# Source code for pvlib.scaling

```
"""
The ``scaling`` module contains functions for manipulating irradiance
or other variables to account for temporal or spatial characteristics.
"""
import numpy as np
import pandas as pd
import scipy.optimize
from scipy.spatial.distance import pdist
[docs]def wvm(clearsky_index, positions, cloud_speed, dt=None):
"""
Compute spatial aggregation time series smoothing on clear sky index based
on the Wavelet Variability model of Lave et al. [1]_, [2]_. Implementation
is basically a port of the Matlab version of the code [3]_.
Parameters
----------
clearsky_index : numeric or pandas.Series
Clear Sky Index time series that will be smoothed.
positions : numeric
Array of coordinate distances as (x,y) pairs representing the
easting, northing of the site positions in meters [m]. Distributed
plants could be simulated by gridded points throughout the plant
footprint.
cloud_speed : numeric
Speed of cloud movement in meters per second [m/s].
dt : float, default None
The time series time delta. By default, is inferred from the
clearsky_index. Must be specified for a time series that doesn't
include an index. Units of seconds [s].
Returns
-------
smoothed : numeric or pandas.Series
The Clear Sky Index time series smoothed for the described plant.
wavelet: numeric
The individual wavelets for the time series before smoothing.
tmscales: numeric
The timescales associated with the wavelets in seconds [s].
References
----------
.. [1] M. Lave, J. Kleissl and J.S. Stein. A Wavelet-Based Variability
Model (WVM) for Solar PV Power Plants. IEEE Transactions on Sustainable
Energy, vol. 4, no. 2, pp. 501-509, 2013.
.. [2] M. Lave and J. Kleissl. Cloud speed impact on solar variability
scaling - Application to the wavelet variability model. Solar Energy,
vol. 91, pp. 11-21, 2013.
.. [3] Wavelet Variability Model - Matlab Code:
https://github.com/sandialabs/wvm
"""
# Added by Joe Ranalli (@jranalli), Penn State Hazleton, 2019
wavelet, tmscales = _compute_wavelet(clearsky_index, dt)
vr = _compute_vr(positions, cloud_speed, tmscales)
# Scale each wavelet by VR (Eq 7 in [1])
wavelet_smooth = np.zeros_like(wavelet)
for i in np.arange(len(tmscales)):
if i < len(tmscales) - 1: # Treat the lowest freq differently
wavelet_smooth[i, :] = wavelet[i, :] / np.sqrt(vr[i])
else:
wavelet_smooth[i, :] = wavelet[i, :]
outsignal = np.sum(wavelet_smooth, 0)
try: # See if there's an index already, if so, return as a pandas Series
smoothed = pd.Series(outsignal, index=clearsky_index.index)
except AttributeError:
smoothed = outsignal # just output the numpy signal
return smoothed, wavelet, tmscales
def _compute_vr(positions, cloud_speed, tmscales):
"""
Compute the variability reduction factors for each wavelet mode for the
Wavelet Variability Model [1-3].
Parameters
----------
positions : numeric
Array of coordinate distances as (x,y) pairs representing the
easting, northing of the site positions in meters [m]. Distributed
plants could be simulated by gridded points throughout the plant
footprint.
cloud_speed : numeric
Speed of cloud movement in meters per second [m/s].
tmscales: numeric
The timescales associated with the wavelets in seconds [s].
Returns
-------
vr : numeric
an array of variability reduction factors for each tmscale.
References
----------
.. [1] M. Lave, J. Kleissl and J.S. Stein. A Wavelet-Based Variability
Model (WVM) for Solar PV Power Plants. IEEE Transactions on Sustainable
Energy, vol. 4, no. 2, pp. 501-509, 2013.
.. [2] M. Lave and J. Kleissl. Cloud speed impact on solar variability
scaling - Application to the wavelet variability model. Solar Energy,
vol. 91, pp. 11-21, 2013.
.. [3] Wavelet Variability Model - Matlab Code:
https://github.com/sandialabs/wvm
"""
# Added by Joe Ranalli (@jranalli), Penn State Hazleton, 2021
pos = np.array(positions)
dist = pdist(pos, 'euclidean')
# Find effective length of position vector, 'dist' is full pairwise
n_pairs = len(dist)
def fn(x):
return np.abs((x ** 2 - x) / 2 - n_pairs)
n_dist = np.round(scipy.optimize.fmin(fn, np.sqrt(n_pairs), disp=False))
# Compute VR
A = cloud_speed / 2 # Resultant fit for A from [2]
vr = np.zeros(tmscales.shape)
for i, tmscale in enumerate(tmscales):
rho = np.exp(-1 / A * dist / tmscale) # Eq 5 from [1]
# 2*rho is because rho_ij = rho_ji. +n_dist accounts for sum(rho_ii=1)
denominator = 2 * np.sum(rho) + n_dist
vr[i] = n_dist ** 2 / denominator # Eq 6 of [1]
return vr
def latlon_to_xy(coordinates):
"""
Convert latitude and longitude in degrees to a coordinate system measured
in meters from zero deg latitude, zero deg longitude.
This is a convenience method to support inputs to wvm. Note that the
methodology used is only suitable for short distances. For conversions of
longer distances, users should consider use of Universal Transverse
Mercator (UTM) or other suitable cartographic projection. Consider
packages built for cartographic projection such as pyproj (e.g.
pyproj.transform()) [2].
Parameters
----------
coordinates : numeric
Array or list of (latitude, longitude) coordinate pairs. Use decimal
degrees notation.
Returns
-------
xypos : numeric
Array of coordinate distances as (x,y) pairs representing the
easting, northing of the position in meters [m].
References
----------
.. [1] H. Moritz. Geodetic Reference System 1980, Journal of Geodesy, vol.
74, no. 1, pp 128–133, 2000.
.. [2] https://pypi.org/project/pyproj/
.. [3] Wavelet Variability Model - Matlab Code:
https://github.com/sandialabs/wvm
"""
# Added by Joe Ranalli (@jranalli), Penn State Hazleton, 2019
r_earth = 6371008.7714 # mean radius of Earth, in meters
m_per_deg_lat = r_earth * np.pi / 180
try:
meanlat = np.mean([lat for (lat, lon) in coordinates]) # Mean latitude
except TypeError: # Assume it's a single value?
meanlat = coordinates[0]
m_per_deg_lon = r_earth * np.cos(np.pi/180 * meanlat) * np.pi/180
# Conversion
pos = coordinates * np.array(m_per_deg_lat, m_per_deg_lon)
# reshape as (x,y) pairs to return
try:
return np.column_stack([pos[:, 1], pos[:, 0]])
except IndexError: # Assume it's a single value, which has a 1D shape
return np.array((pos[1], pos[0]))
def _compute_wavelet(clearsky_index, dt=None):
"""
Compute the wavelet transform on the input clear_sky time series. Uses a
top hat wavelet [-1,1,1,-1] shape, based on the difference of successive
centered moving averages. Smallest scale (filter size of 2) is a degenerate
case that resembles a Haar wavelet. Returns one level of approximation
coefficient (CAn) and n levels of detail coefficients (CD1, CD2, ...,
CDn-1, CDn).
Parameters
----------
clearsky_index : numeric or pandas.Series
Clear Sky Index time series that will be smoothed.
dt : float, default None
The time series time delta. By default, is inferred from the
clearsky_index. Must be specified for a time series that doesn't
include an index. Units of seconds [s].
Returns
-------
wavelet: numeric
The individual wavelets for the time series. Format follows increasing
scale (decreasing frequency): [CD1, CD2, ..., CDn, CAn]
tmscales: numeric
The timescales associated with the wavelets in seconds [s]
References
----------
.. [1] M. Lave, J. Kleissl and J.S. Stein. A Wavelet-Based Variability
Model (WVM) for Solar PV Power Plants. IEEE Transactions on
Sustainable Energy, vol. 4, no. 2, pp. 501-509, 2013.
.. [2] Wavelet Variability Model - Matlab Code:
https://github.com/sandialabs/wvm
"""
# Added by Joe Ranalli (@jranalli), Penn State Hazleton, 2019
try: # Assume it's a pandas type
vals = clearsky_index.values.flatten()
except AttributeError: # Assume it's a numpy type
vals = clearsky_index.flatten()
if dt is None:
raise ValueError("dt must be specified for numpy type inputs.")
else: # flatten() succeeded, thus it's a pandas type, so get its dt
try: # Assume it's a time series type index
dt = clearsky_index.index[1] - clearsky_index.index[0]
dt = dt.seconds + dt.microseconds/1e6
except AttributeError: # It must just be a numeric index
dt = (clearsky_index.index[1] - clearsky_index.index[0])
# Pad the series on both ends in time and place in a dataframe
cs_long = np.pad(vals, (len(vals), len(vals)), 'symmetric')
cs_long = pd.DataFrame(cs_long)
# Compute wavelet time scales
min_tmscale = np.ceil(np.log(dt)/np.log(2)) # Minimum wavelet timescale
max_tmscale = int(13 - min_tmscale) # maximum wavelet timescale
tmscales = np.zeros(max_tmscale)
csi_mean = np.zeros([max_tmscale, len(cs_long)])
# Skip averaging for the 0th scale
csi_mean[0, :] = cs_long.values.flatten()
tmscales[0] = dt
# Loop for all time scales we will consider
for i in np.arange(1, max_tmscale):
tmscales[i] = 2**i * dt # Wavelet integration time scale
intvlen = 2**i # Wavelet integration time series interval
# Rolling average, retains only lower frequencies than interval
# Produces slightly different end effects than the MATLAB version
df = cs_long.rolling(window=intvlen, center=True, min_periods=1).mean()
# Fill nan's in both directions
df = df.fillna(method='bfill').fillna(method='ffill')
# Pop values back out of the dataframe and store
csi_mean[i, :] = df.values.flatten()
# Shift to account for different indexing in MATLAB moving average
csi_mean[i, :] = np.roll(csi_mean[i, :], -1)
csi_mean[i, -1] = csi_mean[i, -2]
# Calculate detail coefficients by difference between successive averages
wavelet_long = np.zeros(csi_mean.shape)
for i in np.arange(0, max_tmscale-1):
wavelet_long[i, :] = csi_mean[i, :] - csi_mean[i+1, :]
wavelet_long[-1, :] = csi_mean[-1, :] # Lowest freq (CAn)
# Clip off the padding and just return the original time window
wavelet = np.zeros([max_tmscale, len(vals)])
for i in np.arange(0, max_tmscale):
wavelet[i, :] = wavelet_long[i, len(vals): 2*len(vals)]
return wavelet, tmscales
```