diff --git a/docs/sphinx/source/api.rst b/docs/sphinx/source/api.rst index cd5b0ced40..3d499eb8c3 100644 --- a/docs/sphinx/source/api.rst +++ b/docs/sphinx/source/api.rst @@ -415,6 +415,8 @@ Functions :toctree: generated/ tracking.singleaxis + tracking.calc_axis_tilt + tracking.calc_cross_axis_tilt .. _iotools: diff --git a/docs/sphinx/source/whatsnew/v0.8.0.rst b/docs/sphinx/source/whatsnew/v0.8.0.rst index 650dbbad1a..f1b0afed92 100644 --- a/docs/sphinx/source/whatsnew/v0.8.0.rst +++ b/docs/sphinx/source/whatsnew/v0.8.0.rst @@ -91,6 +91,15 @@ Enhancements * Added *racking_model*, *module_type*, and *temperature_model_parameters* to PVSystem, LocalizedPVSystem, SingleAxisTracker, and LocalizedSingleAxisTracker repr methods. (:issue:`1027`) +* Added :py:func:`~pvlib.tracking.calc_axis_tilt` to calculate the + tracker axes tilt and :py:func:`~pvlib.tracking.calc_cross_axis_tilt` to + calculate the cross-axis tilt, which is the angle, relative to horizontal, of + the line formed by the intersection between the slope containing the tracker + axes and a plane perpendicular to the tracker axes. (:pull:`823`) +* Added ``cross_axis_tilt`` argument to :py:func:`~pvlib.tracking.singleaxis` + and :py:func:`~pvlib.tracking.SingleAxisTracker` which defaults to zero. Use + :py:func:`~pvlib.tracking.calc_cross_axis_tilt` to calculate the cross-axis + tilt angle if necessary. (:pull:`823`) * Added ability for :py:func:`pvlib.soiling.hsu` to accept arbitrary time intervals. (:pull:`980`) Bug fixes @@ -98,6 +107,7 @@ Bug fixes * Fixed unit and default value errors in :py:func:`pvlib.soiling.hsu`. (:pull:`977`, :pull:`980`) * Handle NUL characters and fix version column dtype in :py:func:`~pvlib.iotools.crn.read_crn`. (:issue:`1025`) +* Fix low sun angle tracker rotation calculation. (:issue:`824`) Testing ~~~~~~~ diff --git a/pvlib/data/singleaxis_tracker_wslope.csv b/pvlib/data/singleaxis_tracker_wslope.csv new file mode 100644 index 0000000000..f6e47ec502 --- /dev/null +++ b/pvlib/data/singleaxis_tracker_wslope.csv @@ -0,0 +1,8761 @@ +timestamp,tracker_theta,aoi,surface_azimuth,surface_tilt +2017-01-01 00:30:00-03:00,,,, +2017-01-01 01:30:00-03:00,,,, +2017-01-01 02:30:00-03:00,,,, +2017-01-01 03:30:00-03:00,,,, +2017-01-01 04:30:00-03:00,,,, +2017-01-01 05:30:00-03:00,31.764595167538808,61.60392068666398,86.38956060349983,31.835172142087643 +2017-01-01 06:30:00-03:00,75.0,19.464202082977415,89.40023282108623,75.01172049497154 +2017-01-01 07:30:00-03:00,64.08760993124552,13.319934295635893,88.91260263389762,64.10886001275362 +2017-01-01 08:30:00-03:00,51.181598879220616,7.902723154729724,88.19965415838165,51.21678334439227 +2017-01-01 09:30:00-03:00,38.07725693987907,3.395610128516448,87.14523715609228,38.13305500669065 +2017-01-01 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+2017-12-31 18:30:00-03:00,1.0292594117617142,83.82993658145304,24.695800586861097,2.464146015169277 +2017-12-31 19:30:00-03:00,,,, +2017-12-31 20:30:00-03:00,,,, +2017-12-31 21:30:00-03:00,,,, +2017-12-31 22:30:00-03:00,,,, +2017-12-31 23:30:00-03:00,,,, diff --git a/pvlib/tests/test_tracking.py b/pvlib/tests/test_tracking.py index 350a108f81..897c64baa0 100644 --- a/pvlib/tests/test_tracking.py +++ b/pvlib/tests/test_tracking.py @@ -6,8 +6,10 @@ from conftest import assert_frame_equal from numpy.testing import assert_allclose +import pvlib from pvlib.location import Location from pvlib import tracking +from conftest import DATA_DIR SINGLEAXIS_COL_ORDER = ['tracker_theta', 'aoi', 'surface_azimuth', 'surface_tilt'] @@ -41,7 +43,7 @@ def test_scalars(): expect = {'tracker_theta': 0, 'aoi': 10, 'surface_azimuth': 90, 'surface_tilt': 0} for k, v in expect.items(): - assert_allclose(tracker_data[k], v) + assert np.isclose(tracker_data[k], v) def test_arrays(): @@ -55,7 +57,7 @@ def test_arrays(): expect = {'tracker_theta': 0, 'aoi': 10, 'surface_azimuth': 90, 'surface_tilt': 0} for k, v in expect.items(): - assert_allclose(tracker_data[k], v) + assert_allclose(tracker_data[k], v, atol=1e-7) def test_nans(): @@ -71,7 +73,7 @@ def test_nans(): 'surface_azimuth': np.array([90, nan, nan]), 'surface_tilt': np.array([0, nan, nan])} for k, v in expect.items(): - assert_allclose(tracker_data[k], v) + assert_allclose(tracker_data[k], v, atol=1e-7) # repeat with Series because nans can differ apparent_zenith = pd.Series(apparent_zenith) @@ -268,7 +270,7 @@ def test_horizon_tilted(): out = tracking.singleaxis(solar_zenith, solar_azimuth, axis_tilt=90, axis_azimuth=180, backtrack=False, max_angle=180) expected = pd.DataFrame(np.array( - [[ 180., 45., 0., 90.], + [[-180., 45., 0., 90.], [ 0., 45., 180., 90.], [ 179., 45., 359., 90.]]), columns=['tracker_theta', 'aoi', 'surface_azimuth', 'surface_tilt']) @@ -276,15 +278,15 @@ def test_horizon_tilted(): def test_low_sun_angles(): - # GH 656 + # GH 656, 824 result = tracking.singleaxis( apparent_zenith=80, apparent_azimuth=338, axis_tilt=30, axis_azimuth=180, max_angle=60, backtrack=True, gcr=0.35) expected = { - 'tracker_theta': np.array([-50.31051385]), - 'aoi': np.array([61.35300178]), - 'surface_azimuth': np.array([112.53615425]), - 'surface_tilt': np.array([56.42233095])} + 'tracker_theta': np.array([60.0]), + 'aoi': np.array([80.420987]), + 'surface_azimuth': np.array([253.897886]), + 'surface_tilt': np.array([64.341094])} for k, v in result.items(): assert_allclose(expected[k], v) @@ -447,6 +449,7 @@ def test_SingleAxisTracker___repr__(): max_angle: 45 backtrack: True gcr: 0.25 + cross_axis_tilt: 0.0 name: None surface_tilt: None surface_azimuth: None @@ -471,6 +474,7 @@ def test_LocalizedSingleAxisTracker___repr__(): max_angle: 90 backtrack: True gcr: 0.25 + cross_axis_tilt: 0.0 name: None surface_tilt: None surface_azimuth: None @@ -486,3 +490,87 @@ def test_LocalizedSingleAxisTracker___repr__(): tz: UTC""" assert localized_system.__repr__() == expected + + +def test_calc_axis_tilt(): + # expected values + expected_axis_tilt = 2.239 # [degrees] + expected_side_slope = 9.86649274360294 # [degrees] + expected = DATA_DIR / 'singleaxis_tracker_wslope.csv' + expected = pd.read_csv(expected, index_col='timestamp', parse_dates=True) + # solar positions + starttime = '2017-01-01T00:30:00-0300' + stoptime = '2017-12-31T23:59:59-0300' + lat, lon = -27.597300, -48.549610 + times = pd.DatetimeIndex(pd.date_range(starttime, stoptime, freq='H')) + solpos = pvlib.solarposition.get_solarposition(times, lat, lon) + # singleaxis tracker w/slope data + slope_azimuth, slope_tilt = 77.34, 10.1149 + axis_azimuth = 0.0 + max_angle = 75.0 + # Note: GCR is relative to horizontal distance between rows + gcr = 0.33292759 # GCR = length / horizontal_pitch = 1.64 / 5 / cos(9.86) + # calculate tracker axis zenith + axis_tilt = tracking.calc_axis_tilt( + slope_azimuth, slope_tilt, axis_azimuth=axis_azimuth) + assert np.isclose(axis_tilt, expected_axis_tilt) + # calculate cross-axis tilt and relative rotation + cross_axis_tilt = tracking.calc_cross_axis_tilt( + slope_azimuth, slope_tilt, axis_azimuth, axis_tilt) + assert np.isclose(cross_axis_tilt, expected_side_slope) + sat = tracking.singleaxis( + solpos.apparent_zenith, solpos.azimuth, axis_tilt, axis_azimuth, + max_angle, backtrack=True, gcr=gcr, cross_axis_tilt=cross_axis_tilt) + np.testing.assert_allclose( + sat['tracker_theta'], expected['tracker_theta'], atol=1e-7) + np.testing.assert_allclose(sat['aoi'], expected['aoi'], atol=1e-7) + np.testing.assert_allclose( + sat['surface_azimuth'], expected['surface_azimuth'], atol=1e-7) + np.testing.assert_allclose( + sat['surface_tilt'], expected['surface_tilt'], atol=1e-7) + + +def test_slope_aware_backtracking(): + """ + Test validation data set from https://www.nrel.gov/docs/fy20osti/76626.pdf + """ + expected_data = np.array( + [('2019-01-01T08:00-0500', 2.404287, 122.79177, -84.440, -10.899), + ('2019-01-01T09:00-0500', 11.263058, 133.288729, -72.604, -25.747), + ('2019-01-01T10:00-0500', 18.733558, 145.285552, -59.861, -59.861), + ('2019-01-01T11:00-0500', 24.109076, 158.939435, -45.578, -45.578), + ('2019-01-01T12:00-0500', 26.810735, 173.931802, -28.764, -28.764), + ('2019-01-01T13:00-0500', 26.482495, 189.371536, -8.475, -8.475), + ('2019-01-01T14:00-0500', 23.170447, 204.13681, 15.120, 15.120), + ('2019-01-01T15:00-0500', 17.296785, 217.446538, 39.562, 39.562), + ('2019-01-01T16:00-0500', 9.461862, 229.102218, 61.587, 32.339), + ('2019-01-01T17:00-0500', 0.524817, 239.330401, 79.530, 5.490)], + dtype=[ + ('Time', ' 1 or apparent_zenith.ndim > 1: raise ValueError('Input dimensions must not exceed 1') - # Calculate sun position x, y, z using coordinate system as in [1], Eq 2. - - # Positive y axis is oriented parallel to earth surface along tracking axis - # (for the purpose of illustration, assume y is oriented to the south); - # positive x axis is orthogonal, 90 deg clockwise from y-axis, and parallel - # to the earth's surface (if y axis is south, x axis is west); - # positive z axis is normal to x, y axes, pointed upward. - - # Equations in [1] assume solar azimuth is relative to reference vector - # pointed south, with clockwise positive. - # Here, the input solar azimuth is degrees East of North, - # i.e., relative to a reference vector pointed - # north with clockwise positive. - # Rotate sun azimuth to coordinate system as in [1] - # to calculate sun position. - - az = apparent_azimuth - 180 - apparent_elevation = 90 - apparent_zenith - x = cosd(apparent_elevation) * sind(az) - y = cosd(apparent_elevation) * cosd(az) - z = sind(apparent_elevation) - - # translate array azimuth from compass bearing to [1] coord system - # wholmgren: strange to see axis_azimuth calculated differently from az, - # (not that it matters, or at least it shouldn't...). - axis_azimuth_south = axis_azimuth - 180 - - # translate input array tilt angle axis_tilt to [1] coordinate system. - - # In [1] coordinates, axis_tilt is a rotation about the x-axis. - # For a system with array azimuth (y-axis) oriented south, - # the x-axis is oriented west, and a positive axis_tilt is a - # counterclockwise rotation, i.e, lifting the north edge of the panel. - # Thus, in [1] coordinate system, in the northern hemisphere a positive - # axis_tilt indicates a rotation toward the equator, - # whereas in the southern hemisphere rotation toward the equator is - # indicated by axis_tilt<0. Here, the input axis_tilt is - # always positive and is a rotation toward the equator. - - # Calculate sun position (xp, yp, zp) in panel-oriented coordinate system: - # positive y-axis is oriented along tracking axis at panel tilt; - # positive x-axis is orthogonal, clockwise, parallel to earth surface; - # positive z-axis is normal to x-y axes, pointed upward. - # Calculate sun position (xp,yp,zp) in panel coordinates using [1] Eq 11 - # note that equation for yp (y' in Eq. 11 of Lorenzo et al 2011) is - # corrected, after conversation with paper's authors. - - xp = x*cosd(axis_azimuth_south) - y*sind(axis_azimuth_south) - yp = (x*cosd(axis_tilt)*sind(axis_azimuth_south) + - y*cosd(axis_tilt)*cosd(axis_azimuth_south) - - z*sind(axis_tilt)) - zp = (x*sind(axis_tilt)*sind(axis_azimuth_south) + - y*sind(axis_tilt)*cosd(axis_azimuth_south) + - z*cosd(axis_tilt)) + # Calculate sun position x, y, z using coordinate system as in [1], Eq 1. + + # NOTE: solar elevation = 90 - solar zenith, then use trig identities: + # sin(90-x) = cos(x) & cos(90-x) = sin(x) + sin_zenith = sind(apparent_zenith) + x = sin_zenith * sind(apparent_azimuth) + y = sin_zenith * cosd(apparent_azimuth) + z = cosd(apparent_zenith) + + # Assume the tracker reference frame is right-handed. Positive y-axis is + # oriented along tracking axis; from north, the y-axis is rotated clockwise + # by the axis azimuth and tilted from horizontal by the axis tilt. The + # positive x-axis is 90 deg clockwise from the y-axis and parallel to + # horizontal (e.g., if the y-axis is south, the x-axis is west); the + # positive z-axis is normal to the x and y axes, pointed upward. + + # Calculate sun position (xp, yp, zp) in tracker coordinate system using + # [1] Eq 4. + + cos_axis_azimuth = cosd(axis_azimuth) + sin_axis_azimuth = sind(axis_azimuth) + cos_axis_tilt = cosd(axis_tilt) + sin_axis_tilt = sind(axis_tilt) + xp = x*cos_axis_azimuth - y*sin_axis_azimuth + yp = (x*cos_axis_tilt*sin_axis_azimuth + + y*cos_axis_tilt*cos_axis_azimuth + - z*sin_axis_tilt) + zp = (x*sin_axis_tilt*sin_axis_azimuth + + y*sin_axis_tilt*cos_axis_azimuth + + z*cos_axis_tilt) # The ideal tracking angle wid is the rotation to place the sun position - # vector (xp, yp, zp) in the (y, z) plane; i.e., normal to the panel and - # containing the axis of rotation. wid = 0 indicates that the panel is - # horizontal. Here, our convention is that a clockwise rotation is + # vector (xp, yp, zp) in the (y, z) plane, which is normal to the panel and + # contains the axis of rotation. wid = 0 indicates that the panel is + # horizontal. Here, our convention is that a clockwise rotation is # positive, to view rotation angles in the same frame of reference as - # azimuth. For example, for a system with tracking axis oriented south, - # a rotation toward the east is negative, and a rotation to the west is - # positive. - - # Use arctan2 and avoid the tmp corrections. - - # angle from x-y plane to projection of sun vector onto x-z plane -# tmp = np.degrees(np.arctan(zp/xp)) - - # Obtain wid by translating tmp to convention for rotation angles. - # Have to account for which quadrant of the x-z plane in which the sun - # vector lies. Complete solution here but probably not necessary to - # consider QIII and QIV. -# wid = pd.Series(index=times) -# wid[(xp>=0) & (zp>=0)] = 90 - tmp[(xp>=0) & (zp>=0)] # QI -# wid[(xp<0) & (zp>=0)] = -90 - tmp[(xp<0) & (zp>=0)] # QII -# wid[(xp<0) & (zp<0)] = -90 - tmp[(xp<0) & (zp<0)] # QIII -# wid[(xp>=0) & (zp<0)] = 90 - tmp[(xp>=0) & (zp<0)] # QIV + # azimuth. For example, for a system with tracking axis oriented south, a + # rotation toward the east is negative, and a rotation to the west is + # positive. This is a right-handed rotation around the tracker y-axis. # Calculate angle from x-y plane to projection of sun vector onto x-z plane - # and then obtain wid by translating tmp to convention for rotation angles. - wid = 90 - np.degrees(np.arctan2(zp, xp)) + # using [1] Eq. 5. + + wid = np.degrees(np.arctan2(xp, zp)) # filter for sun above panel horizon zen_gt_90 = apparent_zenith > 90 wid[zen_gt_90] = np.nan - # Account for backtracking; modified from [1] to account for rotation - # angle convention being used here. + # Account for backtracking if backtrack: - axes_distance = 1/gcr - # clip needed for low angles. GH 656 - temp = np.clip(axes_distance*cosd(wid), -1, 1) + # distance between rows in terms of rack lengths relative to cross-axis + # tilt + axes_distance = 1/(gcr * cosd(cross_axis_tilt)) - # backtrack angle - # (always positive b/c acosd returns values between 0 and 180) - wc = np.degrees(np.arccos(temp)) + # NOTE: account for rare angles below array, see GH 824 + temp = np.abs(axes_distance * cosd(wid - cross_axis_tilt)) - # Eq 4 applied when wid in QIV (wid < 0 evalulates True), QI + # backtrack angle using [1], Eq. 14 with np.errstate(invalid='ignore'): - # errstate for GH 622 - tracker_theta = np.where(wid < 0, wid + wc, wid - wc) + wc = np.degrees(-np.sign(wid)*np.arccos(temp)) + + # NOTE: in the middle of the day, arccos(temp) is out of range because + # there's no row-to-row shade to avoid, & backtracking is unnecessary + # [1], Eqs. 15-16 + with np.errstate(invalid='ignore'): + tracker_theta = wid + np.where(temp < 1, wc, 0) else: tracker_theta = wid - tracker_theta = np.minimum(tracker_theta, max_angle) - tracker_theta = np.maximum(tracker_theta, -max_angle) + # NOTE: max_angle defined relative to zero-point rotation, not the + # system-plane normal + tracker_theta = np.clip(tracker_theta, -max_angle, max_angle) - # calculate panel normal vector in panel-oriented x, y, z coordinates. - # y-axis is axis of tracker rotation. tracker_theta is a compass angle + # Calculate panel normal vector in panel-oriented x, y, z coordinates. + # y-axis is axis of tracker rotation. tracker_theta is a compass angle # (clockwise is positive) rather than a trigonometric angle. - # the *0 is a trick to preserve NaN values. + # NOTE: the *0 is a trick to preserve NaN values. panel_norm = np.array([sind(tracker_theta), tracker_theta*0, cosd(tracker_theta)]) @@ -460,30 +464,28 @@ def singleaxis(apparent_zenith, apparent_azimuth, # calculate angle-of-incidence on panel aoi = np.degrees(np.arccos(np.abs(np.sum(sun_vec*panel_norm, axis=0)))) - # calculate panel tilt and azimuth - # in a coordinate system where the panel tilt is the - # angle from horizontal, and the panel azimuth is - # the compass angle (clockwise from north) to the projection - # of the panel's normal to the earth's surface. - # These outputs are provided for convenience and comparison - # with other PV software which use these angle conventions. + # Calculate panel tilt and azimuth in a coordinate system where the panel + # tilt is the angle from horizontal, and the panel azimuth is the compass + # angle (clockwise from north) to the projection of the panel's normal to + # the earth's surface. These outputs are provided for convenience and + # comparison with other PV software which use these angle conventions. - # project normal vector to earth surface. - # First rotate about x-axis by angle -axis_tilt so that y-axis is - # also parallel to earth surface, then project. + # Project normal vector to earth surface. First rotate about x-axis by + # angle -axis_tilt so that y-axis is also parallel to earth surface, then + # project. # Calculate standard rotation matrix rot_x = np.array([[1, 0, 0], [0, cosd(-axis_tilt), -sind(-axis_tilt)], [0, sind(-axis_tilt), cosd(-axis_tilt)]]) - # panel_norm_earth contains the normal vector - # expressed in earth-surface coordinates - # (z normal to surface, y aligned with tracker axis parallel to earth) + # panel_norm_earth contains the normal vector expressed in earth-surface + # coordinates (z normal to surface, y aligned with tracker axis parallel to + # earth) panel_norm_earth = np.dot(rot_x, panel_norm).T - # projection to plane tangent to earth surface, - # in earth surface coordinates + # projection to plane tangent to earth surface, in earth surface + # coordinates projected_normal = np.array([panel_norm_earth[:, 0], panel_norm_earth[:, 1], panel_norm_earth[:, 2]*0]).T @@ -491,69 +493,22 @@ def singleaxis(apparent_zenith, apparent_azimuth, # calculate vector magnitudes projected_normal_mag = np.sqrt(np.nansum(projected_normal**2, axis=1)) - # renormalize the projected vector - # avoid creating nan values. + # renormalize the projected vector, avoid creating nan values. non_zeros = projected_normal_mag != 0 projected_normal[non_zeros] = (projected_normal[non_zeros].T / projected_normal_mag[non_zeros]).T # calculation of surface_azimuth - # 1. Find the angle. -# surface_azimuth = pd.Series( -# np.degrees(np.arctan(projected_normal[:,1]/projected_normal[:,0])), -# index=times) surface_azimuth = \ np.degrees(np.arctan2(projected_normal[:, 1], projected_normal[:, 0])) - # 2. Clean up atan when x-coord or y-coord is zero -# surface_azimuth[(projected_normal[:,0]==0) & (projected_normal[:,1]>0)] = 90 -# surface_azimuth[(projected_normal[:,0]==0) & (projected_normal[:,1]<0)] = -90 -# surface_azimuth[(projected_normal[:,1]==0) & (projected_normal[:,0]>0)] = 0 -# surface_azimuth[(projected_normal[:,1]==0) & (projected_normal[:,0]<0)] = 180 - - # 3. Correct atan for QII and QIII -# surface_azimuth[(projected_normal[:,0]<0) & (projected_normal[:,1]>0)] += 180 # QII -# surface_azimuth[(projected_normal[:,0]<0) & (projected_normal[:,1]<0)] += 180 # QIII - - # 4. Skip to below - - # at this point surface_azimuth contains angles between -90 and +270, - # where 0 is along the positive x-axis, - # the y-axis is in the direction of the tracker azimuth, - # and positive angles are rotations from the positive x axis towards - # the positive y-axis. - # Adjust to compass angles - # (clockwise rotation from 0 along the positive y-axis) -# surface_azimuth[surface_azimuth<=90] = 90 - surface_azimuth[surface_azimuth<=90] -# surface_azimuth[surface_azimuth>90] = 450 - surface_azimuth[surface_azimuth>90] - - # finally rotate to align y-axis with true north - # PVLIB_MATLAB has this latitude correction, - # but I don't think it's latitude dependent if you always - # specify axis_azimuth with respect to North. -# if latitude > 0 or True: -# surface_azimuth = surface_azimuth - axis_azimuth -# else: -# surface_azimuth = surface_azimuth - axis_azimuth - 180 -# surface_azimuth[surface_azimuth<0] = 360 + surface_azimuth[surface_azimuth<0] - - # the commented code above is mostly part of PVLIB_MATLAB. - # My (wholmgren) take is that it can be done more simply. - # Say that we're pointing along the postive x axis (likely west). - # We just need to rotate 90 degrees to get from the x axis - # to the y axis (likely south), - # and then add the axis_azimuth to get back to North. - # Anything left over is the azimuth that we want, - # and we can map it into the [0,360) domain. - - # 4. Rotate 0 reference from panel's x axis to it's y axis and - # then back to North. + # Rotate 0 reference from panel's x-axis to its y-axis and then back to + # north. surface_azimuth = 90 - surface_azimuth + axis_azimuth - # 5. Map azimuth into [0,360) domain. - # surface_azimuth[surface_azimuth < 0] += 360 - # surface_azimuth[surface_azimuth >= 360] -= 360 - surface_azimuth = surface_azimuth % 360 + # Map azimuth into [0,360) domain. + with np.errstate(invalid='ignore'): + surface_azimuth = surface_azimuth % 360 # Calculate surface_tilt dotproduct = (panel_norm_earth * projected_normal).sum(axis=1) @@ -561,7 +516,7 @@ def singleaxis(apparent_zenith, apparent_azimuth, # Bundle DataFrame for return values and filter for sun below horizon. out = {'tracker_theta': tracker_theta, 'aoi': aoi, - 'surface_azimuth': surface_azimuth, 'surface_tilt': surface_tilt} + 'surface_tilt': surface_tilt, 'surface_azimuth': surface_azimuth} if index is not None: out = pd.DataFrame(out, index=index) out = out[['tracker_theta', 'aoi', 'surface_azimuth', 'surface_tilt']] @@ -570,3 +525,154 @@ def singleaxis(apparent_zenith, apparent_azimuth, out = {k: np.where(zen_gt_90, np.nan, v) for k, v in out.items()} return out + + +def calc_axis_tilt(slope_azimuth, slope_tilt, axis_azimuth): + """ + Calculate tracker axis tilt in the global reference frame when on a sloped + plane. + + Parameters + ---------- + slope_azimuth : float + direction of normal to slope on horizontal [degrees] + slope_tilt : float + tilt of normal to slope relative to vertical [degrees] + axis_azimuth : float + direction of tracker axes on horizontal [degrees] + + Returns + ------- + axis_tilt : float + tilt of tracker [degrees] + + See also + -------- + pvlib.tracking.singleaxis + pvlib.tracking.calc_cross_axis_tilt + + Notes + ----- + See [1]_ for derivation of equations. + + References + ---------- + .. [1] Kevin Anderson and Mark Mikofski, "Slope-Aware Backtracking for + Single-Axis Trackers", Technical Report NREL/TP-5K00-76626, July 2020. + https://www.nrel.gov/docs/fy20osti/76626.pdf + """ + delta_gamma = axis_azimuth - slope_azimuth + # equations 18-19 + tan_axis_tilt = cosd(delta_gamma) * tand(slope_tilt) + return np.degrees(np.arctan(tan_axis_tilt)) + + +def _calc_tracker_norm(ba, bg, dg): + """ + Calculate tracker normal, v, cross product of tracker axis and unit normal, + N, to the system slope plane. + + Parameters + ---------- + ba : float + axis tilt [degrees] + bg : float + ground tilt [degrees] + dg : float + delta gamma, difference between axis and ground azimuths [degrees] + + Returns + ------- + vector : tuple + vx, vy, vz + """ + cos_ba = cosd(ba) + cos_bg = cosd(bg) + sin_bg = sind(bg) + sin_dg = sind(dg) + vx = sin_dg * cos_ba * cos_bg + vy = sind(ba)*sin_bg + cosd(dg)*cos_ba*cos_bg + vz = -sin_dg*sin_bg*cos_ba + return vx, vy, vz + + +def _calc_beta_c(v, dg, ba): + """ + Calculate the cross-axis tilt angle. + + Parameters + ---------- + v : tuple + tracker normal + dg : float + delta gamma, difference between axis and ground azimuths [degrees] + ba : float + axis tilt [degrees] + + Returns + ------- + beta_c : float + cross-axis tilt angle [radians] + """ + vnorm = np.sqrt(np.dot(v, v)) + beta_c = np.arcsin( + ((v[0]*cosd(dg) - v[1]*sind(dg)) * sind(ba) + v[2]*cosd(ba)) / vnorm) + return beta_c + + +def calc_cross_axis_tilt( + slope_azimuth, slope_tilt, axis_azimuth, axis_tilt): + """ + Calculate the angle, relative to horizontal, of the line formed by the + intersection between the slope containing the tracker axes and a plane + perpendicular to the tracker axes. + + Use the cross-axis tilt to avoid row-to-row shade when backtracking on a + slope not parallel with the axis azimuth. Cross-axis tilt should be + specified using a right-handed convention. For example, trackers with axis + azimuth of 180 degrees (heading south) will have a negative cross-axis tilt + if the tracker axes plane slopes down to the east and positive cross-axis + tilt if the tracker axes plane slopes up to the east. + + Parameters + ---------- + slope_azimuth : float + direction of the normal to the slope containing the tracker axes, when + projected on the horizontal [degrees] + slope_tilt : float + angle of the slope containing the tracker axes, relative to horizontal + [degrees] + axis_azimuth : float + direction of tracker axes projected on the horizontal [degrees] + axis_tilt : float + tilt of trackers relative to horizontal [degrees] + + Returns + ------- + cross_axis_tilt : float + angle, relative to horizontal, of the line formed by the intersection + between the slope containing the tracker axes and a plane perpendicular + to the tracker axes [degrees] + + See also + -------- + pvlib.tracking.singleaxis + pvlib.tracking.calc_axis_tilt + + Notes + ----- + See [1]_ for derivation of equations. + + References + ---------- + .. [1] Kevin Anderson and Mark Mikofski, "Slope-Aware Backtracking for + Single-Axis Trackers", Technical Report NREL/TP-5K00-76626, July 2020. + https://www.nrel.gov/docs/fy20osti/76626.pdf + """ + # delta-gamma, difference between axis and slope azimuths + delta_gamma = axis_azimuth - slope_azimuth + # equation 22 + v = _calc_tracker_norm(axis_tilt, slope_tilt, delta_gamma) + # equation 26 + beta_c = _calc_beta_c(v, delta_gamma, axis_tilt) + return np.degrees(beta_c)