Solar forecasting course: forecasting the intermittent solar resource

Cyril Voyant, Research Director, Mines Paris – PSL, Centre O.I.E., Sophia Antipolis

UE14, Mines Paris engineering curriculum (English version of the 2026 course)

Applied research project: modeling and forecasting global solar irradiance

MINES PARIS – PSL

Center O.I.E.

Cite as: Voyant, C. (2026). Forecasting the intermittent solar resource (UE14 course, English version). Mines Paris – PSL. License CC BY 4.0.

Web companion with the worked example: cyrilvoyant.github.io/teaching/solar-forecasting-basics.html

SOPHIA ANTIPOLIS

01

Why forecast solar power?

Grid issues, units, the Ajaccio data

04

Clear sky and clear-sky index

Solis model, CSI, stationarity

Appendices: wide-sense stationarity, conditional expectation, AIC/BIC criteria. Glossary and references at the end.

02 · Solar radiation

Direct, diffuse, atmosphere, Sun position

05

AR model and least squares

Building X and y, least squares, choosing p

03

Time series and forecasting

Components, horizon, a first AR(1) model

06

Evaluation, ELM and online learning

Persistence, metrics, ELM, sliding window, frugality

Context

Solar irradiance is intermittent (clouds).

The power grid must anticipate photovoltaic production.

Need: reliable short-term forecasts (here 1 h).

The full chain we will build

1 · 2 · 3 · GHI · measurements · Clear sky · GHI^cs · CSI index · (difference)

This chain comes back in every chapter; the current step is highlighted.

By the end, you will be able to

compute a clear-sky index (CSI);

build and estimate an AR(p) model by least squares;

build an ELM and measure its variability;

compare models with persistence (MAE, RMSE, skill score);

update a model online and judge its frugality.

4 · 5 · 6 · 7 · Centring

y = CSI − μ

Chronological · split · Model

P · AR · ELM

Forecast · ŷ_t+h · 8 · 9 · Forecast GHI

ŷ + μ + GHI^cs

Evaluation · day · day+night

Convention: bold lower case = vector, bold upper case = matrix, plain = number. Time is always a subscript: y_t, y_t−1, CSI_t+h.

The grid operator balances PV, storage, other plants and consumers.

Solar output varies with clouds. To integrate it into the grid, it must be anticipated.

Grid stability

Anticipate production drops to keep the frequency stable.

Supply / demand balance

Adjust the other plants (hydro, pumped storage*, gas…).

* Pumped-storage hydropower: a reversible dam that stores energy by pumping water uphill.

Storage

Charge or discharge batteries at the right time.

Economic value

Sell production at the best price and cut operating costs.

Irradiance (W/m²)

Power received per m² at a given instant. Example: 1,000 W/m² in full sun at noon.

From irradiance to electrical power

P ≈ η · S · G

Output is proportional to irradiance: forecasting GHI is forecasting production.

Irradiation (Wh/m²)

Energy received per m² over a duration. Over one hour: hourly irradiation in Wh/m² = mean irradiance in W/m².

η: module efficiency (≈ 20 %), S: area (m²), G: irradiance on the module (W/m²).

Example: 10 m² under 800 W/m² → P ≈ 0.20 × 10 × 800 = 1.6 kW.

Our data

Hourly GHI in Wh/m². Météo-France files are in J/cm²: 1 J/cm² = 2.78 Wh/m², so Wh/m² = J/cm² / 0.36.

Hourly GHI measured at Ajaccio in 2007 (Météo-France).

Ten years of measurements: 8 years to learn, 1 year to tune the model, 1 year to test it (chapter 5).

Site: Ajaccio (Corsica), 41.9° N, 8.8° E, 4 m.

Variable: hourly GHI (Wh/m²).

Period: 1998–2007, i.e. 87,600 hours.

Maximum: 1,025 Wh/m².

Daytime mean: 368 Wh/m².

θ_z: zenith angle (between the Sun and the vertical). GHI = Global Horizontal Irradiance.

Direct (DNI)

Received straight from the Sun, measured on a surface perpendicular to the rays (direct normal). Dominant under clear sky.

Diffuse (DHI)

Scattered by air, aerosols and clouds. Dominant under overcast sky.

Reflected (albedo)

Sent back by the ground. Counts only for a tilted surface, not for GHI.

Through the atmosphere, radiation is attenuated by four mechanisms.

Absorption · Rayleigh scattering

Water vapour, ozone, CO₂: part of the energy is absorbed.

By air molecules: it makes the sky blue.

Mie scattering · Clouds

By aerosols (dust, sea salt, pollution): depends on turbidity.

Reflection, scattering and absorption: main cause of intermittency.

Ajaccio, 2007: a clear day and an overcast day.

Under overcast sky, GHI drops here to about 20 % of its clear-sky value.

Solar elevation over the day at Ajaccio, by season.

Zenith angle θ_z: between the Sun and the vertical. The larger it is, the thicker the atmosphere crossed.

Declination δ: angle between the Sun's direction and the equatorial plane; from −23.45° to +23.45° over the year.

Hour angle ω: 15° per hour, zero at solar noon. Latitude φ: 41.9° at Ajaccio.

cos θ_z = sin φ · sin δ + cos φ · cos δ · cos ω

G₀ = I₀ · (d₀/d)² · cos θ_z

G₀: extraterrestrial radiation; I₀ ≈ 1361 W/m² (solar constant; 1367 in older codes).

The Earth–Sun distance d varies by ±1.7 %, so (d₀/d)² varies by ±3.3 %.

Hourly GHI in 2007: hours of the day × days of the year (Ajaccio).

Daily seasonality

Zero at night, maximum around solar noon: a 24-h cycle well described by the clear-sky model.

Annual seasonality

The amplitude of the daily cycle changes with the season (declination, day length): a 365-day cycle.

Raw GHI is not stationary: its mean depends on the hour and the month. Both cycles must be removed before modeling.

A sequence of measurements y₁, y₂, …, y_N of one quantity, taken at a constant time step Δt. Here Δt = 1 h.

Three properties

Regularity: measurements are equally spaced.

Dependence: the past informs the future.

Randomness: each value has an unpredictable part.

What we want for modeling

Stationarity: stable mean and variance over time.

Complete series: no missing values.

Ajaccio: one week of hourly GHI.

Raw GHI is not stationary (day/night and seasonal cycles): that is the whole point of chapter 4.

Trend T_t

Slow long-term change: increasing, decreasing or none.

Cycle C_t

Long, not strictly periodic oscillations (climate phenomena).

For GHI: seasonality dominates, with two cycles (daily and annual).

Seasonality S_t

Regular, repeatable cycles (day, year). Very strong for GHI.

Residual R_t

Unexplained random part. Ideally white noise (independent values, zero mean).

Additive model

y_t = T_t + S_t + C_t + R_t

When the amplitude of fluctuations stays constant.

Multiplicative model

y_t = T_t × S_t × C_t × R_t

When the amplitude grows with the level (GHI: small in winter, large in summer).

Why decompose?

Remove trend and seasonality (the part predictable without effort);

keep a residual that is easier to model;

check that this residual looks like white noise.

Link with what follows: subtracting clear-sky GHI, CSI_t = GHI_t − GHI^cs_t, applies the additive model: both cycles go at once. The ratio (multiplicative model) also exists: chapter 4.

Usual methods by horizon (orders of magnitude).

Time step Δt = 1 h; horizon h in steps: h = 1 → 1 h, h = 3 → 3 h.

Direct forecast: one model trained for each horizon h.

Recursive forecast: forecast t + 1, then feed that forecast back to forecast t + 2, and so on.

In this course: 1-h horizon, direct forecast.

y_t+1 = f(y_t, y_t−1, …, y_t−p+1) + ε_t+1

The predictable part

f sums up what the past tells us about the future. The forecast is ŷ_t+1 = f(y_t, …, y_t−p+1).

The framework suits GHI: regular measurements and strong hour-to-hour dependence (cloud inertia). Further reading: appendix B.

The error

ε_t+1 = y_t+1 − ŷ_t+1 is the unpredictable part. If the model is good, ε looks like white noise.

The goal

Choose f so that errors are as small as possible in mean square.

y_t+1 = w · y_t + ε_t+1

y_t = CSI_t − μ: gap of the CSI to its mean; w: weight of the last gap.

|w| < 1: the series always returns to its mean.

ŷ_t+h = w^h · y_t

Ajaccio: μ ≈ −33 Wh/m² and w ≈ 0.84. A cloud: CSI_t = −300 Wh/m², so y_t = −267.

ŷ_t+1 = 0.84 × (−267) = −224: forecast CSI = −257 Wh/m²; then −221 and −191 at 2 h and 3 h.

The forecast CSI returns to its mean as h grows.

What is it?

The GHI one would measure without clouds, given the Sun's position and a clear atmosphere.

Inputs

site position (latitude, longitude, altitude);

Sun position (θ_z);

water vapour and aerosols (AOD: aerosol optical depth).

Role: GHI^cs_t is the reference used to remove the day/night and seasonal cycles.

Beer–Lambert law: intensity decays exponentially with the path length crossed.

Solis (Ineichen, 2008) is a simplified version of a radiative-transfer model. It starts from the Beer–Lambert law:

GHI^cs ≈ I₀ · exp(−τ / cos θ_z) · cos θ_z

τ: optical depth (air, water vapour, aerosols);

Solis adjusts τ and the exponent for water vapour, aerosols and altitude.

Settings: rural zone, water column 3.79 cm, AOD 0.16, I₀ = 1367 W/m².

The hourly value is the mean of the past hour: the Sun is computed at mid-hour, ω = (i − 12.5) × 15°.

Mean clear-sky error at Ajaccio: ≈ 32 Wh/m².

Ajaccio, 2007: Solis and measurement under clear sky.

CSI_t = GHI_t − GHI^cs_t

≈ 0 under clear sky, and 0 at night;

negative under clouds (down to −GHI^cs);

mean μ ≈ −33 Wh/m²: we center it, y_t = CSI_t − μ.

The procedure

• compute GHI^cs, then the CSI;
• center: y_t = CSI_t − μ;
• forecast ŷ_t+h;
• rebuild GHI^fc_t+h = ŷ_t+h + μ + GHI^cs_t+h (0 at night, never negative).

Ajaccio, 2007: top, GHI; bottom, CSI. The daily cycle has gone from the CSI.

The Solis model removes the cycles in two ways:

Difference (additive model), this course

CSI_t = GHI_t − GHI^cs_t, in Wh/m²;

no division: valid day and night (at night, 0 − 0 = 0);

all hours are kept, no filtering.

For hourly data with nights, the ratio is poorly suited: we use the difference. The ratio is also very sensitive to small timestamp offsets and to the filtering thresholds chosen (Voyant et al., 2025a; 2026).

Ratio (multiplicative model)

k_t = GHI_t / GHI^cs_t, unitless (≈ 1 under clear sky); widely used in the literature;

at night: 0 / 0, undefined; at sunrise and sunset, GHI^cs ≈ 0: values explode;

requires filtering on solar elevation (for instance keep h > 10°).

Mean ± standard deviation by month, daytime hours, Ajaccio 1998–2007.

The CSI is approximately stationary: much more than GHI, though not perfectly. Formal definition: appendix A.

GHI: the mean ranges from 163 to 521 Wh/m² depending on the month.

CSI: the mean stays between −105 and −43 Wh/m², close to 0.

The CSI standard deviation still varies (97 to 185 Wh/m²): more clouds in spring.

How far does GHI depart, on average, from clear sky? We measure the gap ε_t = GHI_t − GHI^cs_t (the CSI):

sCV = (√2 / (1 − d)) · √[ Σ_t (GHI_t − GHI^cs_t)² / Σ_t (GHI^cs_t)² ]

Numerator: typical size of the gaps to clear sky (the clouds).

Denominator: typical size of clear-sky GHI.

√2 / (1 − d) scales the worst case to 1: clear sky, then very overcast, every other hour (d = 0.2: even under overcast sky, at least 20 % of clear sky remains).

sCV = 0: always clear; sCV = 1: maximum variability. It does not depend on the horizon: it is a property of the site.

Ajaccio 1998–2007: sCV and F at 1 h, month by month (Voyant et al., 2026).

Ajaccio: sCV = 0.52 over 1998–2007.

Winter: 0.8 (changeable sky); July: 0.3 (often clear sky).

A variable series is not necessarily unpredictable: if the gaps to clear sky last, they can be anticipated.

ρ(k): correlation between the clear-sky gap now, ε_t, and k hours later, ε_t+k (the memory of clouds).

ρ_max(h): the strongest of these correlations for k between h and 12 h: the memory left when forecasting at horizon h.

F(h) = 1 − sCV · (1 − ρ_max(h))

perfect memory (ρ_max = 1): F = 1, everything is predictable;

no memory (ρ_max = 0): F = 1 − sCV.

Ajaccio 1998–2007: F and remaining memory by horizon (Voyant et al., 2026).

F = 0.92 at 1 h, 0.74 at 3 h, 0.53 at 6 h.

The atmosphere is chaotic: cloud memory fades within a few hours. sCV does not move.

1 · 2 · 3 · GHI · measurements · Clear sky · GHI^cs · CSI index · (difference) · 1 · Measurements

2 · Clear sky

Hourly GHI, converted from J/cm² to Wh/m² (÷ 0.36).

GHI^cs by Solis, Sun at mid-hour, over a typical year.

With the difference, nights cause no problem (0 − 0 = 0): no filtering on solar elevation, unlike the ratio.

4 · 5 · 6 · 7 · Centring

y = CSI − μ

Chronological · split · Model

P · AR · ELM

Forecast · ŷ_t+h · 3 · CSI

CSI_t = GHI_t − GHI^cs_t, every hour.

8 · 9 · Forecast GHI

ŷ + μ + GHI^cs

Evaluation · day · day+night · 4 · Centring

y_t = CSI_t − μ with μ ≈ −33 Wh/m²: series ready for AR and ELM.

An AR(p) model forecasts the next value from the last p values:

y_t+1 = w₁ y_t + w₂ y_t−1 + … + w_p y_t−p+1 + ε_t+1

Matrix form

y = X w + ε

It is a multiple linear regression. Since y is centred, no intercept is needed.

y (N × 1): the values to forecast;

X (N × p): one row per example, with the p lags;

w (p × 1): the coefficients to estimate;

ε (N × 1): the errors.

Five centred CSI values (Wh/m²): y₁ = −10; y₂ = −25; y₃ = −240; y₄ = −180; y₅ = −30. Take p = 2.

The two middle columns form X (3 × 2), the last one forms y (3 × 1).

Each row shifts the previous one by one step.

In practice: all hours are kept, nights included (at night, y = −μ ≈ 33). At Ajaccio: 8 years, about 70,000 rows for training.

We look for the w that minimizes the sum of squared errors:

J(w) = ‖y − Xw‖² = Σ_i (y_i − x_i^T w)²

At the minimum, the derivative is zero:

−2 X^T(y − Xw) = 0 ⟹ X^TX w = X^Ty

Conditions: many more examples than coefficients (N ≫ p);

columns of X not proportional to each other.

ŵ = (X^TX)⁻¹ X^Ty

Exact solution, in one step: no iteration, a single minimum.

Ajaccio, AR(2): ŵ = (0.97; −0.15), computed in 5 ms.

The last gap y_t weighs by far the most.

Xŵ is the point of the “plane of the columns of X” closest to y: its orthogonal projection.

The error ε = y − Xŵ is perpendicular to this plane.

Pythagoras: ‖y‖² = ‖ŷ‖² + ‖ε‖². This is the basis of the R² coefficient.

OLS does not “guess” anything: it projects the measurements onto what the inputs can explain.

Training · 1998 – 2005 · Training

Coefficients are estimated (w, β).

Chronological split, never shuffled: the past is not forecast with the future. After validation, re-estimate on training + validation.

Validation

Hyperparameters (p, L) are chosen without touching the test.

Validation · 2006 · Test · 2007 · time · Test

Performance is measured once, on data never seen.

ACF of the centred CSI, Ajaccio 1998–2005, all hours.

ACF(k): correlation between y_t and y_t−k, the value k hours earlier.

Here ACF(1) = 0.84, ACF(2) = 0.67, ACF(3) = 0.50: it decays slowly.

But this correlation propagates step by step: if y_t depends on y_t−1, which depends on y_t−2, then y_t and y_t−2 are correlated even without a direct link.

The ACF therefore does not tell how many lags are really useful. The PACF is needed.

PACF of the centred CSI, Ajaccio 1998–2005, all hours.

PACF(k): once y_t−1, …, y_t−k+1 are known, what does y_t−k add? It is the last coefficient w_k of an AR(k).

A bar outside the blue band: this lag brings direct information.

For a true AR(p), the PACF vanishes after lag p.

Here PACF(1) = 0.84, PACF(2) = −0.15, then values below 0.08: the PACF suggests p ≈ 2.

But with N ≈ 70,000 points, the ±0.007 threshold is tiny: almost everything is “significant”. Significant ≠ useful.

Filter

look at the data alone, before any model: PACF, AIC / BIC criteria (appendix C);

fast, but indirect.

Wrapper

train an AR(p) for p = 1 to 12 on 1998–2005;

measure the forecast error on validation (2006);

driven by the real objective, but more costly.

Wrapper approach: validation RMSE (2006, daytime hours) versus p.

p = 1: 78.0; p = 2: 77.7; p = 12: 77.3 Wh/m². Parsimony: the smallest p within 0.5 % of the best, i.e. p = 2.

1 · 2 · 3 · GHI · measurements · Clear sky · GHI^cs · CSI index · (difference)

• Compute GHI^cs (Solis), then CSI_t = GHI_t − GHI^cs_t; center it: y_t = CSI_t − μ.
• Split: training / validation / test.
• Choose p on validation.
• Build X and y; estimate ŵ = (X^TX)⁻¹X^Ty.
• Forecast ŷ_t+1 = ŵ^T x_t with x_t = (y_t, …, y_t−p+1).
• Rebuild GHI^fc_t+1 = ŷ_t+1 + μ + GHI^cs_t+1; set 0 at night (Sun below the horizon) and 0 if the forecast is negative.
• Evaluate on 2007: daytime, then day + night.

4 · 5 · 6 · 7 · Centring

y = CSI − μ

Chronological · split · Model

P · AR · ELM

Forecast · ŷ_t+h · 8 · 9 · Forecast GHI

ŷ + μ + GHI^cs

Evaluation · day · day+night

Simple, fast, interpretable: it is the linear reference to beat.

Every model must be compared with a simple reference. If it does not do better, it adds nothing.

Simple persistence

“It will be as it is now.”

GHI^fc_t+1 = GHI_t

Ignores the Sun's course.

Clear sky

“It will be sunny.”

GHI^fc_t+1 = GHI^cs_t+1

Ignores clouds.

Smart persistence

“The sky will stay the same.”

GHI^fc_t+1 = GHI_t − GHI^cs_t + GHI^cs_t+1

Keeps the same gap to clear sky.

Smart persistence = AR(1) with w = 1: the reference of the skill score SS.

Two rules, for every model: 1) at night (Sun below the horizon, θ_z > 90°), forecast = 0; 2) a negative forecast is replaced by 0.

e_t = GHI^fc_t − GHI_t (forecast error)

nMBE = [(1/N) Σ e_t] / mean(GHI_t)

MAE = (1/N) Σ |e_t|

RMSE = √[(1/N) Σ e_t²] ; nRMSE = RMSE / mean(GHI_t)

Essential check: if |nMBE| > 1 %, there is probably a problem, or a very poor model. The bias can be removed, provided this is tested; we do not do it here.

Errors are computed on the rebuilt GHI, over the test year (2007): first on daytime hours, then day + night. Always say which!

nMBE (bias): > 0 if the model over-estimates on average, < 0 if it under-estimates. Errors of opposite sign cancel: it is not a score for ranking models.

MAE: mean error, in Wh/m², easy to interpret.

RMSE: penalizes large errors more (sudden cloud). Always ≥ MAE.

nRMSE: RMSE as % of mean GHI, to compare sites.

SS = 1 − RMSE_model / RMSE_{smart persistence}

Skill score: > 0 if smart persistence is beaten, < 0 otherwise.

Looks only at the RMSE.

Two different references: SS compares with smart persistence, NICE^Σ with simple persistence (Voyant et al., 2025b).

NICE^k = L^k(model) / L^k(simple persistence)

L^k = ((1/N) Σ |e_t|^k)^1/k ; NICE^Σ = (NICE¹ + NICE² + NICE³) / 3

e_t = GHI^fc_t − GHI_t: the forecast error (previous slide).

k = 1: average errors (MAE); k = 2: RMSE; k = 3: large errors;

NICE^Σ < 1: the model beats simple persistence on all three norms at once.

Test 2007. Daytime: 4380 hours, mean GHI 372 Wh/m². Day + night: 8759 hours, mean GHI 186 Wh/m². RMSE in Wh/m²; SS and NICE computed on daytime hours.

Smart persistence divides the RMSE of simple persistence by 1.7. AR gains 3 % over it, ELM 7 %.

79 → 56 · AR RMSE (Wh/m²)

At night (Sun below the horizon), every model forecasts 0 by construction and is right: adding these hours lowers the RMSE (≈ ÷ √2).

But the mean GHI is halved: the nRMSE increases.

The ranking of models does not change.

An error figure without the list of hours used means nothing: always state daytime or day + night.

372 → 186

21 → 30 %

Mean GHI (Wh/m²) · AR nRMSE

Rule 1 (night): when the Sun is below the horizon (θ_z > 90°), the forecast is 0. Rule 2: a negative forecast is replaced by 0.

nMBE: mean bias, as % of mean GHI (see “Measuring the error”). Test 2007.

An nMBE depends on how it is computed: always state daytime or day + night, with or without rules.

How to read this table

No rules, day + night: both persistences have an nMBE of exactly zero.

Rule 2: setting negative forecasts to 0 raises the mean (smart persistence: +0.35 % → +1.25 % in daytime).

Rule 1: the nMBE of simple persistence goes from 0 to −1.02 % over day + night (next slide).

Clear sky: +16 %: it always forecasts sunshine.

second-to-last · daytime hour · last · daytime hour · 150 · 43 · Measurement · … · 150 · Persistence, no rules · … · 150

With rule 1 (night)

Values in Wh/m²; 43 Wh/m² = 2007 mean of the last daytime hour.

A single hour per day, but it weighs heavily: the day + night mean is only 186 Wh/m², because night hours count as zero.

first · night hour

No rules: persistence copies the previous hour's measurement. Over the year, each measurement is used once as measurement and once as forecast: the two sums are equal, nMBE = 0.

With rule 1 (night): at the first night hour the Sun is below the horizon, so the forecast 43 is replaced by 0. Every day, a forecast of about 43 Wh/m² disappears.

Over the year: 365 × 43 ≈ 15,642 Wh/m², plus 1,008 Wh/m² of small values measured at night: 16,650 Wh/m² are missing.

Relative to the GHI measured over the year (1,630,161 Wh/m²): 1.02 %, hence a day + night nMBE of −1.02 %.

0 · 43 · 0

The rule is physically right (no Sun at night), but it changes the bias.

Ajaccio, October 2007: 1-h forecast.

Both models follow regular days and miss sudden changes: an arriving cloud cannot be seen in the past.

AR(p) assumes a linear relation between the CSI and its lags. Often enough at short horizons, but not always.

Difficult situations

fast clear / overcast transitions (cloud fronts);

cloud-edge effects (GHI > GHI^cs);

very variable sky (cumulus);

nonlinear links between lags.

Under-fitting

A linear model cannot capture a curved relation, even with many lags.

Saturation

Beyond some p, adding lags no longer improves anything (slide “Choosing p”).

Idea: transform the inputs nonlinearly, then keep least squares. That is the ELM.

One-hidden-layer ELM (Huang et al., 2006).

A neural network with a single hidden layer.

Input weights W_in and biases b are drawn at random, then frozen.

Each neuron applies a nonlinear function g (tanh, sigmoid…).

Only the output weights β are learnt, by least squares.

No gradient back-propagation: training is done in a single solve.

The p lags of each row go through the hidden layer:

H = g(X W_in + b) (N × L)

Same equation, only the columns change. The small term αI (ridge regression) stabilizes the solve when L is large; at Ajaccio, α = 10⁻³.

Validation RMSE (2006, daytime hours) versus L, for p = 2.

L = 5: 85.2 Wh/m² → under-fitting.

L = 50: 75.9 Wh/m².

Beyond, no further gain: 75.8 at L = 200, 75.9 at L = 500 (risk of over-fitting).

As for p: test a few values (5 to 500) on validation and keep the smallest near-optimal one. Here L = 50.

Input weights are random: two trainings give two different models.

Candidate search: make 20 draws and keep the one with the smallest validation error (2006).

The test (2007) is used only at the end, for the selected draw.

Selected draw: test nRMSE 20.46 %.

The 20 draws: from 20.37 to 20.51 % (mean 20.45 %).

The best on validation is not necessarily the best on test: the gap is small here, but it must be checked.

Each point: one draw. Validation 2006 (horizontal), test 2007 (vertical).

Idea: add calendar variables known in advance to the lags: the hour h and the day of year j.

Problem: coded as a number, 23:00 is “far” from 00:00, although they are neighbouring hours.

Solution: place the hour on a circle, with two variables:

(sin(2π h / 24); cos(2π h / 24)) and (sin(2π j / 365); cos(2π j / 365))

4 more columns in X. The model then “knows” the hour and the season (Voyant et al., 2025a).

Daytime nRMSE, test 2007, same protocol for the 8 variants. “No clear-sky” approach: Voyant et al., 2025a.

AR (linear): needs the CSI. On raw GHI, even with sin/cos, it does worse than smart persistence.

ELM (nonlinear): with sin/cos, it learns the Sun's cycle by itself, from raw GHI.

nRMSE and NICE^Σ give the same ranking.

ELM strengths

fast training, a single solve;

captures nonlinear relations;

same method (OLS) as AR.

About 100 times more coefficients (learnt + random) for 4 more points of skill score. Is the gain worth the complexity?

ELM limits

L must be chosen;

result varies with the draw;

less interpretable than AR;

over-fits if L grows large;

needs a lot of data.

At Ajaccio, 1 h ahead

AR(2): 2 coefficients; nRMSE 21.3 %.

ELM: 51 learnt + 150 random; nRMSE 20.5 %.

Skill score SS: +0.03 (AR) versus +0.07 (ELM).

Forecast · Observe · hour t

hour t + 1

blue: the W training hours; pink: the hour to forecast

Matrix X always keeps W rows: the last W hours.

Every hour, the newest row is added and the oldest removed, then ridge is re-estimated:

ŵ = (X^TX + αI)⁻¹X^Ty

Update

Trick: no need to recompute everything. X^TX ← X^TX + x_newx_new^T − x_oldx_old^T.

Same for the ELM, with H instead of X.

Update: 11 µs for AR, 38 µs for the ELM, per hour.

Useful when the system changes (drifting sensor, soiled panel, new site): the model forgets the distant past.

Test 2007: every hour, ridge re-estimated on the last W hours, from 1 month to 2 years.

Short window: follows recent changes… but learns from little data.

Daytime nRMSE over 2007 versus window length.

AR (2 coefficients) works even with 1 month. The ELM (51 coefficients) degrades clearly with a short window: the more parameters a model has, the more data it needs.

A frugal model reaches good accuracy with little computation, little data and little maintenance.

Timings measured on a laptop: only the orders of magnitude matter.

The right question is not “which model is best?” but “does the gain justify the cost?” (computation, data, maintenance, monitoring).

Grid and units

Forecasting GHI is forecasting production.

Clear sky and CSI

Solis; CSI = GHI − GHI^cs, centred.

Cyril Voyant (cyril.voyant@minesparis.psl.eu) Mines Paris – PSL, Centre O.I.E.

Worked example on the web: cyrilvoyant.github.io/teaching/solar-forecasting-basics.html

Radiation

Direct, diffuse; Sun position; clouds.

AR and OLS

ŵ = (X^TX)⁻¹X^Ty, choice of p.

Time series

Components, horizon, AR(1).

Evaluation, ELM, online

Day / day + night, sliding window, frugality.

Huang G.-B., Zhu Q.-Y., Siew C.-K. (2006). Extreme learning machine: theory and applications. Neurocomputing, 70, 489–501. doi:10.1016/j.neucom.2005.12.126

Ineichen P. (2008). A broadband simplified version of the Solis clear sky model. Solar Energy, 82(8), 758–762. doi:10.1016/j.solener.2008.02.009

Voyant C. et al. (2017). Machine learning methods for solar radiation forecasting: a review. Renewable Energy, 105, 569–582. doi:10.1016/j.renene.2016.12.095

Voyant C. et al. (2022). Benchmarks for solar radiation time series forecasting. Renewable Energy, 191, 747–762. doi:10.1016/j.renene.2022.04.065

Voyant C. et al. (2026). Stochastic coefficient of variation: assessing the variability and forecastability of solar irradiance. Renewable Energy, 256, 123913. doi:10.1016/j.renene.2025.123913

Voyant C., Despotovic M., Notton G., Saint-Drenan Y.-M., Asloune M., Garcia-Gutierrez L. (2025a). On the importance of clearsky model in short-term solar radiation forecasting. Solar Energy, 289, 113490. doi:10.1016/j.solener.2025.113490

Voyant C. et al. (2025b). NICEk metrics: unified and multidimensional framework for evaluating deterministic solar forecasting accuracy. Sustainable Energy Technologies and Assessments, 81, 104588. doi:10.1016/j.seta.2025.104588

Hyndman R. J., Athanasopoulos G. (2021). Forecasting: Principles and Practice, 3rd ed. OTexts. otexts.com/fpp3

Box G. E. P., Jenkins G. M., Reinsel G. C., Ljung G. M. (2015). Time Series Analysis: Forecasting and Control, 5th ed. Wiley.

Data: Météo-France, Ajaccio station. All figures and schematics drawn by the author.

A process (z_t) is wide-sense stationary if:

E[z_t] = μ (constant) and Cov(z_t, z_t+k) = γ(k) (depends only on the lag k)

In practice

the mean is constant over time;

the variance is finite and constant;

the dependence between two instants depends only on their gap.

Simple check: plot the mean and standard deviation month by month (slide “Is the CSI stationary?”). Statistical tests exist but are not needed here.

ŷ_t+h = E[y_t+h | y_t, y_t−1, …, y_t−p+1]

Among all forecasts built from the past, the one that minimizes the mean squared error is the conditional expectation: the mean of possible futures, given the past.

We then write y_t+h = ŷ_t+h + ε_t+h: predictable part + error, with a zero-mean error.

An AR(p) model is a linear approximation of this expectation; the ELM, a nonlinear one.

If the model errors form white noise (non-significant error ACF), no usable information is left in the past.

AIC ≈ N · ln(σ̂²) + 2(p+1)

BIC ≈ N · ln(σ̂²) + (p+1) · ln N

σ̂²: variance of the training errors;

keep the p that gives the smallest criterion.

The first term falls as the model fits better; the second penalizes each added coefficient.

BIC penalizes more than AIC: it picks smaller models.

Advantage: no validation period needed. Drawback: assumes Gaussian errors.