# Synthetic Photovoltaic and Wind Power Forecasting Data

Stephan Vogt
Intelligent Embedded System Lab
Kassel University, Germany
Email<EMAIL_ADDRESS>
& Jens Schreiber
Intelligent Embedded System Lab
Kassel University, Germany
Email<EMAIL_ADDRESS>
& Bernhard Sick
Intelligent Embedded System Lab
Kassel University, Germany
Email<EMAIL_ADDRESS>

###### Abstract

Photovoltaic and wind power forecasts in power systems with a high share of
renewable energy are essential in several applications. These include stable
grid operation, profitable power trading, and forward-looking system planning.
However, there is a lack of publicly available datasets for research on
machine learning-based prediction methods. This paper provides an openly
accessible time series dataset with realistic synthetic power data. Other
publicly and non-publicly available datasets often lack precise geographic
coordinates, timestamps, or static power plant information, e.g., to protect
business secrets. On the opposite, this dataset provides synthetic time-series
measurements, input features, geographic information, timestamps, and physical
properties of a set of individual power plants. The dataset comprises 120
photovoltaic and 273 wind power plants with distinct sides all over Germany
from 500 days in hourly resolution. This large number of available sides
allows forecasting experiments to include spatial correlations and run
experiments in transfer and multi-task learning. Furthermore, the dataset
includes side-specific, power source-dependent, non-synthetic input features
from the Icosahedral Nonhydrostatic-European Union. However, since timestamps
and location are known, combining the synthetic power measurements with other
weather models is also possible. A simulation of virtual power plants with
physical models and actual weather measurements provides synthetic power
measurement time series. These time series correspond to the power output of
virtual power plants at the location of the respective weather measurements.
Since the synthetic time series are based exclusively on weather measurements,
possible errors in the weather forecast are comparable to those in actual
power data. In addition to the data description, we evaluate the quality of
weather-prediction-based power forecasts by comparing simplified physical
models and a machine learning model. This experiment shows that forecasts
errors on the synthetic power data are comparable to real-world historical
power measurements. The machine learning model consists of gradient boosted
regression trees, which show a lower forecast error for wind and photovoltaic
with sufficient training data. In addition, we consider truncated datasets and
find that, at least for photovoltaics, there is an advantage of the physical
approaches when we only use a fraction of the training data. The errors serve
as a baseline for future experiments.

_Keywords_ Wind Power / Photovoltaic Forecasting $\cdot$ Synthetic Dataset
$\cdot$ Machine Learning

##  1 Introduction

Renewable energies are taking on an increasing role in modern energy supply
systems. As their share increases, the energy supply system’s volatility and
dependence on the weather grow. Therefore, energy forecasts are an essential
tool to assure grid stability and reduce uncertainty, e. g., in energy markets
or planning assets of electrical grids. For this reason, the importance of
research on forecasts of wind energy and photovoltaic (PV) systems has
increased.

In renewable energy forecasts, we typically utilize weather forecasts, such as
the wind speed or radiation, from a so called numerical weather prediction
(NWP) model. These predicted weather features are the input to machine
learning (ML) models, predicting the expected power generation, e. g., in day-
ahead forecasts between $24$ and $48$\mathrm{h}$$ into the future. Datasets
must be open for researchers from different fields to develop new forecasting
techniques that reduce forecast error and decrease uncertainty in the energy
system. Open datasets allow to compare newly developed methods, reproduce
these results, and compare results to state of the art.
However, the existing datasets usually have various limitations. In some
cases, the number of sites considered is relatively small (e.g., in [1]), so
it is impossible to make statements with statistical significance. In some
cases, the data is anonymized without an exact geographic location and not
synchronized in time, see [2] and [3]. Such anonymizations prohibit a model
from taking spatiotemporal dependencies into account that are common in the
electrical grid. Further static information, e.g., technical conditions such
as the orientation of PV modules or the rotor-generator ratio of wind
turbines, is often missing. At the same time, when a park has insufficient
data for traditional ML techniques, the utilized physical models require this
static information. Additionaly, these static information are also essential
for, e.g., transfer learning (TL) and zero-shot learning techniques [4] that
are ML methods to handle limited data. In the future, more and more
forecasting techniques only require limited data and provide high-quality
forecasts to compensate for the severe increase in volatile energy resources.
Additionally, in practical application, i.e., in typical systems of forecast
vendors, one has to deal with missing technical information. On the other
hand, in research, it is critical to have a dataset as complete as possible
and thus to be able to analyze how the lack of such information potentially
affects the forecast quality. Therefore, we provide a complete and
open111Currently, the dataset is not accessible through a central data-
repository. Therefore, please write us an e-mail in case you require the data.
accessible dataset that allows researchers to answer various questions in the
field of wind and photovoltaic forecasting with ML techniques.

In particular, we provide a large dataset that comprises synthetic but
realistic 120 photovoltaic and 273 wind virtual power plant sides for roughly
500 days that includes the most relevant static data. This dataset is the
largest open accessible dataset for renewable power forecasts to the best of
our knowledge. We assure that the synthetic power measurements are realistic
by utilizing meteorological weather measurements and physical models. This
consideration is essential as most of the forecast error in energy forecasting
relates to the forecast error caused by the input features from the NWP model.
The period under consideration starts December 8, 2018, and ends June 2, 2020
with hourly time resolution. The length of 500 days allows a model training to
include all seasons in the training set (a year comprises about 75%) and still
have sufficient test data (about 25%).

The remainder of this article is structured as follows. First, in Sec. 2, we
overview the data and the intended use. We also argue why there is a necessity
for such a dataset, e.g., in the context of TL. Afterward, we outline the most
relevant types of data utilized for the generation of the dataset in Sec. 3.
Based on those definitions, we detail the approaches to generate synthetic
power measurements in Sec. 4. Sec. 5 shows our results for power forecasting
techniques on the dataset as a baseline for future research. Finally, we
summarize our work and suggest future work in Sec. 6.

##  2 Data Overview and Intended Use

The data per location is divided into input data (`data_input_<location
id>.csv`) and target data (`data_target_<location id>.csv`). Both CSV files,
input, and target have a timestamp that allows a unique assignment to each
other. This timestamp is the prediction time, i.e., the time the prediction is
valid. The simplest option is to treat the data as i.i.d. samples (even though
they are not). This assumption makes it possible to treat the problem as a
non-linear but time-independent regression problem, and many standard
supervised ML methods can be applied. Here, the input CSV represents a design
matrix (input data matrix) whose individual rows correspond to the rows of the
target CSV and the columns correspond to suitable input features. One might
not use the raw timestamp as an input feature. Otherwise, the model possibly
never generalizes beyond the training period (i.e., the test period, for
example). The target-CSV includes a column ’pw’, which contains the synthetic
power measurement and thus the intended target. We generate these targets
through a physical model based on real-world on-site meteorological weather
measurements. As most of the forecast errors in energy forecasts relate to the
NWP forecasts, by utilizing real-world meteorological weather measurements, we
are capable of providing realistic synthetic power measurements.
Furthermore, we provide a test flag and the output of a physical baseline
model. To communicate a forecasting model’s final test errors, we recommend
the data samples with the test flag. We utilize the same physical model and
parametrization that generated the synthetic power for the baseline. However,
these forecast Icosahedral Nonhydrostatic-European Union (ICON-EU) NWP
forecasts instead of the real-world weather measurements.
Beginning December 1, 2019, a flag marks the beginning of a predefined test
period. The test flag should help to better compare results across research
groups. We suggest using samples between December 8, 2018, and December 1,
2019, for training and validation (model selection). The test data from a
period after December 1, 2019, enables the test of a hypothesis under
consideration as independently as possible of the training period. However, it
is noteworthy that since the test data were published together with the
training data, indirect overfitting may occur unintentionally, for example, if
the test dataset is taken into account in some way in the model selection.
Therefore, methods compared in this way might not work with a similar
advantage on future, previously unknown data. Nevertheless, it can help to
assess the capabilities of investigated methods better.

###  2.1 Forecasting Scenarios

We created the dataset with a day-ahead forecasting scenario in mind, which is
typical for many forecasting vendors. At the same time, day-ahead forecasts
are more challenging than intra-day forecasts, as with an increasing forecast
horizon of the weather model, the error of the input features increases. Based
on current literature [5] and exchanges with experts from the industries, we
identified the following key research areas:

  * •

Single task forecasting is still ongoing research and recent developments from
the field of deep learning allow to reduce the forecast error and thereby
decrease the uncertainty for the electrical grid.

  * •

Regional estimation of wind / PV power production, which take geospatial
dependencies into account, better capture the dynamics in the electrical grid.

  * •

Multi-task learning reduces the forecast error, the number of parameters, and
the training time [6, 7]. Effectively, this technique helps in compensating
for the high demand for forecasts.

  * •

Transfer learning allows for reducing the training time and the required
number of historical power measurements that are often not available [7, 4].

  * •

Meta-learning and similarly zero-shot learning ideally allow to provide power
forecasts without any historical data [4] and are, e.g., an essential
technique to provide forecasts for new power plants without any historical
data.

  * •

Explainable ML allows participants in the energy systems to make justified
decisions.

##  3 Data Basis

Three different data sources play a role in synthesizing power forecasting
data, namely meteorological measurements, numerical weather predictions, and
static (metadata) data for the individual power plants. The static data forms
the parametrization of the physical models. Furthermore, this model simulates
the power output based on meteorological measurements. Besides, the numerical
weather predictions provide the necessary input features, which currently form
our best source for information about future weather conditions in a short-
term range (within a few days).

###  3.1 Meteorological Measurements

Since photovoltaic and wind energy depends highly on the local weather
conditions, meteorological data form the basis for the synthetic data. The
power output of a single plant forms the target variable in many photovoltaic
and wind power forecasting tasks. Typically, power plants have a quick
response to changes in weather conditions, and therefore the variability of
the relevant weather variables is almost directly transferred to the power
output. Meteorological measurements, such as wind speed or global horizontal
irradiation, can provide a realistic degree of variability and randomness as
the basis for the target data.
In addition, meteorological measurements accurately represent spatial and
temporal relationships of the underlying atmospheric state in equal measure.
Figure 1 shows the spatial distribution of the measurement sites. At the same
time, these sides represent the virtual locations of the power plants, i.e.,
wind energy or photovoltaic power plants, which correspond to the synthetic
data. The plot shows that wind speed measurements are also available at all
locations where radiation data is available. The reverse is not applicable, so
less than half of the sites have radiation measurements.

Figure 1: Geographical locations of weather measurements, respective sides of
the synthetic power measurements.

The underlying measurements for solar are from [8] and the wind resource
measurements from [9].

###  3.2 Input Features and Numerical Weather Prediction

We take input features from the ICON-EU weather model as a basis, described in
[10]. These numerical weather predictions were computed and provided by the
German Meteorological Service (_Deutscher Wetterdienst_) and provided for this
study by Fraunhofer IEE. For PV\- and wind energy, several features were
extracted and processed individually. In contrast to the surface synoptic
observation (SYNOP) measurements, the features from the ICON-EU model are
available within the considered forecast horizons (day-ahead scenario) with
hourly resolution. The hourly resolution available here forms the basis for
the final merged dataset.

#### Photovoltaic Input Feature

The features for the PV forecast are listed below, with feature names based on
those in [10]:

fcst_time

Forecast date and time for which the forecast is to be generated and to which
the below-mentioned features refer to in the format `yyyy-mm-dd HH:MM:SS`
[UTC].

nwp_fcst_horiz_hours

forecast horizon in relation to the start of the numerical weather prediction
model run at 00:00 of the previous day [h].

T_HAG_2_M

Air temperature 2m above the earth’s surface [°K].

RELHUM_HAG_2_M

Relative humidity 2m above earth surface [%].

PS_SFC_0_M

Air pressure at the earth’s surface [Pa].

U_GVL_60_HL

Zonal wind speed (wind parallel to latitude) at about 10m above ground [m/s].

V_GVL_60_HL

Meridional wind speed (longitude-parallel wind) at about 10m above ground
[m/s].

ASWDIFDS_SFC_0_M_INSTANT

Instantaneous solar diffuse radiation at the earth’s surface with a time lag
relative to the forecast time of -1h, 0h, or +1h (three features with the
suffixes _m1,_, or _p1) [W/$m^{2}$].

ASWDIRS_SFC_0_M_INSTANT

Instantaneous solar direct radiation at the earth’s surface with a time lag
related to the prediction time by -1h, 0h or +1h (three features with suffixes
_m1,_ or _p1) [W/$m^{2}$].

solar_azimuth

Azimuth angle of the sun position calculated using prediction time and the
toolbox PVLib, described in [11] [°].

solar_zenith

Zenith angle of the sun position calculated using prediction time and Toolbox
PVLib.

Except for the prediction time `fcst_time`, these features can be used
directly as inputs for nonlinear regression models. We use `fcst_time` to
assign the time to the corresponding target value, to other weather models, or
to calculate additional features, e.g., representing the season. Nevertheless,
`fcst_time` is less suitable as a direct input feature in the case of ML
methods since the test data will differ in the considered time and therefore,
an ML model will most likely not generalize well.
Irradiance features ending with `_INSTANT` differ from the original weather
model quantities due to our post-processing of original irradiance values. In
the ICON-EU weather model, these quantities are stored as mean values since
the beginning of the model run. Since we aim to predict the instantaneous
power of the system, we convert these quantities to instantaneous irradiance
features through the differences between temporally adjacent time steps.
In total, the synthetic PV dataset thus contains 15 different features, 14 of
which can be used directly as input to a regression model.

#### Wind Power Input Feature

In the case of the synthetic wind power data, the data are available with 19
comparable input features, which we describe in detail in the following list:

fcst_time

Forecast date and time for which the forecast is to be generated and to which
the below-mentioned features refer to in the format `yyyy-mm-dd HH:MM:SS`
[UTC].

nwp_fcst_horiz_hours

is the forecast horizon in relation to the start of the numerical weather
prediction model run at 00:00 of the previous day [h].

T_HAG_2_M

Air temperature 2m above the earth’s surface [°K].

RELHUM_HAG_2_M

Relative humidity 2m above earth surface [%].

PS_SFC_0_M

Air pressure at the earth’s surface [Pa].

U_GVL_58_HL

Zonal wind speed (wind parallel to latitude) at about 100m above ground with a
time lag related to the forecast time of -1h, 0h or +1h (three features with
suffixes _m1,_ or _p1) [m/s].

V_GVL_58_HL

Meridional wind speed (longitude-parallel wind) at about 100m above ground
with a time lag related to the forecast time of -1h, 0h or +1h (three features
with suffixes _m1,_ or _p1) [m/s].

U_GVL_60_HL

Zonal wind speed (wind parallel to latitude) at about 10m above ground with a
time lag related to the forecast time of -1h, 0h or +1h (three features with
suffixes _m1,_ or _p1) [m/s].

V_GVL_60_HL

Meridional wind speed (longitude-parallel wind) at about 10m above ground with
a time lag related to the forecast time of -1h, 0h or +1h (three features with
suffixes _m1,_ or _p1) [m/s].

ASWDIFDS_SFC_0_M_INSTANT

Instantaneous solar diffuse radiation at the Earth’s surface [W/$m^{2}$].

ASWDIRS_SFC_0_M_INSTANT

Instantaneous solar direct radiation at the Earth’s surface [W/$m^{2}$].

###  3.3 Static Data

We summarize the static data (or metadata) for all locations of the respective
energy source in the file `meta.csv`. The individual rows in the CSV represent
the static data for the individual locations. For both wind and PV, the file
contains the identification number (`loc_id`), the geographical coordinates
(`longitude` or `long` and `latitude` or `lat`), the file names of the
associated input and target data (`input_file_name` and `target_file_name`),
and a brief statistic in form of the number of samples per sample set
(`num_train_samples`, `num_test_samples`). Further static data differ
depending on the energy source, especially the physical parametrization.

#### Photovoltaic Parametrization

In the case of photovoltaics, one finds information about the module, the
inverter, and the wiring (modules per string and strings per inverter).
However, we selected only one configuration for simplicity, namely module:
Canadian Solar CS5P 220M (2009), inverter: ABB Micro 0 25 I OUTD US 208
(208V), with one single module per string, and one string per inverter. We
chose this configuration based on the experience that the exact module and
inverter parameters (e.g., rated power, efficiency, and temperature
coefficients) have a relatively small impact on the prediction quality,
especially after normalization and compared to the errors of the weather
model. Much more important, however, is the exact orientation of the plant.
The orientation parameters are drawn from a random distribution to have a
diverse collection of different orientations in the dataset. We sample the
module tilt angle from a uniform distribution in the interval between 0°
(flat) and 90° (vertical). Similarly, we sample the azimuth angle from a
uniform distribution between 90° (east) and 270° (west), with the south
defined as 180°.

#### Wind Power Parametrization

We summarize similar information about the simulated wind turbines in the
corresponding `meta.csv` file associated with the synthetic wind power
dataset. The file includes the hub height used for vertical interpolation, the
rotor diameter in meters, and the nominal power in kilowatts. Additionally, we
provide information about the respective turbine type. The latter information
is crucial to be able to reproduce the dataset since the corresponding power
curves were stored as manufacturer information in the simulation, similar to
[12]. The turbine power curve and the hub height of each entry in `meta.csv`
were individually selected by randomly choosing one of the possible hub
heights (either maximum or minimum) and one specific turbine type.

##  4 Synthetic Data Generation

This section describes in detail how we assemble the datasets. Fig. 2 depicts
the overall concept. On the left, various data sources are listed. We process
those data sources either through a physical model or feature pre-processing.
The resulting output of the synthetic data generation, namely target and input
data, is given on the right.
The top row, the top grey box, shows the process to synthesize the target
data. In this process, a power model transforms meteorological measurements
(SYNOP) and static plant configurations into a power time series. The used
power models are implemented either as a physical model (PV) or are given as
an empirical look-up table by the power plant manufacturers (power curves of
wind turbines). The resulting time series represents the synthetic power
measurement, which we provide as a target for a ML-based forecasting model.
Note that we do not use numerical weather prediction data to prepare the
target data. This process ensures that the expected forecast errors are
similar in magnitude and characteristics to real-world power forecasts.
The bottom row, the bottom grey box, shows the process of input data
preparation. The raw input features include numerical weather predictions and
the corresponding time stamp information at the nearest weather model grid
node to the measurements side. Depending on the energy source, certain
features are selected and pre-processed. This feature processing consists, for
example, of the time-lags of specific input signals concerning the prediction
time. The motivation for this is to provide samples that can even be used in
regression models that make an i.i.d. assumption. Other features, such as sun
angles in the case of photovoltaics, are also calculated and provided.

Figure 2: Flowchart for synthesing power data from weather measurements and of
the feature extraction process.

Furthermore, on the right side, there is a dashed box. This box symbolizes the
synchronization of the input data and the target variable. This
synchronization takes place according to a day-ahead forecast. Here, a
scenario was set, which assumes the daily creation of a day-ahead forecast.
More precisely, we assume that the 00:00 o’clock model run of the weather
model is delivered to the power forecast provider. Related to the origin of
the 00:00 model run, we synchronize the forecast horizons between 24h and 47h
inclusive with the power values with hourly resolution. The hourly resolution
of the weather prediction is adopted by down-sampling the weather
measurements, which initially have a 10min time resolution. Therefore,
interpolation is not needed. Furthermore, for each geographic power plant
location, the predictions of the nearest point of the NWP grid were extracted.

###  4.1 Photovoltaic Power Model

Radiance measurements are converted via a physical model into power
measurements to generate the synthetic power measurements. The underlying
physical model is the one from the simulation toolbox PVLib [11]. All
simulations use the same photovoltaic module type _Canadian Solar CS5P-220M
(220W) Solar Panel_ which is specified in more detail in [13]. Similarly, the
same type of inverter was used exclusively for all simulations, namely the
_MICRO-0.25/0.3/0.3HV-I-OUTD-US-208/240_ described in [14]. We assume that all
PV power plants contain only a single module as an additional simplification.
The synthetic measurement time series are normalized by the installed
capacity, i.e., the module peak power.
The SYNOP-power measurements include the ten-minute sum of incoming solar
radiation in Joule per cubic Centimeter and the ten-minute sum of the diffuse
solar radiation with the same unit. PVLib expects the global horizontal
irradiance (GHI), the direct normal irradiance (DNI) and the diffuse
horizontal irradiance (DHI) in Watt per square Meter, as well as wind speed
and air temperature. To overcome this difference, we transformed the measured
quantities accordingly by using the methods given in PVLib, i.e.,
_pvlib.irradiance.dni_ to estimate DNI based on GHI, DHI and Zenith angle.
Previously, the zenith angle was computed based on the geographic position and
time, while GHI and DHI are transformed under the assumption of a discrete
approximation of the energy-time derivative, i.e., that the power is the ratio
of a discrete change in energy in a given time interval. Wind speed and air
temperature remained with their PVLib-default values of 0m/s and 20° Celsius
for further simplification.
Finally, the simulation requires information about the location. This
information includes the geographic coordinates, which correspond to the
coordinates of the weather measurement and the altitude. For simplicity, we
assume the altitude to be 0m above sea level. All other model inputs, such as
the albedo, were left at the PVLib default values.
Figure 3(a) and Figure 4 show the simulated PV power in relation to the
predicted direct irradiation on a horizontal plane. An increase in irradiation
leads to an expected increase in the power generation in figure 3(a). At the
same time, the inherent uncertainty of the weather prediction and the missing
consideration of the module orientation leads to a spread of the point cloud
in the y-direction. This difference is also a typical behavior in real-world
datasets.
Figure 4 shows the normalized power output (blue line) of the same plant in a
three-week window together with the normalized horizontal direct irradiation
(yellow line). The slight shift of the blue curve, compared to the yellow
curve, to the right results from a slight west-ward orientation of the
simulated PV module. While sunny days predominantly show agreement in the
yellow and blue curve profile, cloudy days show more extensive and random
deviations between the synthetic power data and the predicted solar resource.
That is typical and results from the NWP difficulties to predict exact cloud
positions. Furthermore, the predicted solar resources appear to fluctuate less
than the synthetic power generation. This difference in fluctuation also
results from the properties of the NWP, which rather forecasts an expected
value of a local patch (several thousand meters in size) compared to the local
point of the single plant and, with that, to smooth out any detailed cloud
structures.

###  4.2 Wind Energy Power Model

A simple power curve model was used to calculate wind power. In the first
step, we extrapolate the measured wind speed $u_{ref}$ from 10m height to the
height of the turbine. For this purpose, the so-called wind profile power law
is applied, described in [15]. With that, we compute the wind speed at the
height of the wind turbine $u_{ref}$ with

$\displaystyle u=u_{ref}\left(\frac{z}{z_{ref}}\right)^{\alpha},$ (1)

where the hub height $z$ of the wind turbine, the measurement height
$z_{ref}=10m$, the measured wind speed $u_{ref}$ and a stratification
coefficient $\alpha=\frac{1}{7}$ (assumption of neutral stratification) are
used. In the next step, a linear interpolation is applied together with a
lookup table of the manufacturer’s power curve as given in [12]. Each power
curve lookup table is defined in terms of reference wind speeds and reference
power measurements, discretized with a resolution of 1 m/s up to a maximum
reference wind speed of 25 m/s. Other possible impact factors, such as air
density, turbulence, or atmospheric stability are ignored for simplicity.
Figure 5 shows the temporal development of the generated wind power output,
normalized by the plant capacity, of a single wind power plant for a three-
week window as an example. Due to the characteristic properties of the power
curve, there is no feed-in power at very low wind speeds (ca. < 1-2 m/s). If
high wind speeds occur (ca. > 12 m/s), the generation is limited by the
nominal power of the wind turbine. These properties form the characteristic
shape of such wind power time series.
At the same time, the yellow curve shows the forecasted wind speed. Here we
can observe that wind speed and power are correlated since higher predicted
wind speeds also tend to have higher power values. This correlation becomes
even more evident in figure 3(b), where the generated power, the synthetic
measurement, is plotted above the predicted wind speed, given by the NWP. The
typical S-shaped course of the power curve appears here. At the same time,
however, there is also a wide spread. The spread, as in the case of PV, is due
to the forecast errors of the weather resource. Analogously to PV, this
uncertainty is close to the real world and thus also shows up in actual wind
power forecast data.

(a) Feed-in of a virtual PV plant plotted against direct-horizontal radiation
(ICON-EU).

(b) The output of a virtual wind turbine over the magnitude of the predicted
wind speed vector (ICON-EU).

Figure 3: Scatter plots of the synthetic power generation (SYNOP based) over
the corresponding predicted weather resource. Figure 4: Exemplary three weekly
course of the synthetic power generation of a virtual PV power plant and the
irradiation normalized with the nominal power and 1000 W/m². The data was
linearly interpolated to a 15-minute resolution. Figure 5: Three-week history
of synthetic power generation from a virtual wind turbine and the predicted
underlying wind speed normalized by the rated power and maximum wind speed,
respectively. The data was linearly interpolated to a 15-minute resolution.

##  5 Experimental Results

The following section conducts an experiment that serves as the baseline for
future research. Moreover, the experiment allows comparing forecast errors of
the proposed synthetic dataset with real-world forecast errors, showing our
dataset’s applicability for future research in renewable power forecasts. We
evaluate physical forecasting techniques on the introduced datasets within the
following experiment. Physical approaches are essential as they are typically
utilized when only limited data is available for training. At the same time,
there is an increasing interest in providing high-quality forecasts through ML
models even with limited data. Therefore, we compare the results of physical
models with the gradient boosting regression tree (GBRT). We choose the GBRT
here as it generalizes well with a limited amount of data and is known to
mitigate the effects of overfitting.

To assess the forecast quality of the GBRT concerning reduced data, as it is
common with TL, we limit the training data to the last $7$,$14$,$30$,$60$ or
$90$ days of training data of a season. Respectively, we repeat this
experiment for each season’s different number of training data to ensure that
results account for seasonal patterns. To increase the number of training data
within this scenario, we linearly interpolate the data to have a $15$-minute
resolution. Additionally, we make one experiment where we consider the
complete training data, indicated through $365$, and compare those forecast
errors with an example from real-world datasets. We consider time steps with
the test flag as test data and the remaining data as training data.

Most often the forecast error, in renewable power forecast, is determined by
the normalized root mean squared error (nRMSE) given by

$\text{nRMSE}=\sqrt{\frac{1}{N}\sum_{i=0}^{i=N}{(y_{i}-\hat{y}_{i})^{2}}},$
(2)

where $y_{i}$ is the $i$-th normalized response from the target and
$\hat{y}_{i}$ is the prediction from a source model on the target. Note that
in the context of renewable power forecasts, we normalize the response $y_{i}$
by the nominal power to assure comparability of the error for different parks.
To express an improvement of one model over another model we utilize the skill
given by

$\text{Skill}=\text{nRMSE}_{gbrt}-\text{nRMSE}_{phy},$ (3)

where $\text{nRMSE}_{gbrt}$ is the nRMSE of the GBRT model and
$\text{nRMSE}_{phy}$ of the physical model, respectively. Consequently, values
below zero indicate an improvement of the GBRT over a physical forecasting
technique.

We optimize the GBRT for each park, season, and number of training data
through a grid search by three-fold cross-validation based on the available
training data. Thereby, the learning rate is evaluated at
$[${10}^{-6}\text{\,}\mathrm{,}$$3.1\text{\times}{10}^{-6}$,\ldots,$3.1\text{\times}{10}^{-1}$,1]$.
The number of estimators is $300$ and the maximum depth is one of $[2,4,6,8]$.
Other values are the default ones of scikit-learn 222https://scikit-
learn.org/, Version 0.24, accessed 2022-02-28.

We utilize the same parametrization for the physical models for creating the
power measurements. The only difference is that we now use the direct
radiation and wind speed, respectively, from the NWP as input. The physical
model for the wind dataset is indicated through Enercon in the following and
PV Physical for the PV dataset. Additionaly, for the wind dataset, we compare
the GBRT results with empirical McLean power curves [16]. This power curve is
well known and considers typical influences by different terrains.

PV day-ahead power forecasting problems often have an nRMSE between $0.06$ and
$0.12$ [17]. The mean nRMSE values of the PV baseline is at $0.085$ on the
proposed synthetic dataset, see Tbl. 1. Forecast errors of the GBRT model are
between $0.12$ and $0.072$ depending on the amount of available training data.
With an increasing amount of training data, the forecast error reduces for
this model. The mean forecast error of the GBRT is only better with the
complete training data. Fig. 6 shows the evaluation concerning the skill for
the PV dataset. Up to $14$ days of training data, the GBRT model has
substantial positive outliers of the skill. With additional training data,
this effect reduces. Beginning from $60$ days of training data, the median
skill is close to zero, indicating the GBRT is as good as the physical
baseline in most cases. With $90$ days of training data and the complete
dataset, the GBRT has improvements over the physical baseline.

Table 1: Mean nRMSE values of the GBRT across all available parks for different amounts of training data in comparison to physical models for the PV dataset. Best values for the amount of training data are highlighted in bold. | GBRT | PV Physical
---|---|---
Number of Days |  |
7 | 0.120 | 0.085
14 | 0.107 | 0.085
30 | 0.092 | 0.085
60 | 0.089 | 0.085
90 | 0.086 | 0.085
365 | 0.072 | 0.085
Figure 6: Skill between the GBRT model and the physical baseline of the
dataset for the PV dataset. Values below zero indicate an improvement of the
GBRT over the physical reference.

Tbl. 2 summarizes the mean nRMSE for the wind dataset. The Enercon baseline
has a mean nRMSE of $0.21$. Wind day-ahead power forecasting problems
typically have an nRMSE between $0.1$ and $0.2$ [17]. Depending on the
utilized empirical McLean power curve, forecast values of this approach range
between $0.212$ and $0.228$. The GBRT forecasting technique has the best mean
nRMSE for all numbers of available training data. With seven days of training
data, the GBRT has an nRMSE of $0.196$, with $90$ days of training data a
nRMSE of $0.147$, and with the complete data, an error of $0.125$. The
improvements of the GBRT over the other models are also given through the
skill in Fig. 7. In contrast to results from the PV dataset, the median skill
is below zero already with seven days of training data. However, with less
than $30$ days of training data, substantial outliers are present compared to
the Enercon baseline. These are not present with more than $30$ days of
training data.

All in all, we can observe that the forecast error of the physical baseline
and, in particular, errors from the ML technique are comparable to those given
by real-world power measurements. For the PV dataset, we see that the physical
baseline is robust, especially in the case of limited data. At the same time,
these excellent results also relate to the fact that the exact physical
characteristics are given, which is most often not the case in real-world
forecasting problems. In the case of the proposed wind dataset, the GBRT
provides a strong baseline for future research. In a comparison of those two
datasets, the physical approach for the PV dataset considers more physical
characteristics in comparison to the Enercon baseline. At the same time, the
PV power forecasting problem can be considered more linear in contrast to wind
power forecasts, which makes it easier to model all dependencies within a
physical baseline. Interestingly, errors of the McLean-Upland empirical power
curve are close to that of the Enercon baseline. This observation lets us
conclude that it is a suitable baseline for limited data when a wind turbine’s
physical characteristics are not given.

Table 2: Mean nRMSE values of the GBRT across all available parks for different amounts of training data in comparison to the physical model and different McLean power curves for the wind dataset. Best values for the amount of training data are highlighted in bold. | GBRT | Enercon | McLean- | McLean- | McLean- | McLean-
---|---|---|---|---|---|---
|  |  | Lowland | Lowland-Regulated | Offshore | Upland
Number of Days |  |  |  |  |  |
7 | 0.196 | 0.210 | 0.228 | 0.216 | 0.239 | 0.212
14 | 0.178 | 0.210 | 0.228 | 0.216 | 0.239 | 0.212
30 | 0.156 | 0.210 | 0.228 | 0.216 | 0.239 | 0.212
60 | 0.154 | 0.210 | 0.228 | 0.216 | 0.239 | 0.212
90 | 0.147 | 0.210 | 0.228 | 0.216 | 0.239 | 0.212
365 | 0.125 | 0.210 | 0.228 | 0.216 | 0.239 | 0.212
Figure 7: Skill between the GBRT model and the physical baseline (Enercon) and
different McLean power curves for the wind dataset. Values below zero indicate
an improvement of the GBRT over those references.

##  6 Conclusion and Future Work

We successfully created a realistic wind and photovoltaic power forecast
dataset within the article. Power measurements are generated based on real-
world weather measurements, ideal for creating power time series through
physical models. At the same time, considering numerical weather prediction as
input features for the ML model, results show that forecast errors are in line
with other real-world renewable power datasets. By providing metadata for each
plant, we provide the possibility for transfer learning, multi-task learning,
and zero-shot learning for future research by considering those physical
characteristics.

Acknowledgments This work has been partially carried out within the project
TRANSFER (01IS20020B) funded by BMBF (German Federal Ministry of Education and
Research) and the research project gridcast (Fkz. 0350004A) funded by the BMBK
(German Federal Ministry for Economic Affairs and Climate Action).
Furthermore, we would like to express our special thanks to the German Weather
Service (DWD) and the Fraunhofer Institute for Energy Economics and Energy
System Technology (IEE) for providing the weather data.

## References

  * [1] Tao Hong, Pierre Pinson, and Shu Fan.  Global energy forecasting competition 2012, 2014.
  * [2] Andre Gensler, Janosch Henze, Bernhard Sick, and Nils Raabe.  Deep Learning for solar power forecasting — An approach using AutoEncoder and LSTM Neural Networks.  In IEEE International Conference on Systems, Man, and Cybernetics (SMC), pages 002858–002865. IEEE, 2016.
  * [3] André Gensler, Bernhard Sick, and Vitali Pankraz.  An analog ensemble-based similarity search technique for solar power forecasting.  In 2016 IEEE International Conference on Systems, Man, and Cybernetics (SMC), pages 002850–002857, 2016.
  * [4] Jens Schreiber, Stephan Vogt, and Bernhard Sick.  Task Embedding Temporal Convolution Networks for Transfer Learning Problems in Renewable Power Time-Series Forecast.  In ECML, 2021.
  * [5] Ghadah Alkhayat and Rashid Mehmood.  A review and taxonomy of wind and solar energy forecasting methods based on deep learning.  Energy and AI, 4:100060, 2021.
  * [6] Jens Schreiber and Bernhard Sick.  Emerging Relation Network and Task Embedding for Multi-Task Regression Problems.  In International Conference on Pattern Recognition, 2020.
  * [7] Stephan Vogt, Axel Braun, Jan Dobschinski, and Bernhard Sick.  Wind Power Forecasting Based on Deep Neural Networks and Transfer Learning.  In 18th Wind Integration Workshop, page 8, 2019.
  * [8] Dwd climate data center (cdc): Historical 10-minute station observations of solar incoming radiation, longwave downward radiation and sunshine duration for germany, version v1.  https://opendata.dwd.de/climate_environment/CDC/observations_germany/climate/10_minutes/solar/.  Accessed: 2020-09-06.
  * [9] Dwd climate data center (cdc): Historical 10-minute station observations of mean wind speed and wind direction for germany, version v1.  https://opendata.dwd.de/climate_environment/CDC/observations_germany/climate/10_minutes/wind/historical/.  Accessed: 2020-09-06.
  * [10] Daniel Reinert, Helmut Frank, and Florian Prill.  ICON database reference manual.  Deutscher Wetterdienst, 2016.
  * [11] William F. Holmgren, Clifford W. Hansen, and Mark A. Mikofski.  pvlib python: a python package for modeling solar energy systems.  Journal of Open Source Software, 3(29):884, 2018.
  * [12] Enercon GmbH.  ENERCON Produktübersicht, 6 2015.
  * [13] SolarDesignTool.  Canadian Solar CS5P-220M, accessed January, 2022.
  * [14] fimer.com.  Product manual MICRO-0.25/0.3/0.3HV-I-OUTD-US-208/240, 2014 (accessed January, 2022).
  * [15] Jawad S Touma.  Dependence of the wind profile power law on stability for various locations.  Journal of the Air Pollution Control Association, 27(9):863–866, 1977.
  * [16] F. Van Hulle, J. O. Tande, K. Uhlen, et al.  Further Developing Europe’s Power Market for Large Scale Integration of Wind Power - Equivalent Wind Power Curves.  European Wind Energy Association (EWEA), 2008.
  * [17] Jens Schreiber, Malte Siefert, Kevin Winter, Arne Wessel, Rafael Fritz, Garrett Good, André Schella, Jewgenija Muraschko, and Stefan Staedler.  Abschlussbericht Projekt Prophesy : Prognoseunsicherheiten von Windenergie und Photovoltaik in zukünftigen Stromversorgungssysteme.  German National Library of Science and Technology, page 168, 2020\.


# First results from BRASS-p broadband searches for hidden photon dark matter

Fayez Bajjali  Sven Dornbusch  Marko Ekmedžić  Dieter Horns  Christoph
Kasemann  Andrei Lobanov  Le Hoang Nguyen  Martin Tluczykont  Gino Tuccari
Johannes Ulrichs  Gundolf Wieching  Anton Zensus

###### Abstract

We discuss first results from hidden photon dark matter searches made with a
prototype of the Broadband Radiometric Axion/ALPs Search Setup (BRASS-p) in
the range of particle mass of $49.63\text{\,}\mathrm{\SIUnitSymbolMicro
eV}74.44\text{\,}\mathrm{\SIUnitSymbolMicro eV}$ (frequency range of
$12\text{\,}\mathrm{GHz}18\text{\,}\mathrm{GHz}$). The conceptual design of
BRASS and a detailed description of its present prototype, BRASS-p, are given,
with a view of the potential application of such setups to hidden photon,
axion, and axion-like particle (ALP) dark matter searches using heterodyne
detectors in the range of particle mass from
$40\text{\,}\mathrm{\SIUnitSymbolMicro eV}$ to
$4000\text{\,}\mathrm{\SIUnitSymbolMicro eV}$ ($10\text{\,}\mathrm{GHz}$ to
$1\text{\,}\mathrm{THz}$). Pioneering measurements made with BRASS-p achieve
the record sensitivity of (0.3–1.0)$\times$$10^{-13}$ to the kinetic mixing
between the normal and hidden photons, assuming the dark matter is made
entirely of unpolarized hidden photons. Based on these results, a discussion
of further prospects for dark matter searches using the BRASS-p apparatus is
presented.

##  1 Introduction

Many well-motivated extensions of the Standard Model (SM) of Particle Physics
contain a hidden (or dark) sector which provides an explanation for the
missing components observed in the Universe while remaining minimally coupled
via gravity to the common forms of matter (see [1] and references therein).
One of such missing, invisible components in the Universe is the so-called
dark matter that is responsible for the largest fraction, 85 %, of the mass
content of the Universe [2]. The best-studied dark matter particle candidate
is a weakly interacting, massive particle (WIMP), that was produced thermally
during the early Universe and is naturally predicted, for instance, as the
lightest supersymmetric particle (LSP) in widely-discussed extensions of the
SM [3].

So far, direct and indirect searches for WIMPs have not revealed the presence
of WIMP-type dark matter. Further improvements in these searches approach some
intrinsic limitations. Direct searches for elastic scattering of WIMPs in
well-shielded underground laboratories are eventually limited by the neutrino
floor of background events from solar and cosmogenic (core-collapse supernova)
neutrinos. Indirect searches (e.g., gamma-ray emission from self-annihilating
LSPs) are limited by the presence of cosmic accelerators that produce an
indistinguishable gamma-ray background. Collider-based searches are limited to
searches for particles that are not predicted in the SM. Even though this is
not to be confused with searching for dark matter, so far no conclusive
evidence for deviations from the SM have been found.

Presently, the notion of WIMP-type dark matter is theoretically well-
established, but there are so far neither experimentally nor observationally
established indications for the existence of WIMPs. Among the non-WIMP
particle-type candidates for dark matter, Weakly Interacting Slim Particles
(WISPs) have gained increased attention in the past years. The two main
representatives of WISPs are the axion or axion-like particles (ALPSs) and
hidden photons (HPs) [4, 5, 6, 7]. Even though the underlying motivations for
the introduction of axions/ALPs and HPs and their respective cosmological
phenomenology are very different, the search strategies in laboratory
experiments share some similarities [8]. An experiment that searches for WISPs
particle sourced from the halo dark matter is called haloscope Almost every
haloscope experiment searching for WISPs relies on a photon signal that
originates from the conversion of WISPs to photons. In the case of pseudo-
scalar axion/ALPs, the inverse Primakoff effect in the presence of an external
magnetic field allows the conversion from WISPs to photon. For HPs, the
relevant process is the kinetic mixing between hidden photons and photons [5,
9].

The resulting signal of the converted photons is often enhanced using a low
dissipation resonant cavity e.g., ADMX [10], WISPDMX [11], HAYSTAC [12, 13],
and various cavity-based experiments in CAPP, Korea [14, 15]. Cavity-based
haloscopes that are operated at a resonant frequency $\nu_{cav}$ are limited
to a WISP-mass $m$ such that $h\nu_{cav}\approx mc^{2}$, neglecting the
kinetic energy of the gravitationally bound dark matter halo. The linear
dimension of the resonating cavity scales approximately with the Compton
wavelength of the dark matter particle, such that with

$\lambda\approx 6\leavevmode\nobreak\
\mathrm{cm}\left(\frac{mc^{2}}{20\leavevmode\nobreak\
\mu\mathrm{eV}}\right)^{-1},$ (1.1)

usable volume becomes progressively smaller for higher masses. For the mass
range beyond tens of $\mu$eV and up to a few meV, experiments with resonant
enhancement from metamaterials [16, 17] or boosting from dielectric material
stacks [18, 19] are under development.

A different approach to search for DM makes use of the conversion of HPs into
an electromagnetic wave at the surface of a conductor [20]. For HP with a
mass, $m$, and a photon coupling, $\chi$, the conversion results from
excitation of free electrons on the conductor surface by the residual electric
field $\textbf{E}_{\mathrm{hp}}=\chi\,m\,\textbf{X}_{\mathrm{hp}}$ vested in
the HP dark matter field, $\textbf{X}_{\mathrm{hp}}$, propagating through the
conductor. For conversion of an axion/ALPs with a mass, $m$, the respective
electric field is induced by a transverse magnetic field,
$B_{\mathrm{\perp}}$, (the field component oriented parallel to the conducting
surface) extending up to at least a wavelength, $\lambda_{\mathrm{a}}=2\pi/m$,
of the electromagnetic wave generated by the conversion. With such settings,
the measurement sensitivity to the HP dark matter signal is proportional to
the area $A_{\mathrm{conv}}$ of the conversion surface, while for the
axion/ALPs dark matter it is $\propto
A_{\mathrm{conv}}B_{\mathrm{\perp}}^{2}$.

Adopting this approach, we describe here the first prototype of a Broadband
Radiometric axion/ALPs Search Setup (BRASS) which will ultimately aim to
employ a permanently magnetised conversion surface with
$A_{\mathrm{conv}}\approx 100\leavevmode\nobreak\ \mathrm{m}^{2}$ and
$B_{\mathrm{\perp}}\approx 1$ T. The BRASS prototype (BRASS-p) presently
operates with $A_{\mathrm{conv}}=6.0\leavevmode\nobreak\ \mathrm{m}^{2}$ and
does not have the permanent magnet panels in place, which limits its current
application to searches for HP dark matter. The physical foundations and the
conceptual design of BRASS-p measurements are described in Section 2. The
current BRASS-p setup is presented and discussed in Section 3. Section 3.7
describes the system noise calibration procedures, and the data taking run
from the HP dark matter search with BRASS-p in the extended frequency-range
from $\nu=12\leavevmode\nobreak\ \mathrm{GHz}$ to $\nu=18\leavevmode\nobreak\
\mathrm{GHz}$, which corresponds to the HP mass range from
$m=41.35\leavevmode\nobreak\ \mu\mathrm{eV}$ to $m=74.44\leavevmode\nobreak\
\mu\mathrm{eV}$.

##  2 Hidden photon dark matter signal in BRASS

###  2.1 Hidden photon conversion

The conversion of HP dark matter to detectable photons is driven by the
kinetic mixing [21, 22] of the HP $\tilde{X}^{\mu}$ (field
$\tilde{X}^{\mu\nu}$) with photons $A^{\mu}$ (field $F^{\mu\nu}$). The
strength of the mixing is related to the mixing angle $\chi\ll 1$ that is
predicted to typical values between $\chi\approx 10^{-12}$ and $\chi\approx
10^{-3}$ (see reference [20] and references therein).

The electric field that results from the kinetic mixing is given by

$\mathbf{E}_{\text{hp}}=\chi m\mathbf{X}_{\text{hp}},$ (2.1)

with $\sqrt{|\mathbf{X}_{\text{hp}}|^{2}}=m\sqrt{2\rho_{DM}}$ for a given
local dark matter density $\rho_{DM}$.***The natural units with $c=\hbar=1$
are assumed throughout the text.

The resulting (time-averaged) amplitude of the electric field is

$\sqrt{\langle|\mathbf{E}_{\text{hp}}|^{2}\rangle}=\chi\sqrt{2\rho_{DM}}\approx
3.3\times
10^{-9}\frac{\mathrm{V}}{\text{m}}\left(\frac{\chi}{10^{-12}}\right)\left(\frac{\rho_{DM}}{0.3\leavevmode\nobreak\
\mathrm{GeV}/\mathrm{cm}^{3}}\right).$ (2.2)

At the boundary of a conductor surface, the tangential component of the
electric field $\bf{E}_{\text{hp}}$ that results from the mixing with the HP
field oscillates at the frequency $\nu\approx 4.8\leavevmode\nobreak\
\mathrm{GHz}(m/20\leavevmode\nobreak\ \mu\mathrm{eV})$. The oscillating
electrons in the conductor will emit radiation perpendicular to the conductor
surface [20].

The strength of the field that is relevant for conversion is therefore the
component tangential $\mathbf{E}_{hp,\parallel}$ to the surface of the
conductor. This component is maximal for a polarisation of $\bf{X}_{hp}$
parallel to the surface ($\vartheta=0$) and disappears for a perpendicular
polarisation ($\vartheta=\pi/2$). In general,
$|\mathbf{E}_{hp,\parallel}|=\chi m|\cos\vartheta||\mathbf{X}_{hp}|$ depends
on the orientation of the polarisation of the HP $\bf{X}_{hp}$.

In the following, we consider the scenario in which the hidden photon dark
matter particles behave like a gas, and their polarisation direction points to
random directions everywhere in space. Hence, the averaged emitted power
$P_{\text{hp}}$ over the surface area of the conversion surface $A$ is given
by

$P_{\text{hp}}=A\langle|\mathbf{E}_{hp,\parallel}|^{2}\rangle=A\frac{2}{3}\chi^{2}\rho_{DM}\,.$
(2.3)

In DM search experiments, this expectation is matched against the actual
output power $P_{\mathrm{det}}$ or (in case of non-detection) the standard
deviation sensitivity (noise power) $\sigma_{\mathrm{det}}$ of the
measurement, thereby constraining the HP coupling strength, $\chi$. In
experiments in which individual photons cannot be counted, the measured power
and sensitivity of detection are described most generally by the radiometer
equation

$P_{\mathrm{det}}=(\mathrm{S/N})\,\sigma_{\mathrm{det}}=(\mathrm{S/N})\,k_{\mathrm{B}}T_{\mathrm{sys}}\,\mathrm{BW}^{1/2}\,\tau^{-1/2}\,,$
(2.4)

where S/N is the signal-to-noise level of detection, or exclusion, of the
signal, $k_{\mathrm{B}}$ is the Boltzmann constant, $T_{\mathrm{sys}}$ is the
system noise temperature of the measurement apparatus, and BW and $\tau$ are
the measurement bandwidth and duration.

It is worthwhile mentioning, that the expected signal is at all times fully
linearly polarised [23]. Intriguing enough, the direction of the polarisation
can change smoothly over time for the case where the orientation of
$\bf{X}_{hp}$ remains constant over some extended region in space such that
the local polarisation direction in the rotating laboratory frame changes
smoothly and in a predictable way with time. However, we are not considering
these more model-dependent scenarios here and leave it for future
investigations and future data-taking.

For the DM material located in a virialised Galactic halo, the received signal
is expected to have a Maxwellian profile with a characteristic fractional
width of ${\Delta\nu}/{\nu}\sim 10^{-6}$. For non-virialised scenarios
involving DM streams, signals with ${\Delta\nu}/{\nu}\sim 10^{-7}$ or even
smaller can be expected [24, 25]. To optimise the detection probability of
such signals, measurements with the fractional spectral resolution down to
$\sim 10^{-9}$ may be needed.

###  2.2 Conceptual design of BRASS

The conceptual optical design of BRASS (Fig. 1) comprises a shielded chamber
with a flat, conducting conversion surface at which the electromagnetic signal
from HP or axion/ALPs DM is generated and a parabolic mirror concentrating
this signal onto a detector. Operations with multiple chambers are also
principally feasible, allowing for enhancing the detection and directional
sensitivity. The actual setup of the BRASS prototype experiment operates with
a single chamber and employs a $D=2.5$ m reflector with a focal distance of
$f=4.8$ m.

$\mathcal{A}$$\mathcal{D}$$\gamma_{\text{hp}}$$\gamma_{\text{hp}}$ Figure 1:
Conceptual optical scheme of BRASS. A flat, conducting, and permanently
magnetised conversion surface ($\mathcal{A}$) is employed for generating an
electromagnetic signal arising in response to the electric field of the dark
matter propagating through. This electric field is either inherent to the DM
particles themselves (for hidden photons) or induced by the magnetic field
near the conversion surface (for axions/ALPs). The resulting electromagnetic
signal is focused by the parabolic reflector and detected using a heterodyne
receiver ($\mathcal{D}$) with a broadband digital backend.

The outgoing electromagnetic wave generated at the conversion surface by the
passage of DM particles can be treated as a plane wave if the electric field
$\mathbf{E}_{\mathrm{hp}}$ oscillates in phase over the full area of the
conversion surface. This conditions is satisfied for as long as the coherence
length of the HP field is large against the linear dimension of the conversion
surface. The coherence length can be approximated by the de Broglie
wavelength,

$\lambda_{d.B.}\approx\frac{2\pi}{m\,\beta}\approx 60\leavevmode\nobreak\
\mathrm{m}\frac{20\leavevmode\nobreak\
\mu\mathrm{eV}}{m}\frac{10^{-3}}{\beta}\,,$ (2.5)

where $\beta\approx 10^{-3}$ the velocity of the dark matter flow. For the
conversion surface of BRASS-p with a linear dimension
$d=2.5\leavevmode\nobreak\ \mathrm{m}$, the oscillations will be quasi-
coherent for masses $m\lesssim 0.5\leavevmode\nobreak\ \mathrm{meV}$.

To quantify the signal properties, taking into account the diffraction and
reflections within the apparatus, we simulate the optical setup of BRASS-p
using the finite element method (FEM) implemented in the COMSOL
Multiphysics${}^{\text{\textregistered}}$ package. We adopt the spatial
geometry of BRASS and simplify the problem to two dimensions, so that the
essential components (the conversion surface and the parabolic reflector) with
their respective geometry are embedded through matching boundary conditions
that mimic an infinite free space. The simulation setup is constructed
assuming that the electric field ($\mathbf{E}_{\mathrm{hp}}$) is generated at
the boundary of the conversion surface and it excites a plane wave propagating
outward. The surface of the parabolic mirror is simulated as a perfect
electric conductor. The entire setup is spatially enclosed by a perfect
absorbing surface which approximates the free space conditions.

(a)(b)

Electric Field (V/m)

PEC

Figure 2: HP signal propagation. The coupling factor $\chi=10^{-13}$ is
selected for the purpose of demonstration. (a) The $z$ component of the
electric field is excited by the conversion of hidden photon dark matter at
the surface of the conductor. (b) This compound figure shows the propagation
of the plane waves in two scenarios: with (left) and without (right) the
parabolic mirror (the green edge at the top right). The dashed box indicates
the region presented by (a).

The simulation is performed for a frequency of 14 GHz, and the magnitude of
the simulated electric field corresponds to the signal strength expected from
the hidden photon dark matter with the photon coupling $\chi=10^{-13}$. The
maximum field strength obtained in the simulation is $\approx 5\times
10^{-10}\leavevmode\nobreak\ \mathrm{V/m}$, which is consistent with the
expected time averaged field strength given by Eqn. 2.2.

An example of the resulting instantaneous electromagnetic field near the
conversion surface is shown in Fig. 2a, where the diffraction at the edges of
the surface can be clearly seen. The effect of the parabolic reflector on the
electric field distribution can be assessed from Fig. 2b, which compares the
generated field freely propagating outward (left panel), to the one focused by
the parabolic mirror onto the detector (right panel).

The overall optical efficiency, $\eta_{\mathrm{opt}}$, of the signal focusing
can be estimated by integrating the total power in the freely outgoing
electromagnetic wave to the total power at the surface of the receiver
aperture. For the simulations shown in Fig. 2, $\eta_{\mathrm{opt}}\approx
0.98$ is obtained. The simulation shows a good approximation of the BRASS-p
optical efficiency in the entire 12 GHz to 18 GHz frequency range for which
$\lambda_{\mathrm{d.B.}}/D=6.4$–$11.6$.

##  3 BRASS-p

The present prototype setup BRASS-p is shown in Fig. 3 and its main parameters
are summarised in Table 1. BRASS-p is located in a climatised and shielded
room with a 60 dB radiation shielding effectiveness up to 18 GHz. The
conversion surface of BRASS-p consists of 24 $50\times 50$ cm aluminium panels
The 24 panels are arranged in a $5\times 5$ square pattern, with the central
slot left empty for the microwave beam throughput to the receiver, while the
gap between the panels is 2 cm. Ultimately, the effective conversion area
where the emitted signal arrives to the mirror is
$A_{0}=4.44\leavevmode\nobreak\ \text{m}^{2}$.

The conversion surface and the parabolic reflector are separated by 4.5 metres
and mounted on two separate frames. The frame of the conversion surface also
supports a broadband $12\text{\,}\mathrm{GHz}18\text{\,}\mathrm{GHz}$ receiver
located behind the conversion surface. The rest of the detection apparatus is
placed outside of the shielded room, in order to reduce the radio frequency
interference (RFI) produced by various electronic components.

Figure 3: BRASS-p setup, currently under operation in a radiation-shielded laboratory at University of Hamburg. Conversion surface | Panel Size | 50 cm $\times$ 50 cm
---|---|---
Number of Panel | 24
Effective Conversion Area | 4.44 m2
| Direction | 120° (SE)
Parabolic mirror | Diameter | 2.5 m
Focal Length | 4.8 m
Surface RMS | 4-6 $\mu$m
Receiver frontend | Frequency | 12-18 GHz
Polarization | Dual Linear (H+V)
Receiver Temperature | 15 K
Output IF Bandwidth | 4096 MHz
Digital backend | Maximum Sampling Rate | 8192 Msamples/s
Maximum Output Bandwidth | 4096 MHz/polarisation
Spectral Resolution | 625 Hz, 125 Hz, 25 Hz
Location | 53°34’40"N,9°53’8"E
Table 1: Overview of the BRASS-p key components and their parameters.

###  3.1 Mechanical and optical setup

The parabolic reflector of BRASS-p is mounted on a fixed metal frame anchored
to the floor. It is assembled from 13 aluminium panels, with each panel having
a surface roughness of less than 2 $\mu$m. A laser scan of the entire
reflector has shown that its surface deviations from the designed parabolic
geometry have an r.m.s of 4 $\mu$m–6 $\mu$m, thus allowing for its to be
operated at frequencies of up to $\approx$4\text{\,}\mathrm{THz}$$ for which
the standard antenna formalism [26] yields a reduction of the gain by 3 dB.

The other two main optical components of BRASS-p (the conversion surface and
the corrugated horn feed of the receiver) are aligned with respect to the
parabolic reflector. The receiver feed is oriented along the optical axis of
the reflector and placed at its focal point. The conversion surface is
orthogonally aligned with the optical axis of the reflector, with the
alignment made both for each of the 24 individual panels and for the overall
surface. Dedicated ray-tracing calculations have demonstrated that the angular
accuracy of $\approx 1^{\circ}$ is required for the alignment to ensure the
required optical performance [27].

The frame of the converter surface is mounted on wheels, and at the first step
of the alignment the frame is aligned orthogonally to the optical axis of the
reflector. Each of the converter surface panels is then aligned individually
with respect to the frame. The orientation of each panel has been verified
with two independent methods. First, a commercial 3D-laser scanning device
(Leica RTC360) has been deployed in the middle point of the lab to record the
positions of the parabolic reflector as well as of the panels. Fitting a flat
profile to the point cloud of each panel returns the deviation of 0.3 mm. The
surface deviation is due to the bending from the mechanical stress between the
actuators that attached to the panels. In addition to that, the orientation of
each panel was examined using a custom-made alignment set attached to it which
comprises three ranging lidars and a CCD camera to register the positions of
the lidar spots on the parabolic dish [27]. Geometric analysis of the
measurements made with these two approaches confirms that the surface normal
of the panels and the optical axis of the reflector have been aligned within
$\approx 0.2^{\circ}$.

In order to align the receiver feed, a cap with a laser is mounted on the
opening of the feed. The laser traces the optical axis of the feed. The feed
is then adjusted so that the laser spot is visible behind the small hole ($2$
mm in diameter) at the centre of the parabolic reflector. With this approach,
the optical axes of the receiver feed and the parabolic reflector have been
aligned to within $\approx 0.02^{\circ}$. The offset between the focal point
and the phase centre is estimated at $\pm 0.1$ mm.

###  3.2 Receiver

BRASS-p operates with a Ku-band receiver designed and manufactured at the
MPIfR, Bonn. The receiver covers a frequency range of
$12\text{\,}\mathrm{GHz}18\text{\,}\mathrm{GHz}$, with horizontal and vertical
polarisation channels providing measurements in dual linear polarisation mode.
The corrugated shape of the receiver feed ensures a low cross-talk (less than
$-40\text{\,}\mathrm{dB}$) between the two polarisation channels and high gain
over the target frequency. The signal path through the receiver frontend is
shown in Fig. 4.

The orthomode transducer (OMT), the calibration noise source, and the high-
electron mobility transistor (HEMT) used as low noise amplifier (LNA) are
operated at a temperature of $15\text{\,}\mathrm{K}$. The noise contributed by
these components determines the sensitivity of the receiver.

The outputs of the LNA are connected via coaxial cables to the first
heterodyne mixing stage operated at a room temperature. The heterodyne mixes
the signal with a tuneable local oscillator (LO) from
$8\text{\,}\mathrm{GHz}10\text{\,}\mathrm{GHz}$, this LO is synthesised by an
external reference signal of 100 MHz. This way, the signal is shifted from the
Ku band ($12\text{\,}\mathrm{GHz}18\text{\,}\mathrm{GHz}$) to the intermediate
frequency (IF) range from $4\text{\,}\mathrm{GHz}8\text{\,}\mathrm{GHz}$ with
a bandwidth of $4\text{\,}\mathrm{GHz}$. The second amplifying stage and
equalizer are applied to the IF signal before the second mixing stage and
digitizer integrated in the third generation digital backend system.

The receiver performance is characterised using an internal calibration source
(noise diode) with a noise temperature of $14\text{\,}\mathrm{K}$. This
procedure yields $T_{\mathrm{rec}}=$10\text{\,}\mathrm{K}$$ for the average
receiver temperature over the full
$12\text{\,}\mathrm{GHz}18\text{\,}\mathrm{GHz}$ frequency range.

The receiver is controlled by a control computer outfitted with a graphical
user interface (GUI) for setting the LO frequency, operating the cryostat
valves, and monitoring a number of peripheral parameters of the receiver such
as the output power, bias currents, amplifier gains, and operational
temperature.

Figure 4: The receiver frontend of BRASS-p.The aperture of the corrugated Ku-
band feed is operated at room temperature. The OMT unit separates the two
polarisation signals and guides the respective signals to the two LNAs. The
LNA is a wideband HEMT amplifier for the
$12\text{\,}\mathrm{GHz}18\text{\,}\mathrm{GHz}$ band, coupled to the external
noise source for calibration. The OMT and the LNA are hosted inside a Dewar
cryostat at 15 K. The signal is guided outside the cryostat via coaxial cable
to the downconverter, providing the 4–$8\text{\,}\mathrm{GHz}$ IF output. The
IF output signal undergoes further gain amplification and bandpass
equalisation before being output through an SMA port.

###  3.3 Aperture efficiency

Aperture efficiency, $\eta_{\mathrm{a}}$, of the the BRASS setup describes the
effective collecting area of its optical apparatus,
$A_{\mathrm{eff}}=\eta_{\mathrm{a}}A_{0}$. The aperture efficiency is a
product of individual factors pertaining to the conversion surface, the
parabolic reflector, and the feed antenna of the receiver:

$\eta_{\mathrm{a}}=\eta_{\mathrm{sf}}\cdot\eta_{\mathrm{il}}\cdot\eta_{\mathrm{sp}}\cdot\eta_{\mathrm{opt}}\cdot\eta_{\mathrm{ph}}\,,$
(3.1)

where $\eta_{\mathrm{sf}}$ is the surface efficiency, $\eta_{\mathrm{il}}$ and
$\eta_{\mathrm{sp}}$ are the illumination and spillover efficiencies of the
receiver feed, $\eta_{\mathrm{opt}}$ is the optical diffraction efficiency,
and $\eta_{\mathrm{ph}}$ is the phase centre efficiency.

The surface efficiency, $\eta_{\mathrm{sf}}$, is the cumulative product of all
deviations of the conversion surface and the parabolic reflector from their
ideal shapes. The surface rms for both of them is better than
$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ which does not lead to any
degradation of the signal in the
$12\text{\,}\mathrm{GHz}18\text{\,}\mathrm{GHz}$ frequency range. The surface
efficiency is therefore effectively determined by the magnitude of bending of
the conversion panels ($\sigma_{s}=$0.3\text{\,}\mathrm{mm}$$), and the
resulting $\eta_{\mathrm{sf}}$ ranges from 0.97 to 0.95 in the frequency range
of $12\text{\,}\mathrm{GHz}18\text{\,}\mathrm{GHz}$ (top panel of Fig. 5).

The product of the illumination efficiency, $\eta_{\mathrm{il}}$, and the
spillover efficiency $\eta_{\mathrm{sp}}$ of the receiver feed, optimised by a
$-$10 dB taper, is shown in the middle panel of Fig. 5. The optical efficiency
$\eta_{\mathrm{opt}}=0.98$ is estimated from the FEM simulation described in
Section 2.2.

The phase centre offset efficiency $\eta_{\mathrm{ph}}$ describes the effect
of an offset between the phase centre of the receiver horn and the focal point
of the parabolic reflector. Measurements made with a laser scanner show this
offset to not exceed $\pm 0.1\leavevmode\nobreak\ \text{mm}$, thus not
introducing any signal degradation [28] and allowing for
$\eta_{\mathrm{ph}}=1$ to be applied.

The resulting total aperture efficiency, shown in the bottom panel of Fig. 5,
is $\eta_{\mathrm{a}}\approx 0.72$ and varies only weakly over the
$12\text{\,}\mathrm{GHz}18\text{\,}\mathrm{GHz}$ frequency range. With this
efficiency BRASS-p has an effective collecting area
$A_{\mathrm{eff}}=3.22\leavevmode\nobreak\ \mathrm{m}^{2}$.

Figure 5: Aperture efficiency, $\eta_{\mathrm{a}}$ of BRASS-p as described by
the equation 3.1 (bottom panel) and two of its components: the surface (top
panel) and feed illumination and spillover (middle panel) efficiencies.

###  3.4 Data acquisition system

For each of the two polarisation channels, the IF signal delivered by the
receiver is processed as shown in Fig. 6. The signal is first digitised in a
digital baseband converter unit DBBC3 [29] designed and manufactured for
BRASS-p by HAT-Lab (www.hat-lab.cloud) and MPIfR, Bonn. The DBBC3 unit
digitises up to eight input signal streams simultaneously and delivers an
instantaneous bandwidth of 4 GHz per polarisation. In this process, the two IF
signals from the two polarisation channels are downconverted to the 0–4 GHz
band and digitised by four interleaved analogue-to-digital converter (ADC)
modules, each operating at 2048 megasamples/s (Msps). In order to optimize the
gain stages in each ADC, the each ADC’s power is measured, and a correction
factor is applied in dB fractions to get from each of ADCs a very similar
value with a precision of less than 1 %. A novel technique was employed to
finely adjust delay sampling between consecutive ADCs by utilizing the digital
control capabilities of the device’s input clock delay, synchronizing the
sampling time of two adjacent ADCs, and followed by the cross-correlation of
the obtained data. Maximizing this correlation provides precise information
for setting the delay control to achieve the desired final delay. With this
method, the delay time error is down to a fraction of the picosecond, thus
minimizing sampling artefacts.

During BRASS-p data taking runs, the digitisation was performed over a fixed
integration time and the digitised output was stored and processed in a
dedicated computer.

Figure 6: Signal processing path for a single polarisation channel of BRASS-p.
The 4–8 GHz IF signal from the receiver frontend is first downconverted to the
0–4 GHz band and then split into four streams digitised separately in the
respective analog-to-digital converter (ADC) modules. Each ADC module operates
at 2048 Megasamples/s (Msps) and is phase shifted by $\frac{\pi}{2}$ with
respect to the adjacent ADC modules. The digitally sampled output of the ADC
is packed, timestamped, and sent to the acquisition computer via four fibre
links. The samples are placed in a temporary memory storage from which they
are accesed by a GPU-based program that matches the timestamps and interleaves
the samples to obtain the true time series data at 8192 Msps. The interleaved
time series is then concatenated and sent to the fast Fourier transform (FFT)
module. The resulting power spectrum is stacked and saved to disk.

The data processing comprised interleaving and concatenation of the input
streams, followed by a fast Fourier transform (FFT) of the concatenated data
stream and subsequent storing and averaging of the resulting power spectra.
The FFT processing was parallelised using a graphical computing unit (GPU)
with high performance shader modules coupled with video random access memory
(VRAM). The parallelisation used the algorithm developed for the WISPDMX
experiment [11]. This allows the FFT to operate at a time efficiency of 60%
when providing 160 million spectral channels (frequency resolution of
$25\text{\,}\mathrm{Hz}$) across the $4\text{\,}\mathrm{GHz}$ bandwidth. For
the BRASS-p $12\text{\,}\mathrm{GHz}18\text{\,}\mathrm{GHz}$ frequency band,
this corresponds to ${\Delta\nu}/{\nu}=(1.4$–$2.1)\times 10^{-9}$, which is
fully sufficient for addressing both the standard halo scenario and the DM
stream models. Further acceleration of the data processing can be achieved by
parallelising the FFT with the data sample acquisition, which can be
implemented after an upgrade of the data acquisition computer. Presently, a
single data acquisition segment of BRASS-p lasts 19.8 seconds. The data stream
was arranged into 2475 fragments with 65536000 samples each. At the sampling
rate of 8192 Msps, the respective frequency resolution is
$125\text{\,}\mathrm{Hz}$.

###  3.5 Measurements

Data taking runs of BRASS-p consist of multiple data taking segments described
above. These segments are set up before a given data taking run and then
executed automatically. First, the receiver frontend is configured, using its
GUI interface to control the peripheral parameters of the receiver. The
frontend GUI is subsequently used for monitoring receiver performance during
the data taking run. Next, the DBBC3 (cyan block in Fig. 6) is set up, using
its client terminal interface which controls and monitors the signal gain and
the output sample distribution. At the final step, the control computer is
engaged to run a script that manages the data taking run, including the
collection and processing of the DBBC3 output by two desktop computers, the
FFT processing of the digitised time series data, and the subsequent storage
of the processed spectra.

Fig. 7 shows an example of raw power spectra obtained from a single
$19.8\text{\,}\mathrm{s}$ segment of BRASS measurements. The narrow-band
spikes seen in the spectra result from interference introduced by various
electronic components of the setup. Most of these peaks arise from the
interference of the $10\text{\,}\mathrm{MHz}$ and have a width of exactly one
spectral channel ($125\text{\,}\mathrm{Hz}$). These spikes will normally be
excised during the dedicated signal searches. The oscillatory pattern visible
in Fig. 7 in the $200\text{\,}\mathrm{MHz}$ zoom in the spectra results from
standing waves produced through multiple reflections between different optical
components of the setup. This pattern can be partially corrected during the
system noise calibration of the setup.

Figure 7: Examples of output spectra for the two polarisations measured by
BRASS-p in a single $19.8\text{\,}\mathrm{s}$ segment. The top panel shows the
full $4\text{\,}\mathrm{GHz}$ spectra and the bottom panel shows a zoom over a
$200\text{\,}\mathrm{MHz}$ part of the spectra. Narrow band spikes seen in the
spectra result from interference with various electronic components of the
setup, and these spikes will have to be excised during the detailed search for
a DM signal. The oscillatory pattern visible in the $200\text{\,}\mathrm{MHz}$
fragment of the spectra is produced by standing waves due to multiple
reflections between the main optical components of the setup.

###  3.6 Total power and system noise calibration

The total power and system noise calibration of BRASS-p measurements is
carried out by measuring the frequency-dependent system temperature
$T_{\mathrm{sys}}(\nu)$ of the setup from on/off spectra obtained with the
$T_{\mathrm{cal}}$ signal of the noise diode, which yields

$T_{\mathrm{sys}}(\nu)=T_{\mathrm{cal}}\left(\frac{P_{\mathrm{on}}(\nu)}{P_{\mathrm{off}}(\nu)}-1\right)^{-1}\,,$
(3.2)

where $P_{\mathrm{on}}$ and $P_{\mathrm{off}}$ are the power spectra measured
with and without the $T_{\mathrm{cal}}$ signal added. The total power
calibration factor, $\kappa(\nu)$ is then given by

$\kappa(\nu)=\frac{k_{\mathrm{B}}\,T_{\mathrm{sys}}(\nu)\,\Delta\nu^{1/2}\,\tau^{-1/2}}{\sigma_{\mathrm{off}}(\nu)}\,,$
(3.3)

where $\sigma_{\mathrm{off}}(\nu)$ is the standard deviation (or noise power)
in the raw spectrum. The calibration procedure is illustrated in Fig. 8
showing a $240\text{\,}\mathrm{MHz}$ fragment of the on/off spectra and the
resulting $T_{\mathrm{sys}}(\nu)$ and $\kappa(\nu)$ obtained from these
spectra. This calibration procedure also provides at least partial correction
for the standing waves, since the $T_{\mathrm{cal}}$ signal from the noise
diode undergoes the same reflections as the incoming radiation generated at
the conversion surface. This can be observed in the regular patterns also
appearing in the $T_{\mathrm{sys}}(\nu)$ and $\kappa(\nu)$ plotted in Fig. 8.

Figure 8: Total power calibration procedure using the on/off measurements with
the noise diode, illustrated with a $240\text{\,}\mathrm{MHz}$ fragment of the
spectral measurements. Shown are the on/off power spectra,
$P_{\mathrm{on}}(\nu)$ and $P_{\mathrm{off}}(\nu)$, measured in the
V-polarisation channel (top panel) and the resulting system temperature,
$T_{\mathrm{sys}}(\nu)$ (middle panel) and calibration factor, $\kappa(\nu)$
(bottom panel).

.

Figure 9: Calibrated noise power per spectral channel,
$\sigma_{\mathrm{det}}(\nu)$, for the frequency range of
$12\text{\,}\mathrm{GHz}16\text{\,}\mathrm{GHz}$, obtained for the vertical
(V) and horizontal (H) polarisation channels from a single
$19.8\text{\,}\mathrm{s}$ measurement of BRASS-p. In each polarisation
channel, a detection sensitivity of
$\sigma_{\mathrm{det}}\approx$1\text{\times}{10}^{-21}\text{\,}\mathrm{W}$$ is
reached.

The average $T_{\mathrm{sys}}$ obtained from the on/off calibration is about
$30\text{\,}\mathrm{K}$, which is close to what can be expected after
accounting for the major factors contributing to it. These factors are the
receiver noise temperature, $T_{\mathrm{rec}}\approx$10\text{\,}\mathrm{K}$$,
the effective antenna temperature $T_{\mathrm{ant}}=(1-{\cal
R})\,T_{\mathrm{room}}\lesssim$3\text{\,}\mathrm{K}$$ (for the antenna
reflectivity ${\cal R}=0.99$ and the room temperature
$T_{\mathrm{room}}=$20\text{\,}\mathrm{\SIUnitSymbolCelsius}$$), and the
spillover temperature,
$T_{\mathrm{spill}}=\epsilon_{\mathrm{f}}\,T_{\mathrm{room}}\approx$11\text{\,}\mathrm{K}$$,
of thermal radiation from the room also entering the receiver feed, for which
$\epsilon_{\mathrm{f}}=P_{\mathrm{lobe}}/P_{\mathrm{beam}}\approx 0.04$ can be
estimated. Summed up, these three factors yield a temperature of
$24\text{\,}\mathrm{K}$, and we expect that the remaining difference of
$\approx$6\text{\,}\mathrm{K}$$ between this value and the average measured
$T_{\mathrm{sys}}$ can result from the multiple reflections responsible for
the standing waves.

The respective calibrated noise spectra, $\sigma_{\mathrm{det}}(\nu)$,
obtained from a single $19.8\text{\,}\mathrm{s}$ segment are shown in Fig. 9
for the $12\text{\,}\mathrm{GHz}16\text{\,}\mathrm{GHz}$ band, indicating that
the lowest detectable power of
$\sigma_{\mathrm{det}}\approx${10}^{-21}\text{\,}\mathrm{W}$$ can be achieved
by BRASS-p in such a measurement.

Figure 10: Stacked noise power per spectral channel,
$\sigma_{\mathrm{det}}(\nu)$, obtained from BRASS-p first science measurement
runs SR1a (left panels) and SR1b (right panels) with the total measurement
times of $59.99\text{\,}\mathrm{h}$ and $66.07\text{\,}\mathrm{h}$,
respectively. A detection sensitivity of $\sigma_{\mathrm{det}}\approx
10^{-23}$ W is reached in these measurements.

###  3.7 Results from the first science run of BRASS-p

The first science run (SR1) of BRASS-p was carried out in November and
December 2022, and it consisted of two data taking periods (SR1a and SR1b) in
which the BRASS-p setup was used for making respective measurements in the
frequency ranges of $12\text{\,}\mathrm{GHz}16\text{\,}\mathrm{GHz}$ and
$14\text{\,}\mathrm{GHz}18\text{\,}\mathrm{GHz}$ frequency ranges. Table 2
provides a summary of these runs.

| SR1a | SR1b
---|---|---
Frequency band: | 12-16 GHz | 14-18 GHz
Number of polarization channels: | 2 | 2
Spectral resolution: | 125 Hz | 125 Hz
Starting date: | 13 Nov 2022 | 24 Nov 2022
Ending date: | 19 Nov 2022 | 2 Dec 2022
Number of spectra collected: | 9998 | 12012
Total measurement time: | 54.989 hours | 66.066 hours
Table 2: Summary of BRASS-p measurement runs SR1a and SR1b

For each of the data taking runs, the individual spectra have been averaged,
yielding cumulative noise spectra. These spectra are plotted in Fig. 10
separately for the two polarisation channels. As expected, the cumulative
spectra reach a noise level of $\sigma_{\mathrm{det}}\approx 10^{-23}$ W,
signifying a factor of $\approx 100$ improvement over the noise measured in
single $19.8\text{\,}\mathrm{s}$ data acquisitions.

First inspections of the spectra obtained do not reveal an obvious signal
candidate, which following the Maxwellian distribution of the dark matter
particle’s velocity. Deferring detailed signal searches to a later
publication, we apply here the cumulative noise spectra for calculating the
effective detection sensitivity of the BRASS-p SR1 measurements for the
kinetic mixing, $\chi(\nu)$, of hidden photons. Applying the measured noise
power per spectral channel, $\sigma_{\mathrm{det}}(\nu)$, to the equation 2.3,
and taking into account that a virialized DM signal should be distributed over
$N_{\mathrm{ch}}(\nu)$ channels [30], we obtain

$\chi(\nu)=3.2\times
10^{-14}\left(\frac{\sigma_{\mathrm{det}}(\nu)N_{\mathrm{ch}}(\nu)^{-1/2}}{10^{-23}\text{
W}}\right)^{\frac{1}{2}}\left(\frac{0.3\text{
GeV}/\text{cm}^{3}}{\rho_{\text{DM}}}\right)^{\frac{1}{2}}\left(\frac{1\text{
m}^{2}}{A_{\mathrm{eff}}}\right)^{\frac{1}{2}}\left(\frac{\sqrt{2/3}}{\alpha}\right)\,,$
(3.4)

where $\alpha=1/3$ for a single polarisation channel and $\alpha=2/3$ for the
two polarisation channels combined.

The resulting $\chi(\nu)$ are combined for the entire range of the hidden
photon mass probed by the SR1 measurements of BRASS-p and plotted in Fig. 11
for each polarisation channel. Similar sensitivities are achieved in both
polarisation channels, benchmarking on $\chi\lesssim 10^{-13}$ over the full
range of measurements. Further improvement by a factor of $\approx\sqrt{2}$
can be achieved by combining the two polarisation channels together, and this
will be done together with the planned detailed signal search.

Figure 11: Combined detection sensitivity of BRASS-p measurements plotted for
each polarisation channel over the entire range of the hidden photon mass
probed by the SR1 measurements.

To put the BRASS-p SR1 measurements in the broader context of hidden photon
searches, the detection sensitivity obtained for the vertical polarisation
channel is compared in Fig. 12 to exclusion limits obtained in other
experiments. This comparison shows that BRASS-p reaches the best sensitivity
to the kinetic mixing of hidden photon dark matter over the entire range of
particle mass probed by its SR1 measurements.

Figure 12: Detection sensitivity of the BRASS-p SR1 measurements in the
vertical polarisation channel, compared to exclusion limits obtained in other
hidden photon detection experiments [31].

##  4 Conclusions

This paper, and the BRASS-p experiment described in it, demonstrate the
overall feasibility of searches for dark matter signals generated at a
conducting converter surface [20] to be employed for broadband radiometric
searches for hidden photon and axion DM in the radio-to-submillimetre regime,
from $\sim$$1\text{\,}\mathrm{GHz}$ to $\sim$$1\text{\,}\mathrm{THz}$, in
which heterodyne receivers can be effectively applied for detecting signals
over broad bands and at a spectral resolution $\Delta\nu/\nu\lesssim 10^{-6}$.
The BRASS-p setup deals successfully with all the critical aspects of
generating and detecting an electromagnetic signal from dark matter particles
passing through the converter surface, and the first BRASS-p measurements
reach the power sensitivity of $10^{-23}$ W in the
$12\text{\,}\mathrm{GHz}18\text{\,}\mathrm{GHz}$ frequency range, which
translate into the world record sensitivity of $\chi<10^{-13}$ to the kinetic
mixing of DM hidden photons.

The initial results from BRASS-p measurements presented in this paper will be
further refined through a dedicated signal search which would adapt and expand
the methodology developed for such searches in the broadband part of the
haloscope experiment WISPDMX [30]. The data acquisition will be updated to
include a digital FX (DiFX) [32] correlation of the two polarisation channels,
thus enabling the full polarisation reconstruction and an effective feed
rotation [33] for probing all possible electric vector orientations of the
signal and hence expanding substantially the capabilities of HP dark matter
searches with BRASS-p. Multiple epoch observations would further explore
seasonal effects expected due to the Earth orbital motion [34] as well as the
possibility for caustic focusing of the dark matter signal [35].

Hidden photon DM searches with BRASS-p can be now readily extended over a
number of bands in the $1\text{\,}\mathrm{GHz}$–$1\text{\,}\mathrm{THz}$
frequency range, thanks to excellent synergies with the technologies already
available and under current development for radio astronomical measurements.
Presently, nine different receivers constructed for the Effelsberg and APEX
telescopes can be reproduced and used with the existing BRASS-p setup,
delivering excellent sensitivity to the hidden photon dark matter. Fig. 13
shows the expected kinetic mixing sensitivity that can be achieved by BRASS-p
with these receivers, assuming a 10-day measurement run for each of them. A
number of novel ultra-wideband receivers including BRAND [36] and the ALMA
receiver set [37] will allow for an even wider exploration of the
frequency/particle mass parameter space.

Figure 13: BRASS-p SR1 measurements complemented with the expected detection
sensitivities (shown with dashed lines) which can be achieved in 10-day
measurement runs using the existing BRASS-p setup with one of nine broadband
receivers constructed for the Effelsberg 100-metre telescope and the 15-metre
APEX telescope.

BRASS measurements will benefit strongly from further synergies with radio
astronomical development in the area of data acquisition, and in particular
from the DBBC4 developments [36] which would provide the possibility to
increase the recording bandwidth to $64\text{\,}\mathrm{GHz}$, thus allowing
to cover most of the receiver bands within a single measurement run.

Outfitted with permanently magnetised panels composed of NdFeB magnets [38]
providing an $\approx$$1\text{\,}\mathrm{T}$ magnetic field over a
$\approx$$1\text{\,}\mathrm{cm}$ scale height, BRASS-p will also be used for
carrying out searches for the ALP dark matter at frequencies above
$\approx$$10\text{\,}\mathrm{GHz}$. BRASS-p will not have a sufficient
sensitivity to probe the QCD axion band. The latter would require making a 100
day measurement run with an instrument for which the product
$B^{2}\,A_{\mathrm{eff}}$, of the squared magnetic field strength and the
effective collecting area should reach $\approx 100\leavevmode\nobreak\
\mathrm{T}^{2}\,\mathrm{m}^{2}$. BRASS-p will be operating at
$B^{2}\,A_{\mathrm{eff}}\approx 3\leavevmode\nobreak\
\mathrm{T}^{2}\,\mathrm{m}^{2}$ thus requiring an about 30-fold increase of
this figure of merit for targeting the axion dark matter. This increase will
be the main aim of further BRASS developments, and this aim can be achieved
either by enlarging the collecting area of the setup (potentially also
employing multiple chambers for that) or employing stronger permanent magnets
which may become available.

Based on the discussion presented in this paper, we can conclude that the
successful demonstration of feasibility of broadband DM searches provided by
BRASS-p establishes it as a robust experimental facility for hidden photon DM
searches in the $1\text{\,}\mathrm{GHz}$ to $1\text{\,}\mathrm{THz}$ frequency
range and builds a strong foundation for further developments of this approach
and its ultimate transformation into a long-term facility for very efficient
HP, ALP, and axion dark matter searches.

## Acknowledgments

We thank Marjolein Verkouter, Uwe Bach, Jan Wagner, and Oliver Polch for their
useful technical support and Alan Roy for useful comments and suggestions for
the manuscript. DH, LHN, and MT acknowledge the support by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy – EXC 2121 „Quantum Universe“ – 390833306.

## References

  * [1] M. Battaglieri, A. Belloni, A. Chou, P. Cushman, B. Echenard, R. Essig et al., _US Cosmic Visions: New Ideas in Dark Matter 2017: Community Report_ , _arXiv e-prints_ (2017) arXiv:1707.04591 [1707.04591].
  * [2] Planck Collaboration, N. Aghanim, Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi et al., _Planck 2018 results. VI. Cosmological parameters_ , _Astronomy and Astrophysics_ 641 (2020) A6 [1807.06209].
  * [3] G. Jungman, M. Kamionkowski and K. Griest, _Supersymmetric dark matter_ , _Phys. Rep._ 267 (1996) 195 [hep-ph/9506380].
  * [4] R. Essig, J.A. Jaros, W. Wester, P.H. Adrian, S. Andreas, T. Averett et al., _Dark Sectors and New, Light, Weakly-Coupled Particles_ , .
  * [5] P. Arias, _Cold Dark Matter from the Hidden Sector_ , in _-_ , vol. 59, pp. 56–59, Oct., 2012, DOI.
  * [6] A. Ringwald, _Axions and axion-like particles_ , in _Proceedings of the 49th Rencontres de Moriond - 2014 Electroweak Interactions and Unified Theories, EW 2014_ , pp. 223–230, 2014.
  * [7] L. Di Luzio, M. Giannotti, E. Nardi and L. Visinelli, _Corrigendum to “The landscape of QCD axion models” [Phys. Rep. 870 (2020) 1-117]_ , _Phys.Reports_ 870 (2020) 1 [2003.01100].
  * [8] P. Sikivie, _Experimental tests of the invisible axion_ , _Physical Review Letters_ 51 (1983) 1415.
  * [9] S.A. Abel, M.D. Goodsell, J. Jaeckel, V.V. Khoze and A. Ringwald, _Kinetic mixing of the photon with hidden U(1)s in string phenomenology_ , _Journal of High Energy Physics_ 2008 (2008) 124.
  * [10] N. Du, N. Force, R. Khatiwada, E. Lentz, R. Ottens, L.J. Rosenberg et al., _Search for Invisible Axion Dark Matter with the Axion Dark Matter Experiment_ , _Physical Review Letters_ 120 (2018) 151301.
  * [11] L.H. Nguyen, A. Lobanov and D. Horns, _First results from the WISPDMX radio frequency cavity searches for hidden photon dark matter_ , _JCAP_ 10 (2019) 014.
  * [12] B.M. Brubaker, L. Zhong, Y.V. Gurevich, S.B. Cahn, S.K. Lamoreaux, M. Simanovskaia et al., _First Results from a Microwave Cavity Axion Search at 24 $\mu$eV_, _Physical Review Letters_ 118 (2017) 61302.
  * [13] K.M. Backes, D.A. Palken, S.A. Kenany, B.M. Brubaker, S.B. Cahn, A. Droster et al., _A quantum enhanced search for dark matter axions_ , _Nature_ 590 (2021) 238.
  * [14] Y.K. Semertzidis, J.E. Kim, S. Youn, J. Choi, W. Chung, S. Haciomeroglu et al., _Axion Dark Matter Research with IBS/CAPP_ , .
  * [15] O. Kwon, D. Lee, W. Chung, D. Ahn, H.S. Byun, F. Caspers et al., _First Results from an Axion Haloscope at CAPP around 10.7 $\mu$eV_, _Physical Review Letters_ 126 (2021) 191802.
  * [16] M. Lawson, A.J. Millar, M. Pancaldi, E. Vitagliano and F. Wilczek, _Tunable Axion Plasma Haloscopes_ , _Physical Review Letters_ 123 (2019) 141802.
  * [17] J. Schütte-Engel, D.J. Marsh, A.J. Millar, A. Sekine, F. Chadha-Day, S. Hoof et al., _Axion quasiparticles for axion dark matter detection_ , _Journal of Cosmology and Astroparticle Physics_ 2021 (2021) .
  * [18] P. Brun, A. Caldwell, L. Chevalier, G. Dvali, P. Freire, E. Garutti et al., _A new experimental approach to probe QCD axion dark matter in the mass range above 40 $\mu$eV_, _European Physical Journal C_ 79 (2019) 186.
  * [19] B. Majorovits, _MADMAX: A new road to axion dark matter detection_ , _Journal of Physics: Conference Series_ 1342 (2020) 12098.
  * [20] D. Horns, J. Jaeckel, A. Lindner, A. Lobanov, J. Redondo and A. Ringwald, _Searching for WISPy cold dark matter with a dish antenna_ , _Journal of Cosmology and Astroparticle Physics_ 2013 (2013) .
  * [21] B. Holdom, _Two u(1)’s and $\varepsilon$ charge shifts_, _Physics Letters B_ 166 (1986) 196.
  * [22] R. Foot, H. Lew and R. Volkas, _A model with fundamental improper spacetime symmetries_ , _Physics Letters B_ 272 (1991) 67.
  * [23] A. Caputo, A.J. Millar, C.A.J. O’Hare and E. Vitagliano, _Dark photon limits: A handbook_ , .
  * [24] C.A.J. O’Hare and A.M. Green, _Axion astronomy with microwave cavity experiments_ , _Phys. Rev. D_ 95 (2017) 063017 [1701.03118].
  * [25] S.S. Chakrabarty, Y. Han, A.H. Gonzalez and P. Sikivie, _Implications of triangular features in the Gaia skymap for the Caustic Ring Model of the Milky Way halo_ , _Phys. Dark Univ._ 33 (2021) 100838 [2007.10509].
  * [26] J. Ruze, _Antenna Tolerance Theory – A Review_ , _IEEE Proceedings_ 54 (1966) 633.
  * [27] T. Flehmke, _Calibration of the optical system of brass_ , Master’s thesis, Universität Hamburg, Luruper Chaussee 149, 11, 2021.
  * [28] P. Ingerson and W. Rusch, _Radiation from a paraboloid with an axially defocused feed_ , _IEEE Transactions on Antennas and Propagation_ 21 (1973) 104.
  * [29] G. Tuccari, S. Buttaccio, W. Alef, M. Wunderlich, D.A. Graham, A. Bertarini et al., _DBBC3: VLBI at 32 gbits per second_ , in _Proceedings of Science_ , vol. 2012-Octob, Jan., 2012, DOI.
  * [30] L.H. Nguyen, A. Lobanov and D. Horns, _First results from the WISPDMX radio frequency cavity searches for hidden photon dark matter_ , _Journal of Cosmology and Astroparticle Physics_ 2019 (2019) 14.
  * [31] C. O’Hare, “cajohare/axionlimits: Axionlimits.” https://cajohare.github.io/AxionLimits/, July, 2020.  10.5281/zenodo.3932430.
  * [32] A.T. Deller, W.F. Brisken, C.J. Phillips, J. Morgan, W. Alef, R. Cappallo et al., _DiFX-2: A More Flexible, Efficient, Robust, and Powerful Software Correlator_ , _Publications of the Astronomical Society of the Pacific_ 123 (2011) 275 [1101.0885].
  * [33] T. Robishaw and C. Heiles, _The Measurement of Polarization in Radio Astronomy_ , in _The WSPC Handbook of Astronomical Instrumentation, Volume 1: Radio Astronomic al Instrumentation_ , A. Wolszczan, ed., pp. 127–158, World Scientific (2021), DOI.
  * [34] A. Caputo, A.J. Millar, C.A.\\.e.J. O’Hare and E. Vitagliano, _Dark photon limits: A handbook_ , _Physical Review D_ 104 (2021) .
  * [35] H. Kim and A. Lenoci, _Gravitational focusing of wave dark matter_ , _Phys. Rev. D_ 105 (2022) 063032.
  * [36] G. Tuccari, W. Alef, H. Rottmann, S. Dornbush, A. Felke, A. Roy et al., _From BRAND to DBBC4_ , in _International VLBI Service for Geodesy and Astrometry 2022 General Meeting Proceedings_ , K.L. Armstrong, D. Behrend and K.D. Baver, eds., pp. 72–76, Jan., 2023\.
  * [37] A. Remijan, A. Biggs, P.A. Cortes, B. Dent, J. Di Franceso, E. Fomalont et al., “ALMA Technical Handbook,ALMA Doc. 7.3, ver. 1.1.” 2019, ALMA Technical Handbook,ALMA Doc. 7.3, ver. 1.1ISBN 978-3-923524-66-2, June, 2019.  10.5281/zenodo.4511522.
  * [38] D. Brown, B.-M. Ma and Z. Chen, _Developments in the processing and properties of NdFeb-type permanent magnets_ , _Journal of Magnetism and Magnetic Materials_ 248 (2002) 432.