOV augmentation (Sec. 3) we use random
horizontal flip augmentation, like many prior works.

Sampler. We use the DDPM [23] sampler with eight denoising steps for indoor
datasets. For outdoor datasets we find that two denoising steps suffice. We
report results using a mean of eight samples, following [40]; as shown in
Table 6, sample averaging leads to improved performance.

Evaluation. We adopt the evaluation protocol of ZoeDepth [5]. We report in-
distribution performance on the NYU [42] and KITTI [19] datasets, and
generalization performance on eight unseen datasets [5], namely, SunRGBD [44],
iBims-1 [44], DIODE Indoor [46], Hypersim [36] for indoors, and Virtual KITTI
2 [6], DDAD [20], DIML Outdoor [10], DIODE Outdoor [46] for outdoors. We
closely follow the evaluation protocol, including depth ranges and cropping,
used in [5] and report results using the standard error and accuracy metrics
that are used in literature.

|  | Indoors | Outdoors
---|---|---|---
Metric | # samples | NYU | SunRGBD | DIODE Indoor | ibims-1 | Hypersim | KITTI | DIML Outdoor | DIODE Outdoor | Virtual KITTI 2 | DDAD
REL | 1 | 0.077 | 0.110 | 0.304 | 0.135 | 0.383 | 0.055 | 0.311 | 0.634 | 0.093 | 0.122
8 | 0.076 | 0.109 | 0.298 | 0.130 | 0.382 | 0.055 | 0.300 | 0.628 | 0.093 | 0.122
RMS | 1 | 0.321 | 0.311 | 1.426 | 0.579 | 5.521 | 2.544 | 2.572 | 9.602 | 4.842 | 6.755
8 | 0.313 | 0.306 | 1.407 | 0.563 | 5.527 | 2.527 | 2.522 | 9.577 | 4.828 | 6.740
Table 6: Averaging multiple samples leads to small but consistent improvement
on both REL and RMS.

###  4.3 Results

Zero-shot. Tables 1 and 2 report zero-shot performance on eight OOD datasets.
DMD-NK permits fair comparison with ZoeDepth, as both are fine-tuned on NYU
and KITTI. With these datasets, DMD outperforms ZoeDepth across all but DIODE
Outdoor, where DMD-NK outperforms ZoeDepth on relative error but not RMSE and
$\delta_{1}$. Importantly, but perhaps not surprisingly, further performance
improvements are obtained by fine-tuning on larger amounts of data. Our
variant trained on a mixture with KITTI, NYU, nuScenes, and Taskonomy, dubbed
DMD-MIX, generalizes much better on OOD test data, establishing an even
stronger new state of the art.

Figures 2 and 3 visualize qualitative comparison between DMD and ZoeDepth on
indoor and outdoor datasets respectively. Our method captures more accurate
metric-scale depth with both indoor and outdoor scenes. Fig. 5 illustrates the
qualitative differences in depth maps from DMD-NK and DMD-MIX. By virtue of
training on a larger dataset, DMD-MIX significantly improves the depth scale
and fine-grained depth details near object boundaries.

In-distribution. Table 4 reports results on the KITTI and NYU datasets. On
KITTI, DMD-MIX outperforms ZoeDepth on all metrics except RMSE where it is
competitive. On NYU DMD-MIX substantially outperforms ZoeDepth on relative
error and is competitive on other metrics. Interestingly, DMD-MIX outperforms
DMD-NK on in-distribution data.

|  | Indoors | Outdoors
---|---|---|---
Metric | Experiment | NYU | SunRGBD | DIODE Indoor | ibims-1 | Hypersim | KITTI | DIML Outdoor | DIODE Outdoor | Virtual KITTI 2 | DDAD
REL | No FOV aug or cond | 0.074 | 0.124 | 0.337 | 0.180 | 0.479 | 0.055 | 1.399 | 0.615 | 0.095 | 0.116
W/ FOV aug and cond | 0.076 | 0.109 | 0.298 | 0.130 | 0.382 | 0.055 | 0.300 | 0.628 | 0.093 | 0.122
RMS | No FOV aug or cond | 0.310 | 0.348 | 1.535 | 0.722 | 5.247 | 2.597 | 5.919 | 8.529 | 4.874 | 6.476
W/ FOV aug and cond | 0.313 | 0.306 | 1.407 | 0.563 | 5.527 | 2.527 | 2.522 | 9.577 | 4.828 | 6.740
Table 7: Ablation showing that training without FOV augmentations and
conditioning hurts generalization to out-of-domain data due to overfitting on
the training data intrinsics.

###  4.4 Ablations

We next consider several ablations to test different components of the model,
all with DMD-NK for expedience.

Log vs linearly scaled depth. Table 3 shows that parameterizing depth in log
scale (Sec. 3.2) improves quantitative performance. As expected, this is
beneficial for datasets of indoor scenes and also for datasets of outdoor
scenes with shallower depths, like DIML Outdoor and DIODE Outdoor.
Furthermore, as shown in Fig. 4, using linear-scaling leads to noise artifacts
in the depth estimates for indoor scenes which is resolved with log-depth
parameterization.

Field-of-view conditioning. Table 5 shows that FOV conditioning achieves the
best performance. Fig. 6 perturbs the conditioning FOV signal during
inference, showing that optimal performance occurs at or close to the true
FOV.

| REL $\downarrow$ | RMSE $\downarrow$
---|---|---
| NYU | KITTI | NYU | KITTI
ZoeDepth Auto Router | 0.102 | 0.075 | 0.377 | 2.584
Ours no fov aug / cond | 0.074 | 0.055 | 0.310 | 2.597
Table 8: Comparison against ZoeDepth without scene type supervision. ZoeDepth
performance degrades significantly when the scene type (indoor or outdoor) is
not provided. DMD learns well without such supervision.

No FOV augmentation or conditioning. ZoeDepth found that without scene-type
supervision for the experts (i.e., Auto Router), ZoeDepth’s performance
degrades, even for in-domain data. To compare against ZoeDepth in this
setting, we fine-tune a model on NYU and KITTI without FOV augmentations or
conditioning. Interestingly, DMD performs relatively well in this setting for
in-domain data (Table 8). Nevertheless, as shown in Table 7, OOD performance
is better with FOV augmentation and conditioning.

${\bm{\epsilon}}$ vs ${\bm{v}}$ diffusion parameterization. Inference latency
is a concern with diffusion models for vision. DDVM [40], for example, uses
128 denoising steps for depth estimation which can be prohibitive. We find
that using the $v$-parameterization dramatically reduces the number of
denoising steps required for good performance. As shown in Table 9,
${\bm{\epsilon}}$-parameterization requires 64 denoising steps to match the
performance of a model with ${\bm{v}}$-parameterization using only 1 denoising
step. Intuitively, ${\bm{v}}$-parameterization ensures that the model
accurately recovers the signal at both ends of the noise schedule, unlike
${\bm{\epsilon}}$-parameterization where estimating the noise is easy for low
SNR inputs.

| NYU | KITTI
---|---|---
Num denoising steps | ${\bm{\epsilon}}$ | ${\bm{v}}$ | ${\bm{\epsilon}}$ | ${\bm{v}}$
1 | 2.374 | 0.077 | 0.596 | 0.056
4 | 1.484 | 0.075 | 0.406 | 0.055
16 | 0.409 | 0.074 | 0.141 | 0.055
64 | 0.077 | 0.074 | 0.056 | 0.055
Table 9: Comparison of relative error on NYU and KITTI for models trained with
${\bm{\epsilon}}$\- and ${\bm{v}}$\- parameterization. Both are fine-tuned on
NYU and KITTI without FOV augmentation or conditioning.

##  5 Conclusion

We propose a generic diffusion-based monocular metric depth generator with no
task-specific inductive biases and no specialized architectures for handling
diverse indoor and outdoor scenes. Our log-scale depth parameterization
adequately allocates representation capacity to different depth ranges. We
advocate augmenting the FOV of training data through simple
cropping/uncropping to enable generalization to fields-of-view beyond those in
the training datasets and show that simply uncropping with noise padding is
effective for simulating a larger FOV. We find that conditioning on the FOV is
essential for disambiguating depth-scale. We further propose a new fine-tuning
dataset mixture that dramatically improves performance. With these innovations
combined, we establish a new state-of-the-art outperforming the prior work of
ZoeDepth across all zero-shot and in-domain datasets by a substantial margin.

### Acknowledgements

We thank Jon Shlens, Kevin Swersky, Shekoofeh Azizi and Cristina Vasconcelos
for their detailed feedback and help with figures. We also thank Ting Chen,
Daniel Watson, Forrester Cole and others in Google DeepMind and Google
Research for helpful discussions.

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| Image | Ground Truth |  ZoeDepth [5] | DMD (ours)
---|---|---|---|---
DIODE Indoor |  |  |  |
Hypersim |  |  |  |
ibims-1 |  |  |  |
NYU |  |  |  |
SunRGBD |  |  |  |
Figure 7: Qualitative comparison between our method and ZoeDepth [5] on indoor
scenes. Compared with ZoeDepth, our method estimates depths at more accurate
scale over diverse datasets.

##  Appendix A Additional qualitative examples

In Fig. 7 and Fig. 8, we provide additional qualitative comparison between DMD
and ZoeDepth on indoor and outdoor datasets respectively. Our method
consistently captures more accurate metric-scale depth in various indoor and
outdoor scenes.

| Image | Ground Truth |  ZoeDepth [5] | DMD (ours)
---|---|---|---|---
DDAD |  |  |  |
DIML Outdoor |  |  |  |
DIODE Outdoor |  |  |  |
KITTI |  |  |  |
Virtual KITTI 2 |  |  |  |
Figure 8: Qualitative comparison between DMD and ZoeDepth [5] on outdoor scenes. Compared with ZoeDepth [5], our method is able to estimate a more accurate depth scale. | Indoors | Outdoors
---|---|---
Experiment | NYU | SunRGBD | DIODE Indoor | ibims-1 | Hypersim | KITTI | DIML Outdoor | DIODE Outdoor | Virtual KITTI 2 | DDAD
DMD-NK true FOV | 0.076 | 0.109 | 0.298 | 0.130 | 0.382 | 0.055 | 0.300 | 0.628 | 0.093 | 0.122
DMD-NK est. FOV | 0.077 | 0.124 | 0.302 | 0.140 | 0.407 | 0.062 | 1.490 | 0.639 | 0.114 | 0.132
DMD-MIX true FOV | 0.072 | 0.091 | 0.291 | 0.118 | 0.318 | 0.053 | 0.190 | 0.553 | 0.092 | 0.108
DMD-MIX est. FOV | 0.072 | 0.136 | 0.291 | 0.114 | 0.361 | 0.061 | 1.049 | 0.560 | 0.131 | 0.119
FOV est. error (degrees) | 1.520 | 5.114 | 0.735 | 2.339 | 4.224 | 0.842 | 27.998 | 3.459 | 4.142 | 4.384
Table 10: Comparison of relative error (REL) using an estimated (est.) FOV
versus the true FOV. Even with a simple FOV estimator the depth estimation
results using an estimated FOV are competitive to those with the true FOV,
except for DIML Outdoor where our FOV estimator struggles to predict correct
FOV as shown by the large error in FOV estimation.

##  Appendix B Handling unknown field-of-view

While RGB camera intrinsics are available for most practical uses of monocular
depth estimators (e.g. cell phones, robot platforms or self-driving cars),
they may sometimes be unknown (e.g. internet images or generative imagery).
One solution to handle the unknown FOV would be to estimate the camera
intrinsics from the RGB image.

To test this, we train a simple model for predicting the vertical field-of-
view (FOV). The model consists of the encoder of a pre-trained Palette [37]
model followed by a spatial average pooling layer and a linear head that
predicts a scalar. It is trained to regress to $\tan(\theta/2)$, where
$\theta$ is the vertical FOV, using a $L_{1}$ loss. We train on a mix of the
NYU and KITTI datasets employing the same FOV augmentation strategy as used
for DMD-NK.

Table 10 compares the depth estimation performance of our models when using
the estimated FOV versus the true FOV. Despite the simplistic design of our
FOV estimator, the depth estimation performance when using the estimated FOV
is competitive to that when using the true FOV, except for the DIML outdoor
dataset, where the large error in FOV estimation leads to a much worse
relative error for depth. This might be due to the significantly larger FOV of
DIML outdoor, compared to other outdoor datasets, which could be challenging
for our simple FOV regressor to generalize to. We hypothesize that
incorporating more sophisticated camera intrinsic prediction models, such as
those proposed by [25, 27], could lead to further improvements in FOV
estimation accuracy, and thereby better metric depth estimates. However, we
defer a thorough investigation of this approach to future work.

##  Appendix C Additional training details

We perform the first stage of supervised training for 1.5M steps with a
learning rate of $1\times 10^{-4}$ and the second (final) stage for 50k steps
with a learning rate of $3\times 10^{-5}$. For the FOV augmentation in the
second stage we randomly uniformly sample a scale in [0.8, 1.5] to crop/uncrop
the image and depth maps.


# Optimal DG allocation and sizing in power system networks using swarm-based
algorithms

Kayode E. Adetunji
School of Electrical and Information Engineering
University of the Witwatersrand
Johannesburg
<EMAIL_ADDRESS>
&Ivan Hofsajer
School of Electrical and Information Engineering
University of the Witwatersrand
Johannesburg
<EMAIL_ADDRESS>
&Ling Cheng
School of Electrical and Information Engineering
University of the Witwatersrand
Johannesburg
<EMAIL_ADDRESS>

###### Abstract

Distributed generation (DG) units are power generating plants that are very
important to the architecture of present power system networks. The benefit of
the addition of these DG units is to increase the power supply to a network.
However, the installation of these DG units can cause an adverse effect if not
properly allocated and/or sized. Therefore, there is a need to optimally
allocate and size them to avoid cases such as voltage instability and
expensive investment costs. In this paper, two swarm-based meta-heuristic
algorithms, particle swarm optimization (PSO) and whale optimization algorithm
(WOA) were developed to solve optimal placement and sizing of DG units in the
quest for transmission network planning. A supportive technique, loss
sensitivity factors (LSF) was used to identify potential buses for optimal
location of DG units. The feasibility of the algorithms was confirmed on two
IEEE bus test systems (14- and 30-bus). Comparison results showed that both
algorithms produce good solutions and they outperform each other in different
metrics. The WOA real power loss reduction considering techno-economic factors
in the IEEE 14-bus and 30-bus test system are 6.14 MW and 10.77 MW, compared
to the PSOs’ 6.47 MW and 11.73 MW respectively. The PSO has a more reduced
total DG unit size in both bus systems with 133.45 MW and 82.44 MW compared to
WOAs’ 152.21 MW and 82.44 MW respectively. The paper unveils the strengths and
weaknesses of the PSO and the WOA in the application of optimal sizing of DG
units in transmission networks.

_K_ eywords Swarm intelligence $\cdot$ Optimization algorithms $\cdot$ Power
system networks $\cdot$ Real and reactive power loss minimization $\cdot$
Voltage profile improvement $\cdot$ Economic constraints

##  1 Introduction

The planning of Power System Networks (PSN) with the integration of
Distribution Generation (DG) is an important feature in the modern-day
electric utility grid. There are certain parameters to be considered such as
quantity and size of DG units, the best location, bus configuration, and even
the most suitable DG unit technology to be used [1]. A common and central
problem is the placing and sizing of DG units [1]. The wrong installation of
DG units can have an adverse effect on power flow and voltage stability which
will in turn cause an upsurge in line losses [2, 3], thereby inferring an
increase in economic costs [4]. It is therefore important to find the optimal
location and size of DG units in a PSN to solve certain objectives. The
installation of DG units is not a linear problem. This means that the increase
in the number of DG units does not directly improve grid performance.

The most reported objective in literature is the power loss minimization.
Several methodologies have been proposed for improving common objective
functions such as power loss minimization and voltage profile improvement.
Analytical approaches (mathematical methods) and metaheuristic methods.
Metaheuristic methods have been more consistent because of the combinatorial
nature of the placing and sizing problem. Hence, better computational time and
efficiency rate can be achieved with less expensive computation resources. In
addition to the solving the complex problem, some techniques have been
developed to reduce the complexity of the problem. Examples are power loss
index (PLI), voltage stability index (VSI), and most recently loss sensitivity
factor (LSF). These techniques have also been researched on, and improved
overtime [5].

Over the last decade, meta-heuristic methods have been developed with or
without supplementary techniques for optimal location and sizing of DG units
in PSNs. Deghanian et al [6] presented a Non-dominated sorting genetic
algorithm (NSGA-II) for finding optimal location of DG units with objective
functions of system reliability improvement, total system cost reduction, and
network loss minimization in distribution power system. The authors used the
probabilistic load flow to simulate uncertainties in a power system. The
algorithm was tested on a IEEE 37-bus test system.

Prakash and Lakshminarayana [7] presented PSO algorithm to place multiple DG
units with the objective of minimizing power loss. Jamian et al [8] also
presented a rank evolutionary PSO for optimal multiple DG placement for
maximum power output. The results was compared other variants of the PSO.

Moradi and Abedini [9] proposed a hybrid algorithm, Intelligent Water Drops
(IWD) and GA for optimally sizing and allocating DGs in a microgrid, for
solving objectives such as power loss minimization and improvement of voltage
stability. Voltage Stability Indices (VSI) was used for the identification of
candidate buses. The hybrid algorithm was tested on a 33-bus and 69-bus
system, and has a record computational time, which increases linearly with
number of DG units.

Kaur at al [10] developed a mixed integer nonlinear programming (MINLP) for
finding optimal placement and sizing of DG units with an objective function of
loss minimization in distribution systems. The deterministic approach was
carried out through branch and bound method using sequential programming.
Combined loss sensitivity was used to shortlist candidate buses which was
followed by the implementation of the algorithm. This method differs from
approaches where the algorithm is solely used for optimal sizing while the
technique would have catered for optimal placement. Scalfati et al [11] used a
similar (but mixed integer linear programming) approach to optimally size
energy sources in a DC microgrid, given the objective to minimize total cost
of ownership. This was implemented for the planning and management phase of
the microgrid.

Rastgou et al [12] implemented an improved harmony serach algorithm with
sensitivity analysis for network expansion and planning in a DG-present grid
network. Karimi et al [5] also worked on optimal planning using NSGA-II. The
objectives were maximizing voltage stability, minimizing costs, minimizing
voltage deviation, and reducing emission.

HassanzadehFard and Jalilian [13] developed a PSO algorithm to optimally size
and site RE-based DG units, considering total cost minimization, total network
power loss minimization, voltage stability improvement, and emission
reduction, under varying loads. A 13-bus test radial network was used for
evaluation.

Gampa and Das [14] presented a two-stage framework to sequentially find
optimal location of DG units using sensitivity analysis and a fuzzy-heuristic
to optimally size the DG units. This was done for reconfiguration of feeders
considering real power loss reduction and voltage profile improvement.

Jeddi et al [15] implemented harmony algorithm for planning distribution
networks. Profit maximization and reliability benefit were considered. Home-
Ortiz et al [16] proposed a mixed integer conic programming (MICP) to
optimally find type, size, and location of RE-based DG units.

Ali et al [17] used the ant lion optimization (ALO) algorithm alongside loss
sensitivity factors (LSF) to optimal allocate and size DG units in a
distribution network. This was done considering loading conditions and voltage
profile improvement. The algorithm was tested on 33 a- and 69-bus test
systems. Reddy et al [18] also used the ALO but alongside index vector
technique to optimally place and size DG units. The technique was used to
select buses for DG locations. The problem was solved considering power loss
minimization. Their algorithm was tested on four IEEE test bus systems. The
index vector was also used alongside whale optimization algorithm (WOA) [19]
to sequentially place and size DG units, considering power loss minimization.

Rammoorthy and Ramachandran [20] applied the PSO to the optimal sizing of
multiple DG units in transmission systems. The algorithm was implemented
alongside the LSF technique to reduce power loss and enhance the voltage
profile. The IEEE 30-bus system was used for evaluation. Mustaffa et al [21]
also implemented an Adaptive Evolutionary Programming method on the IEEE
30-bus system, to reduce the total real power loss. The algorithm outperforms
the PSO on four different scenarios.

To the best of the author’s knowledge, researchers have not studied the
comparison of the application of the PSO and WOA to the optimal sizing problem
in a meshed transmission network. More studies have been carried out on RDNs
as compared to transmission networks. Most RDN studies consider a void
network. On the other hand, transmission networks are considered as an
existing network that comprises of units such as transformers, synchronous
machines, and synchronous condensers [22]. This study endeavours to uncover
the distinctive difference in performance between the implementation of the
PSO and the WOA to the optimal sizing problem. Specifically, this paper
implements the Loss Sensitivity Factor (LSF) alongside the WOA and the PSO for
the optimal location and sizing of DG units in a meshed transmission network.
The objective is to compare the total real and reactive power loss
minimization and the voltage profile improvement results from the PSO and the
WOA.

The rest of the paper is organized as follows. Section 2 discusses the
modelling of a meshed transmission network where the important parameters and
the load flow model are addressed. Section 3 discusses the addition of DG
units to the network. The section further explains the technique for the
selection criteria. The objective function for the sizing problem is discussed
in Section 4, where the technical, economic cost and constraints are
formulated. In Section 5, the PSO and the WOA are presented as the robust
swarm intelligence for the sizing problem. Section 6 presents the results from
the analysis. In doing so, the IEEE 14- and 30-bus was used as the testbed for
evaluation. Finally, the conclusion is entailed in Section 7.

##  2 Mathematical Modelling of Meshed Transmission Network

The load flow problem is majorly formulated to analyze the normal operation
state for power system networks. A practical representation of a meshed
Transmission Network (TN) is illustrated in Figure 1. TNs can consist of
electrical components such as transformers, capacitors, transmission lines,
and impedance loads [22, 23]. A vital feature for the modelling of a TN is the
use of nodes and branches, which represent the buses and transmission lines
respectively. Nodes are classified into three major types namely $PQ$, $PV$
and slack nodes. The $PQ$ nodes possess the real and reactive ($P,Q$)
parameters and voltage magnitude and the angle will be calculated. The $PV$
nodes are taken as known real power and voltage magnitude, $V$. The slack node
is considered as a special bus because of its singularity, i.e. there is only
one slack bus in a TN. The known parameters for this node is voltage magnitude
and angle ($V,\delta$).

Figure 1: Schematic representation of a simple meshed TN

From the diagram, $P_{i}^{T}$ and $Q_{i}^{T}$ represents an installed
substation transformers’ real and reactive power from node $i$. $P_{k}^{G}$
and $Q_{k}^{G}$ represents an installed generators’ real and reactive power
from node $j$. The impedance on the line is represented as $R_{i,j}+X_{i,j}$.
The power equation can be expressed as follows;

$P_{i}+Q_{i}\;=\;\dot{V}\sum_{j\neq
i}{Y}_{ij}{V}_{j}\;(i\;=\;1,\;2\;,...\;,\;n),$ (1)

where $P_{i}$ and $Q_{i}$ are the injected real and reactive power at node $i$
respectively. Here, ${\widehat{Y}}_{ij}$ is an element of an admittance matrix
from node $i$ to $j$ and ${\widehat{V}}_{j}$ is the injected voltage at node
$j$.

The nodal equation can be written in terms of injected current $I_{i}$

$I_{i}\;=\;\sum_{j\neq i}{Y}_{ij}{V}_{j}\;(i\;=\;1,\;2\;,...\;,\;n).$ (2)

###  2.1 Load Flow Studies

This study implemented the Newton-Raphson (N-R) algorithm to compute the bus
voltage magnitudes and angles, real and reactive power of DGs, and real and
reactive power loss of the PSN. The N-R algorithm is an iterative method which
approximately converts non-linear simultaneous equations to linear
simultaneous equations. N-R is known for a fast convergence characteristic,
and performs better than most deterministic load flow technique [24]. For the
N-R load flow, the injected current can be represented as shown in Equation
(3),

$I_{i}\;=\;\sum_{j\neq i}^{N}{Y}_{ij}{V}_{j}\angle(\theta_{ij}+\delta_{j}).$
(3)

The injected current can be substituted for real and reactive power units as

$P_{i}-Q_{i}\;=\;\sum_{j\neq
i}^{N}{Y}_{ij}{V}_{i}{V}_{j}\angle(\theta_{ij}+\delta_{j}-\delta_{i}).$ (4)

Therefore, real and reactive power can be derived by splitting Equation (4) as

$P_{i}\;=\;\sum_{j\neq
i}^{N}{Y}_{ij}{V}_{i}{V}_{j}\cos(\theta_{ij}+\delta_{j}-\delta_{i})$ (5)

and

$Q_{i}\;=\;\sum_{j\neq
i}^{N}{Y}_{ij}{V}_{i}{V}_{j}\sin(\theta_{ij}+\delta_{j}-\delta_{i}).$ (6)

The two equations are written in terms of $V$ and $\delta$, and can be easily
manipulated to solve for the Jacobian matrix as shown in Equation (7) [25,
26],

$\begin{bmatrix}\Delta P\\\ \Delta
Q\end{bmatrix}\;=\;\begin{bmatrix}J_{1}&J_{3}\\\
J_{2}&J_{4}\end{bmatrix}\begin{bmatrix}\Delta\delta\\\ \Delta V\end{bmatrix},$
(7)

where $J_{1},J_{2},J_{3}$, and $J_{4}$ are elements in the Jacobian matrix,
and $\Delta P$ and $\Delta Q$ are the respective real and reactive power
residuals from the scheduled and calculated values [26, 27]. They are
represented as

$\Delta P_{i}^{k}=P_{i}^{sch}-P_{i}^{k},$ (8)

and

$\Delta Q_{i}^{k}=Q_{i}^{sch}-Q_{i}^{k}.$ (9)

The bus voltages and magnitudes can be estimated as

$V^{k+1}=V_{i}^{k}+\Delta V_{i}^{k},$ (10)

and

$\delta^{k+1}=\delta_{i}^{k}+\Delta\delta_{i}^{k}.$ (11)

##  3 Addition of DG Units to Power System Networks

The addition of DG units in a bus network is determined by the characteristics
of each bus under normal load conditions. The DG unit supplies real power [28]
and can be represented as the difference between present real power and real
power demand at the same node:

$P_{i}=P^{\text{DG}}_{i}-P^{\text{D}}_{i},$ (12)

where $P^{\text{D}}_{i}$ is the load demand at node $i$ while
$P^{\text{DG}}_{i}$ is the real power injection from DG unit at node $i$.

###  3.1 Loss Sensitivity Factors

Loss sensitivity factor technique is an analytical approach to determine a
proper location of DG units in a power system network. LSF is one method to
lessen the job of an optimization algorithm, hence it is computed to reduce
the solutions search space. This method was firstly developed for optimal
capacitor placement and is based on the principle of linearization [29]. Its
application for DG placement is not far-off. Sensitivity values are calculated
(from base case) for each bus and are ranked in a descending order to form a
priority list. The rate of change in real power loss (from Equation (17)) to
the real power injection at a particular bus should be equals zero. The LSF
equation [30, 31] is shown in (12).

Here,

$\frac{{\delta P}_{L}}{{\delta P}_{j}}=2{\sum_{j=1,j\neq
i}^{N}{(\alpha_{ij}P_{j}-\;\beta_{ij}Q_{j})}}$ (13)

where

$P_{i}=\;\frac{1}{\alpha_{k}}\left[\beta_{k}Q_{i}+{\sum_{j=1,j\neq
i}^{N}{(\alpha_{ij}P_{j}-\;\beta_{ij}Q_{j})}}\right].$ (14)

Here, $P_{i}$ is the real power injected at node $i$, which is the difference
the generated real power and real potential load demand (from Equation (12)).
Therefore, the real power injected from $i^{th}$ DG units.

$P_{i}^{\text{DG}}=P^{\text{D}}_{i}-\;\frac{1}{\alpha_{k}}\left[\beta_{k}Q_{i}+{\sum_{j=1,j\neq
i}^{N}{(\alpha_{ij}P_{j}-\;\beta_{ij}Q_{j})}}\right].$ (15)

The sensitivity values is normalized to values between 0 and 1, using Equation
(16) [32].

$LSF_{j}=\frac{LSF_{j}-LSF_{min}}{LSF_{max}-LSF_{min}}\;\;\;j\;\forall\;\\{2,\;3,\;.\;.\;.\;,\;N_{b}\\}$
(16)

##  4 Objective Function Formulation for the Problem

Optimal DG sizing in RDN reduces the real power loss and improves the voltage
profile. The power loss can be represented as [33]:

$P_{L}=\;{\sum_{i=1}^{N}{{\sum_{j=1}^{N}{\left[\alpha_{i,j}(P_{i}P_{j}+\;Q_{i}Q_{j})\right]+\;}}}}\beta_{i,j}(Q_{i}P_{j}+\;P_{i}Q_{j}),$
(17)

where

$\alpha_{i,j}=\frac{r_{i,j}}{V_{i}V_{j}}{{cos}{\left(\delta_{i}-\;\delta_{j}\right)}}$
(18)

and

$\beta_{i,j}=\frac{r_{i,j}}{V_{i}V_{j}}{{sin}{\left(\delta_{i}-\;\delta_{j}\right)}}.$
(19)

Here, $i$ and $j$ are the sending and the receiving bus respectively, while
$Z_{i,j}=\;r_{i,j}+\;{jx}_{i,j}$ represent the branch impedance from bus $i$
to bus $j$. Decrease in real power loss leads to the minimization of DG unit
cost. However, increase in DG size may cause an adverse effect on the
investment cost. Therefore, there is a need to handle the size of DGs at the
potential buses (the location). The addition of DG units can also affect the
voltage quality of the network, hence a voltage deviation technique is
formulated as

$VD=\sum_{i=2}^{{N_{PQ}}}|V_{i}-V_{ref}|,$ (20)

to check the voltage profile of each bus. The reference voltage, $V_{ref}$ is
set at zero and $V_{i}$ represents the voltage at each PQ bus after the
addition of DG units. The objective functions are subjected to constraints.

###  4.1 Economic Objective Function

The cost generated while sizing the DGs is very important. Therefore, there is
a need to minimize it simultaneously with the sizing objective [34, 35, 36,
37].

$DG_{\text{cost}}=\;\sum_{i=1}^{N}\sum_{j=1}^{N_{DG}}P_{i}K_{conn}TC^{\text{DG}}P_{j}^{\text{DG}},$
(21)

Where the cost of DG unit is represented as $C^{\text{DG}}$, and $T$
represents the electricity tariff from the DG unit source. Here, $K_{conn}$ is
the connection factor to the grid, $P_{i}$ is the injected power at $i^{th}$
number of selected buses for the DG unit placement. The size of the $j^{th}$
DG unit is represented as $P_{j}^{\text{DG}}$.

The eco-technical objective function is formulated as the sum of the power
loss objective ($P_{L}$) and the DG cost objective ($DG_{\text{cost}}$).

###  4.2 Constraints

The operating voltage at every bus must satisfy the range at all buses. The
bus voltage limit is formulated as

$V_{i}^{min}\leq\;V_{i}\leq V_{i}^{max}\;\;i=1,2,...,N.$ (22)

where $V_{i}$ is the current voltage at bus $i$. Then $V_{i}^{min}$ and
$V_{i}^{max}$ are 0.98 and 1.01 respectively.

Rated thermal capacity must not be exceeded at each branch. Therefore, the
branch current limit is defined as

$|I_{i,i+1}|\leq I_{i,i+1}^{rated}$ (23)

where $i$ and $i+1$ represent the sending and receiving nodes respectively.

####  4.2.1 Power Flow Balance

The total real and reactive power generation (from all DG units) must be equal
to the total real and reactive load and the total real and reactive power loss
[33]. Therefore, a balance of the power flow is calculated as shown below

${\sum_{i=1}^{N}{P^{\text{DG}}_{i}=\;{\sum_{i=1}^{N}{P^{\text{LOSS}}_{i}}}+\;{\sum_{i=1}^{N}{P^{\text{LOAD}}_{i}}}}}$
(24)
${\sum_{i=1}^{N}{Q^{\text{DG}}_{i}=\;{\sum_{i=1}^{N}{Q^{\text{LOSS}}_{i}}}+\;{\sum_{i=1}^{N}{Q^{\text{LOAD}}_{i}}}}}$
(25)

From Equation (17),

$\begin{split}\sum({P^{\text{DG}}_{i}-{P^{\text{LOAD}}_{i})}}=\sum_{j=1}^{{N}}V_{i}V_{j}[G_{ij}cos(\delta_{i}-\delta_{j})\\\
+B_{ij}sin(\delta_{i}-\delta_{j})]\end{split}$ (26)
$\displaystyle\sum({Q^{\text{DG}}_{i}-{Q^{\text{LOAD}}_{i})}}=\sum_{j=1}^{N}V_{i}V_{j}[G_{ij}sin(\delta_{i}-\delta_{j})$
(27) $\displaystyle+B_{ij}cos(\delta_{i}-\delta_{j})]$

$i=1,2,...,N$
where $G_{k}$ and $B_{k}$ is the conductance and susceptance of the
transmission line $k$ respectively, $V_{i}$ and $V_{j}$ are the voltage
magnitudes at sending node $i$ and receiving node $j$, $\delta_{i}$ and
$\delta_{j}$ are the voltage angles wrt $V_{i}$ and $V_{j}$.

##  5 Proposed Algorithms

###  5.1 Particle Swarm Optimization

PSO is a computationally intelligent algorithm that uses the behavior of
organisms (especially birds and fishes). The synergy and coordination of the
behaviour of such organisms is based on the manipulation of distances to reach
a desired goal. The organisms are represented as particles and they keep a
memory of their coordinates [8]. During every iteration, the particles
communicate and learn from each other. The best position of each particle is
then evaluated and called the personal best (pbest). Then, the overall best
position from the personal best of every particle is calculated as the global
best (gbest). Every particle moves with respect to a velocity. This velocity
is calculated using Equation (29). The PSO is dependent on the number of
particles that will be needed for the search of solution space (swarm),
weight, cognitive parameter, and social parameter. The weighting factors of
the PSO is very important, as it determines the algorithms’ performance. In
most cases, the parameters are replicated from previous successful studies.
Cognitive, social, and weight are the parameters of the PSO. The random
factors are induced for the stochastic nature of the algorithm. Each
particles’ pbest is calculated as

$x_{i(t+1)}=\;x_{i(t)}+\;V_{i(t+1)},$ (28)

where

$V_{t+1}=wV_{t}+C_{1}r_{1}\left(P_{i\left(t\right)}-x_{i\left(t\right)}\right)+C_{2}r_{2}(g_{\left(t\right)}-x_{i\left(t\right)}).$
(29)

Here, $x_{i(t)}$ is the position of particle $i$ in time step $t$ and
$V_{i(t)}$ is the velocity of particle $i$ in time step $t$. The best position
of each particles in time step $t$ is represented as $P_{i(t)}$ while
$g_{(t)}$ is the global best of all particles in the swarm in time step $t$.
$C_{1}$ and $C_{2}$ represents the cognitive and social parameter while
$r_{1}$ and $r_{2}$ represents the induced random parameter. The $g_{(t)}$
does not have the index $i$ because it belongs to a whole swarm and not a
particle-specific experience. The velocity equation is comprised of components
related to swarm behavior. The weight, $w$ is updated in such a way that it
decays with respect to maximum number of iterations. This behavior is
illustrated as [38]

$w=\;w_{max}-\frac{w_{max}-w_{min}}{t_{max}}t.$ (30)

Figure 2: Flowchart of the PSO implementation

The implementation of the PSO algorithm for optimal sizing is well illustrated
in Algorithm 1.

Algorithm 1 PSO Implementation

Input bus and branch data parameters and run the load flow algorithm to update
the bus and branch data.
Solve the load flow problem to obtain the real and reactive power loss and the
voltage (pu) at each bus.
Identify the candidate buses by using Equation (13) to get the sensitivity
values.
Initialize: a population of particles with values and positions from the
candidate bus matrix, cognitive and social parameters, number of iterations
maxIter, and counter t.

while $t<maxIter$ do

for Each particle $i$ do

Adapt velocity $V_{n}$ of the particle using (30)

Update the position $x_{n}$ of the particle using Equation (31)

Evaluate the objective functions $f(\overrightarrow{X}_{n})$ \- Equation (17)
and (20).

if $f(\overrightarrow{X}_{n})<f(\overrightarrow{P}_{n})$ then

$\overrightarrow{P}_{i}\leftarrow\overrightarrow{X}_{i}$

end if

if $f(\overrightarrow{X}_{n})<f(\overrightarrow{P}_{g})$ then

$\overrightarrow{P}_{g}\leftarrow\overrightarrow{X}_{i}$

end if

end for

end while

return $\overrightarrow{P}_{g}$

###  5.2 Whale Optimization Algorithm

The Whale optimization algorithm (WOA) was proposed by Mirjalili [39]. The
algorithm has been successfully applied to many optimization problems
including the optimal sizing problem. The algorithm involves the feeding
nature of whales (specifically humpback whales). According to the intelligent
behavioural nature of whales, they use a specific technique to target small
fish that are close to the sea surface. This technique is called the bubble
net feeding, where whales swim round a school of fish (prey) to form a
9-shaped bubble trail. In this algorithm, an initial solution is termed as the
objective prey and assumed as the current best solution. During the course of
iterations, other whales update their positions towards the best whale
position. The WOA is mathematically modelled in three sections. (i) encircling
prey (ii) bubble net hunting method and (iii) search the prey.

####  5.2.1 Encircling prey

This form of hunting is based on a circular positioning around a prey. The
position of the whale moves towards the prey in a step-wise manner. It can be
modelled as

$\overrightarrow{X}_{t+1}={\overrightarrow{X}}_{t}^{\ast}-A\cdot\overrightarrow{D},$
(31)

where $A=2a\;\cdot\;r\;\;-\;a$ and $a$ is linearly reduced from 2 to 0, and
$r\in[0,1]$ where $r$ can take up random in $0$ and $1$ interval.
$\overrightarrow{D}=\;|C\;\;\cdot\;\;\overrightarrow{X_{t}^{\ast}}-\;\overrightarrow{X_{t}}|,$
where $C=2\cdot\;r.$
The current iteration is denoted by subscript $t$ and $t+1$ for the next
iteration. $\overrightarrow{X}$ represents the position vector and
$\overrightarrow{X^{\ast}}$ represents the current best solution.

####  5.2.2 Bubble net hunting method

This method is the exploitative phase of the algorithm. Figure 3 illustrates
the mechanism of the shrinking approach. This behaviour is attained by
decreasing the value of $a$, as explained in Section 5.2.1. This behaviour is
modelled as shown in Equation (31). The spiral hunting approach is a helic-
shaped movement of whales from a position $(X,Y)$ to hunt prey (the best
solution) at a position $(X*,Y*)$. This is illustrated in Figure 4. The spiral
equation is created as follows:

$\overrightarrow{X}_{t+1}\;=\;\;\overrightarrow{D}^{\prime}\cdot\;e^{bl}\cdot{{cos}{\left(2\pi
l\right)}}+\;\overrightarrow{X}^{\ast}.$ (32)

The component $cos(2\pi l)$ simulates the spiral shape of the whale’s path, as
shown in Figure 4. $l$ takes a value between [-1,1], and $b$ is a constant
(usually 1) that gives the spiral shape regular definition. Since the whales
can be shrink towards a prey or move spirally, there is a need to model this
behaviour. A probability of 50% is choosen for both feeding mechanisms. The
model is as follows:

$\overrightarrow{X}_{t+1}\;=\;\left\\{\begin{array}[]{lc}\;\overrightarrow{X}_{t}^{\ast}-\;\overrightarrow{A}\cdot\overrightarrow{D},&p<0.5\\\
\overrightarrow{D}\cdot\;e^{bl}\cdot\cos(2\pi
l)+\;\overrightarrow{X}_{t}^{\ast}&p\geq 0.5\end{array}\right.$ (33)

where $p$ is a random number between $[0,1]$.

Figure 3: Shrinking mechanism of the WOA (adapted from [7]) Figure 4: Spiral
hunting simulation of the WOA (adapted from [7])

####  5.2.3 Search for prey

The WOA uses this technique to mainly overcome a possible local optimum. This
is called the exploration phase, where whales search randomly according to
check other possible solutions. By this, if the best solution is not global
the search agent can still fall out to find other better solution in the
global space. The use of this approach is dependent on the value of $A$. If
the value of $A$ is lesser than 1, Equation (31) is triggered, else Equation
(34) is triggered. Equation 34 shows the model.

$\overrightarrow{X_{t+1}}=\overrightarrow{X_{rand}}-A\cdot\overrightarrow{D}$
(34)

where
$\overrightarrow{D}=\;\left|C\;\;\cdot\;\overrightarrow{X_{rand}}\;-\;\overrightarrow{X_{t}}\right|$,
$\overrightarrow{X_{rand}}$ represents the random whales’ position in current
iteration. $\overrightarrow{D}$ is the distance between current whale
position, $\overrightarrow{X_{t}}$ and the randomly selected whales’ position.

Figure 5: Flowchart of the WOA implementation

The implementation of the WOA algorithm for optimal sizing is well illustrated
in Algorithm 2.

Algorithm 2 WOA Implementation

Input bus and branch data parameters and run the load flow algorithm to update
the bus and branch data.
Solve the load flow problem to obtain the real and reactive power loss and the
voltage (pu) at each bus.
Identify the candidate buses by using Equation (13) to get the sensitivity
values. Initialize: a population of whales with values and positions from the
candidate bus matrix, number of iterations maxIter, and counter t.
Initialize a, A, C, l and p

Evaluate the objective functions for each whale $f(\overrightarrow{X}_{i})$.

Obtain the best whale, $\overrightarrow{X}^{*}$

while $t<maxIter$ do

for Each particle $i$ do

if $p<0.5$ then

if $|A|<1$ then

Update the position of the current whale by the equation (31)

else[$|A|\geq 1$]

Update the position of the current whale by the equation (34)

end if

else[$p\geq 0.5$]

Update the position of the current whale by the equation (32)

end if

Calculate the fitness of each whale, $\overrightarrow{X}_{i}$

if $\overrightarrow{X}_{i}>\overrightarrow{X}^{*}$ then

$\overrightarrow{X}^{*}\leftarrow\overrightarrow{X}_{i}$

end if

end for

$t++$

end while

return $\overrightarrow{X}^{*}$

##  6 Implementation Results and Discussion

The algorithms were developed in the MATLAB environment (R2017a version) and
simulations were run on an Intel(R) Core(TM) i7-3520M @ 2.90 GHz. The outcomes
from the algorithms are presented in this section, and have been evaluated on
the IEEE 14-bus and 30-bus test RDN system.

Parameters used in the algorithms are presented in Table 1. The metric is
based on power loss minimization and voltage profile improvement. The approach
uses a two-stage procedure for the optimal location and sizing of DG units.
The LSF technique was used to identify the potential buses for placing the DG
units. Then the meta-heuristic optimization algorithm was applied to calculate
the optimal sizes of the DG units with respect to the potential locations.
This two-stage approach is necessary for reducing the solution search space
for the algorithms. The main work of the algorithms was done by the N-R load
flow algorithm, which mainly uses the bus and branch data to compute the power
losses. The parameter values of the algorithms are shown in Table 1 [40].

Figure 6: IEEE 14-Bus distribution test system [41] Table 1: Algorithm parameters PSO |
---|---
Parameters | Description | Value
POP | Population of bird flocks | 50
dim | Dimension | 4
maxIter | Number of iterations | 150
$w_{1}$ | minimum weight | 0.4
$w_{2}$ | maximum weight | 0.9
$c_{1}$ | cognitive parameter | 2
$c_{2}$ | social parameter | 2
WOA |
POP | Population of whales | 50
dim | Dimension | 4
maxIter | Number of iterations | 150

###  6.1 IEEE 14-Bus test system results

The DG unit sizes were initially chosen randomly within the DG limits
$(1MW\leq\ DGsize\leq\ 50MW)$ for the potential bus locations. Thereafter, the
objective function was computed based on the defined constraints. The total
power losses were calculated for each algorithm. The base case was 100 MVA and
11 kV, with voltage magnitude limits between 1.01 and 0.98 pu, with the first
bus assumed at 1.0 pu. The total load demand of the test system is 259 MW and
73.5 MVaR, with the real and reactive power loss as 13.593 MW and 56.910
respectively.

Figure 7: Real power loss on each branch before and after DG integration for 14-bus test system.  Figure 8: Total real power loss of the IEEE-14 bus system Figure 9: Reactive power loss improvement of IEEE 14-bus test system Figure 10: Voltage profile of IEEE 14-bus test system  Table 2: Real and reactive power loss values and improvement for 14-bus test system Real power | MW | Performance (%)
---|---|---
WOA Power loss technical | 8.00 | 69.76%
WOA Power loss techno-economic | 6.14 | 121.50%
PSO Power loss technical | 6.43 | 111.49%
PSO Power loss techno-economic | 6.47 | 110.41%
Reactive power |  |
WOA Power loss technical | 27.59 | 07.94%
WOA Power loss techno-economic | 28.44 | 120.58 %
PSO Power loss technical | 25.80 | 113.65%
PSO Power loss techno-economic | 31.64 | 110.36%
Table 3: Comparison of the algorithms’ real and reactive power loss values for 14-bus test system Real power | Performance (%)
---|---
WOA improvement technical | -19.73%
WOA improvement techno-economic | 5.41%
Reactive power |
WOA improvement technical | -12.73%
WOA improvement techno-economic | 5.41%

The LSF was implemented to identify the candidate bus locations for optimal
sizing of the DG units. The potential bus locations are 2, 8, 9, 10. These bus
locations are selected based on the values close to zero, and are prioritized
in a descending manner. The first four indices of the bus configuration are
selected.

Figure 7 illustrates the real power loss on each bus. The PSO while
considering techno-economic objective, reduces the real power loss. The WOA
comes close but only with the consideration of the technical objective. A
summary of the total power loss reduction is illustrated in Figure 8, which
shows that the PSO averagely performs better than the WOA. Table 2 further
elucidates the values and improvement level of the algorithms. Each parameter
is compared to the base case to calculate the improvement percentage. The PSO,
considering the technical and techno-economic objectives, reduced the real
power loss to 6.43 MW and 6.47 MW respectively, while the WOA considering the
technical and techno-economic objectives, reduced the real power loss to
8.0069 MW and 6.14 MW respectively. The PSO, considering the technical and
techno-economic objectives, reduces the reactive power loss to 25.80 MVaR and
31.64 MVaR respectively, while the WOA considering the technical and techno-
economic objectives, reduces the reactive power loss to 27.59 MVaR and 28.44
MVaR respectively. Table 4 shows the comparison of the WOA to the PSO in the
optimal sizing of DG units. In other words, the PSO is the base case. It is
more evident that the PSO performs better than the WOA when considering only
the technical objective.

From Table 5, the WOA produces a lower DG unit size when considering only
technical objective. However, the PSO performs better than the WOA in terms of
the techno-economic objective. This is synonymous to WOA results in [42]. The
real and reactive power loss from the PSO is 6.47 MW and 28.43 MVaR, while the
WOA is 6.14 MW and 25.80 MVaR respectively. From Figure 9, it is observed that
the WOA when considering the techno-economic objective, has the highest
reactive power loss reduction on the bus network. Figure 10 shows the voltage
profile of the IEEE 14-bus test system, with WOA considering techno-economic
constraints having the most improved profile.

Table 4: DG units sizes for candidate buses Bus no. | PSO | PSO (Eco) | WOA | WOA (Eco)
---|---|---|---|---
2 | 41.191 | 39.545 | 20.343 | 49.328
8 | 42.133 | 43.167 | 20.552 | 38.826
9 | 13.823 | 9.880 | 22.645 | 32.311
10 | 38.777 | 37.937 | 20.612 | 33.540
TOTAL | 135.924 | 130.529 | 84.152 | 154.005

###  6.2 IEEE 30-Bus test system results

The DG unit sizes were initially chosen randomly within the DG limits $(1\
MW\leq\ DGsize\leq\ 50\ MW)$ for the potential bus locations. Thereafter, the
objective function was computed based on defined constraints. The total power
losses are calculated for each algorithm. For the base case of 100 MVA and 11
kV, with voltage magnitude limits between 1.01 and 1.1 pu, the real power loss
is 13.5929 MW. The total load demand of the test system is 283.4 MW and 126.2
MVaR.

Figure 11: IEEE 30-Bus distribution test system [43] Figure 12: Real power loss on each branch before and after DG integration for 14-bus test system.  Figure 13: Total real power loss of the IEEE-30 bus system Figure 14: Reactive power loss improvement of the IEEE-30 bus test system Figure 15: Voltage profile of IEEE 30-bus test system Table 5: Real and reactive power loss values and improvement for 30-bus test system Real power | MW | Performance (%)
---|---|---
WOA Power loss technical | 7.54 | 62.75%
WOA Power loss techno-economic | 10.77 | 132.38%
PSO Power loss technical | 11.56 | 49.35%
PSO Power loss techno-economic | 11.73 | 51.57%
Reactive power |  |
WOA Power loss technical | 34.77 | 50.17%
WOA Power loss techno-economic | 45.87 | 98.15%
PSO Power loss technical | 48.57 | 40.11%
PSO Power loss techno-economic | 49.17 | 41.82%
Table 6: Comparison of the algorithms’ real and reactive power loss values for 30-bus test system Real power | Performance (%)
---|---
WOA improvement technical | 53.31%
WOA improvement techno-economic | 8.97%
Reactive power |
WOA improvement technical | 39.71%
WOA improvement techno-economic | 7.18%

The LSF was implemented to identify the candidate bus locations for optimal
sizing of DGs. The potential bus locations are 2, 6, and 7. These bus
locations are selected based on the values close to zero. The first three
indices of the bus configuration are selected.

Figure 12 illustrates the real power loss on each bus for each algorithm. It
is observed that the WOA performs better than the PSO when considering only
the technical objective. With the techno-economic objective, the WOA still
performs better but by a small margin. Figure 13 further shows the summary of
results on power loss reduction from the optimization algorithms for the IEEE
30-Bus test system. It is clearly seen that the WOA performs better than the
PSO when considering both technical and techno-economic objectives. It is
observed from Table 5, that both algorithms decrease the total real power
loss. The PSO, under the technical and techno-economic constraints, reduced
the real power loss to 11.564 MW and 11.736 MW respectively, while the WOA
reduced it to 7.543 MW and 10.770 MW respectively.

For the optimal sizing, the PSO when considering the technical objective,
performs better than the WOA in the sizing of DG units, with the total DG
unit’s capacity as 58.106 MW compared to the PSOs’ 137.547 MW. The WOA has a
better total DG unit size when considering the addition of economical
objective, with 3 MW lesser than the PSO. However, the PSO produces a lower
total DG size, considering the techno-economic objective. However, the WOA
produces a lower power loss than the PSO. This is synonymous to WOA results
obtained in [18]. The total real and reactive power loss from the PSO is
11.736 MW and 49.167 MVaR, while the WOA is 10.769 MW and 45.870 MVaR
respectively. From Figure 14, it is observed that the WOA with techno-economic
objective, has a better reactive power loss on the bus network.

Figure 15 shows the voltage profile of the IEEE 30-bus test system. The WOA,
with the consideration of techno-economic constraints improved the voltage
profile of the bus network.

Table 7: DG units sizes for candidate buses Bus no. | PSO | PSO (Eco) | WOA | WOA (Eco)
---|---|---|---|---
2 | 8.542 | 33.454 | 45.323 | 23.102
6 | 15.324 | 16.541 | 44.121 | 20.732
7 | 34.240 | 32.330 | 48.103 | 35.442
TOTAL | 58.106 | 82.325 | 137.547 | 79.276

##  7 Conclusion

In this paper, swarm intelligence-based algorithms, the PSO and the WOA were
applied to the sizing of DG units for optimal planning of a meshed
transmission network. The paper reports a proper evaluation of the application
of both algorithms on the IEEE bus test systems (14- and 30-bus). Technical
and techno-economic (addition of both technical and economical) objectives
were considered, and were simulated separately for more observations.

The algorithms outperform each other in different metrics. For instance, the
WOA when considering the techno-economic objective, performs better in the
total real power loss minimization for both the IEEE 14- and 30-bus test
systems. The WOA also performs better in the voltage profile improvement of
the IEEE 14-bus network while the PSO ranks higher in performance on the
voltage profile improvement for the IEEE 30-bus network. However, the PSO
performs better when considering only the technical objective for the total
power loss minimization. The reactive power loss is most minimized by the WOA
when considering the techno-economic objective for the IEEE 14-bus network.
However, the WOA performs better on the 30-bus network, when the technical
objective is only considered.

The PSO performs better in the sizing of the DG units, when considering the
technical and techno-economic objectives for the IEEE 14- and 30-bus network.
The only exception (where the WOA performs better) is the consideration of the
technical objective for the IEEE 14-bus network.

The consideration of the technical and techno-economic objectives have their
differences. The technical objective can be more generalized since it is more
scientific while the addition of the economical objective (which is techno-
economic) makes it less generalizable. This is because economic parameters are
dependent on a country’s economic factors. In some cases, the economic state
of provinces or states in a country may vary. Furthermore, economic factors
are also time-dependent.

Results also proved that the PSO and the WOA remain successful in optimal
placement and sizing problem even in transmission networks. Both algorithms
produced a remarkable result from the solutions. PSO produces a lesser total
DG units size from the IEEE 14-bus test system evaluation. The WOA had the
most minimal real and reactive power loss values on both bus configuration.
The WOA also performs better but by a few margin in the voltage profile
improvement.

For future studies, newly modified or hybridized algorithms can be developed
according to the specific lapses in present results.

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