Multi-Day-Ahead Price Forecasting Analysis

1. Introduction

Day-Ahead (DA) price or demand forecasts, including Day-Ahead Market (DAM) and Day-Ahead Real-Time (DA RT) price forecasts, are important inputs to electricity market planning, bidding, scheduling, and risk management. While next-day DA price or demand forecasts are widely used, many operational and analytical decisions require price expectations several days before the delivery day. Multi-day-ahead forecasting provides an earlier assessment of expected market conditions and allows forecast information to be incorporated into planning and decision-making before the delivery period approaches.

However, extending the forecast horizon introduces a fundamental forecasting challenge. For a given delivery day D, the information available to generate a forecast changes substantially depending on when the forecast is generated. A forecast generated on D-1 has access to information that is not available on D-3 or D-7. Weather forecasts may be revised, demand expectations may change, generation availability may evolve, and other market fundamentals may be updated or become available as the delivery day approaches. Therefore, forecasts generated for the same delivery day at different points in time represent different information states.

This creates two fundamental questions for multi-day-ahead DA forecasting:

• How does forecast accuracy change as the forecast lead time increases?
• Does forecast uncertainty appropriately reflect the increasing uncertainty at longer forecast horizons?

A meaningful evaluation therefore requires more than comparing forecasted and observed prices. Each forecast must be evaluated using the information set available at its forecast origin. For example, a D-7 forecast should be evaluated as it was originally generated using the information available at D-7, rather than reconstructed later using information that became available between D-7 and D-1.

This case study examines these questions using a PJM 7DA DAM system price forecasting example. The analysis evaluates multi-day-ahead DAM price forecasts across an extended forecast horizon, focusing on both forecast accuracy and probabilistic forecast uncertainty as a function of forecast lead time.

The vintage forecasting framework presented here is implemented using QR AMPS (AI Market Prediction System). The same presentation remains valid for demand forecasting via QR ALPS (AI Load Prediction System). The framework can be applied to both DAM and DA RT price/demand forecasting; however, the numerical analysis presented in this case study focuses specifically on PJM DAM system price forecasting.

2. Multi-Day-Ahead DAM Forecasting Challenge

2.1 Forward & Backward Perspectives

A multi-day-ahead forecast differs fundamentally from a conventional next-day forecast. For a delivery day D, forecasts can be generated several times before the market period occurs. For example, a forecast generated seven days before delivery is based only on information available at that time, while a forecast generated one day before delivery benefits from several additional days of market, weather, load, and other available information. To clarify the concepts we introduce the notation

F_L (D | X_L)

to represent a forecast for day D, performed on day L, conditional to the information X_L.

Typically for price forecasting, L = D-1, D-2, … D-7.

and for demand forecasting, L = D-1, D-2, …, D-14

Each forecast therefore represents a different information state X_L for the same future delivery day D. This distinction is important because evaluating a D-7 forecast using information that became available on D-6, D-5, or later would introduce information that was not available when the original forecast was generated. Multi-day-ahead forecasts must therefore be evaluated using the information set available at their respective forecast origins.

This relationship can be viewed from two complementary perspectives:

• Forward, the forecasting process starts from a given forecast origin and generates forecasts for progressively later delivery days.
• Backward or Vintage, a fixed delivery day D is selected and the historical forecast vintages generated on D-1, D-2, …, D-7 are examined to evaluate how the forecast evolved as the delivery date approached.

Together, the Forward Forecast and Forecast Vintage perspectives provide two complementary ways of analyzing multi-day-ahead forecasting: the forward perspective examines forecasts across increasing delivery horizons from a fixed forecast origin, while the vintage perspective examines how forecasts for a fixed delivery day evolve as new information becomes available.

The framework is not restricted to a seven-day horizon. The maximum forecast horizon can be configured according to the forecasting application and the availability of sufficiently informative forward-looking input data. In this case study, the D+1 through D+7 terminology is used to illustrate an extended multi-day-ahead forecasting horizon.

2.2 Forward Forecast

A Forward Forecast is generated from a fixed forecast origin and extends into future delivery days. For a forecast generated on day D, the delivery days may extend from the next day through the configured maximum forecast horizon:

D → D+1
…
D → D+3
….
D → D+7

On day D, conditional to all known features and information X_D, we forecast every day D+i, up to 7 DA:

F_D (D+i | X_D), i = 1,…,7

The forward-forecast perspective therefore evaluates how forecast uncertainty and other forecast characteristics change as the delivery date moves further into the future while the forecast origin remains fixed. The D+1 forecast is generated with the same information state as the D+7 forecast, but the D+7 delivery period is further into the future and therefore subject to greater uncertainty and potential changes in market conditions.

2.3 Forecast Vintage

A Forecast Vintage reverses the perspective. Instead of fixing the forecast origin and extending the delivery date forward, the delivery day D is fixed and forecasts generated at different points before that delivery day are collected: Forecast Origin → Delivery Day

D – 7 → D
…
D – 3 → D
…
D – 1 → D

Forecast Vintage is the collection of forecasts generated for day D, every prior day D-i, using the information state X_D-i of the execution day:

F_D-i (D | X_D-i), i = 1,…,7

The forecast-vintage perspective allows the evolution of a forecast to be examined as the delivery day approaches. The D-7 forecast represents the earliest forecast in the example, while the D-1 forecast incorporates additional information that became available during the intervening days.

3. Multi-horizon Forecasting Data Availability

A key practical challenge in multi-day-ahead forecasting is that not all forward-looking market information is available at every forecast horizon. Market operators may publish different forecasts, projections, and market datasets with different look-ahead periods and update schedules. Consequently, the information available to generate a D+1 forecast may differ substantially from the information available for D+3, D+7, or longer-horizon forecasts.

The information set X_D may therefore contain a combination of forward-looking variables with different availability horizons. For example, depending on the market and the specific data products available, the information set on forecast day D may include:

• Two-Day-Ahead (2DA) projections: information extending through D+2
• Seven-Day-Ahead (7DA) or Week-Ahead projections: information extending through D+7

The information set X_D may therefore contain a combination of forward-looking variables with different availability horizons. For example:

X_D = (D+2 demand, D+2 solar, D+2 wind, D+7 demand, D+7 Solar, D+7 Wind, D+7 Outages),

or

X_D = (D+2 demand, D+2 solar, D+2 wind, D+7 demand, D+7 outages).

This means that the feature set available to the price forecasting model is not necessarily identical across the entire forecast horizon. A feature with a D+2 availability horizon can be used directly for D+1 and D+2 forecasts, but it cannot provide a genuine forward-looking value for D+5 or D+7 unless an appropriate methodology is used to extend or replace it.

The forecasting architecture must therefore be able to manage feature availability according to the forecast horizon. In the QR AI platform, supporting both QR ALPS and QR AMPS, forward-looking features can be configured so that the model uses only information available for the corresponding delivery horizon. For example, a D+7 forecast configuration may apply different feature-handling strategies:

These configurations illustrate an important principle of multi-day-ahead forecasting: the AI model must account explicitly for the availability horizon of each forward-looking feature. A forecast for D+7 should not use a D+7 value that was published or updated only after the forecast was generated, nor should a D+7 forecast be reconstructed using information that became available at D+3, D+2, or D+1.

The appropriate treatment depends on the characteristics and availability of each variable. A feature may be replaced by another forward-looking source, excluded beyond its availability horizon, or extended using a deterministic pattern when such an approach is appropriate. In all cases, the objective is to preserve the information state that would have been available when the forecast was originally generated.

This feature-availability framework is particularly important for forecast-vintage analysis. When evaluating forecasts generated at D-7, D-5, or D-1 for the same delivery day, each forecast must use the forward-looking information that was actually available at its respective forecast origin. Later revisions or newly published forecasts cannot be introduced retrospectively.

4. Forecast-Vintage Evaluation Methodology

To replicate real-world operational conditions, this study uses the forecast-vintage approach from QR AMPS and, QR ALPS. Under this methodology, each model execution is preserved together with its original execution timestamp, model configuration, input data, and forecast output. Forecasts can therefore be evaluated according to the information that was actually available when each forecast was generated.

Rather than reconstructing a forecast retrospectively from today’s historical dataset, the analysis uses the original forecast created at its execution time. This distinction is essential because market, weather, load, generation, and other input data may be revised or updated after the forecast was generated.

For a given delivery day, this results in a sequence of forecast vintages:

D-7 Forecast → D-6 Forecast →… → D-2 Forecast → D-1 Forecast → Actual DAM Price

Each vintage represents a separate forecast execution for the same delivery day, generated at a different forecast origin and therefore based on a different information set. The collection of vintages makes it possible to examine both the accuracy of forecasts at different lead times and the way forecasts evolve as new information becomes available.

For a given delivery day, the forecast-vintage analysis can therefore address questions such as:

• How accurate was the forecast one day before delivery?
• How did forecast accuracy change at longer lead times?
• How did the forecast trajectory change as the delivery day approached?
• How did the forecast respond to information that became available between forecast vintages?
• Did probabilistic forecast uncertainty increase appropriately with forecast horizon?

The forecast-vintage approach therefore provides a consistent framework for evaluating forecast accuracy, forecast evolution, and forecast uncertainty across lead times, while preserving the information state associated with each original model execution.

5. Probabilistic Forecast Uncertainty

A probabilistic forecast provides additional information by estimating a distribution of forecast outcomes rather than a single point estimate. In QR AMPS and, QR ALPS. probabilistic forecasts are produced at multiple quantile levels, for example: P5, P10, P25, P50, P75, P90, and P95 where P50 represents the median forecast. A quantile represents a specified percentile of the forecast distribution. For example, actual prices are equal or lower/higher than the P5/P95 forecast with a 5% probability etc.

The probabilistic multi-day-ahead forecast is evaluated across three complementary dimensions that characterize both the accuracy of the central forecast and the behavior of the forecast distribution.

• Standard error metrics such as MAE, RMSE, MAPE or Directional Win Rate can be used to evaluate the error of the middle forecast for day D, performed on day D-7, D-6, …, D-1 vs the actual DAM price of day D. This measures the numerical accuracy of the central forecast at each forecast horizon. Forecast accuracy will generally be expected to deteriorate as the forecast horizon increases because the forecast is generated with less complete information about the eventual market conditions. However, the magnitude and consistency of this degradation are empirical and can vary by delivery day and market conditions.

Point-forecast metrics alone do not measure the quality of the full probabilistic forecast distribution.

• Price-Level-Dependent and Proportional Forecast Uncertainty. The probabilistic forecast captures the fact that forecast uncertainty is not constant across the price range. Uncertainty can increase with the expected price level, producing wider forecast distributions during higher-price periods and narrower distributions during lower-price periods. This represents a nonlinear relationship between expected price and modeled forecast uncertainty, rather than a constant absolute error band.

For example, as illustrated in the figure above for PJM System DAM prices on September 21, 2026, the probabilistic bands become wider around the morning and evening peak periods, where forecasted prices are higher, while they are comparatively narrower during lower-price periods.

This proportional behavior is important because a fixed absolute uncertainty band would imply the same forecast dispersion regardless of the expected price level. The quantile forecast instead allows uncertainty or volatility to scale with the expected price, providing information not only about the expected P50 price but also about how the magnitude of forecast uncertainty changes across the 24-hour delivery profile.

• Uncertainty Across the Forecast Horizon: A further dimension of probabilistic forecast evaluation is how the forecast distribution evolves as the delivery date moves further into the future. For a fixed forecast origin, the forecast for D+1 is generated for a delivery period that is closer in time, while the D+7 forecast concerns a delivery period several days further ahead. As the forecast horizon increases, the available information about future market conditions generally becomes less complete and more subject to revision.

This can be evaluated by comparing the probabilistic forecast bands across the forecast horizon. The figure below illustrates forecasts generated on September 22, 2026, for delivery days D+1 through D+7, corresponding to September 23 through September 29. The displayed quantiles are P5, P10, P25, P50, P75, P90, and P95.

One simple measure of the width of the forecast distribution is the P95-P5 spread:

Uncertainty Width = |P95 − P5|

A larger P95 – P5 spread indicates a wider predictive distribution and therefore greater modeled forecast uncertainty for that delivery period. The analysis can therefore illustrate the expected relationship between forecast horizon and predictive uncertainty:

• Shorter Horizon → More Available Information → Narrower Forecast Distribution
• Longer Horizon → Less Complete Information → Greater Forecast Uncertainty.

A numerical example is fully explored in section 6.4 below.

In QR AMPS and, QR ALPS, quantile forecasting incorporates algorithmic mechanisms for modeling forecast uncertainty across the prediction horizon. A dedicated parameter, “Expand forecast uncertainty with horizon,” controls how the width of the probabilistic forecast bands expands as the delivery period moves further into the future.

When enabled and appropriately configured, the parameter progressively increases the forecast distribution width with lead time, allowing the probabilistic forecast to reflect the greater uncertainty associated with more distant delivery periods. The resulting D+1 through D+7 forecasts provide a direct view of how the modeled forecast distribution evolves with horizon – from a relatively concentrated distribution at shorter horizons to a wider range of plausible prices at longer horizons.

6. PJM DAM System Price Case Study

6.1 Analysis Setup

As explained in section 3, the following PJM data was used in producing 7-DA Probabilistic PJM DAM system price forecasts:

• The publicly available 7-Day Load Forecast provides hourly forecasts by region and is updated on an hourly basis.
• PJM also maintains longer-horizon renewable generation forecasts. Its wind and solar forecasting procedures specify hourly forecasts extending up to 168 hours ahead, with the forecast itself updated hourly.

This multi-day-ahead analysis is implemented using the QR AMPS forecasting platform, which provides an environment for developing, executing, monitoring, and evaluating electricity price forecasting models. A key capability for this case study is the preservation of individual forecast executions and their associated timestamps. This enables the forecast-vintage methodology described in section 4 and allows historical forecasts to be evaluated according to their original forecasting context.

QR AMPS also supports probabilistic forecasting, allowing the forecasting system to generate configurable quantile forecasts in addition to the central forecast. The platform provides analytical capabilities for examining:

• point forecasts;
• probabilistic forecasts;
• forecast vintages;
• forecast error metrics;
• feature engineering and selection.

The feature engineering and selection capabilities can also be used to construct the model’s input set while maintaining a clear distinction between historical information (training set) and forward-looking information available at each forecast origin (feature set). Additionally, market-published variables with a limited forward availability horizon are incorporated only over the period for which they are available. Beyond that horizon, the corresponding features are either automatically excluded from the model input set or extended using an appropriate deterministic pattern, such as a daily profile, depending on the characteristics of the variable, refer to the table in section 3.

6.2 Forecast Accuracy by Horizon

Day D = September 21, 2026:

To evaluate forecast accuracy across lead times, we consider successive forecasts for the PJM DAM system price for the same delivery day D = September 21, 2026, generated on each preceding day, as described in Section 4.

D-5 (September 16): 5 DA forecast for September 21
…

D-1 (September 20): DA Forecast for September 21

Formally, the forecasts are represented as:

F_D-i (D | X_D-i), i = 1,…,5

where X_D-i represents the information available on day D-i, when the forecast was generated.

The average daily error of each forecast against the actual hourly DAM prices on September 21 is summarized below. For each metric, the forecast generated at D-i is compared with the corresponding actual hourly DAM price for delivery day D. Results are gathered in the table below, where the Column D-i shows the

Error metrics of the forecast (F_D-i (D | X_D-i) for day D performed on day D-i vs the actual price of day D

For this delivery day, the MAE, RMSE, and MAPE generally increase as the forecast origin moves farther from the delivery day. The D-1 forecast has the lowest error across all three numerical error metrics, while the D-5 forecast has the highest. This is consistent with the expectation that forecasts generated farther in advance are based on a less complete information set.

The progression, however, is nonlinear. The relative error compared with the D-1 forecast is shown below:

Relative to the D-1 forecast, the D-5 forecast has approximately 55% higher MAE, 58% higher RMSE, and 54% higher MAPE. The increase is therefore not proportional to the number of days in the forecast horizon; the degradation becomes more pronounced at the longer horizon in this example.

Directional Win Rate provides a complementary measure because it evaluates whether the forecast correctly captures the direction of price movement. Unlike MAE, RMSE, and MAPE, a higher Directional Win Rate indicates better performance. In this example, the D-1 forecast achieves a 91.67% win rate, while the D-5 forecast achieves 70.83%. The intermediate vintages do not follow a strictly monotonic progression, illustrating that forecast performance can vary from one horizon to another even when the general accuracy trend deteriorates with increasing lead time.

Versioning Report

QR AMPS and QR ALPS provide a visual versioning report in which a target delivery date D is selected and the forecasts generated for that delivery day at successive earlier forecast origins can be viewed sequentially as a rolling film. Plotting these forecast trajectories makes it possible to observe how the expected hourly price profile evolves as the delivery period approaches.

Combined with the corresponding error metrics, this qualitative visual analysis provides a practical basis for assessing how far in advance a forecast can provide useful guidance and how much forecast detail or precision can reasonably be relied upon at each horizon.

The actual DAM price series for day D is shown in black, while the forecast trajectories are shown in blue. In the September 21, 2026 example, the D-1 forecast more closely matches the actual DAM price profile, particularly around the peak periods. At the longer D-5 horizon, the forecast still captures important structural characteristics of the delivery day, including the overall hourly price pattern, but with less sharpness in the timing and magnitude of the price movements. As the delivery day approaches, successive forecast vintages refine this broader price pattern and increasingly capture the actual shape of the DAM price profile.

6.3 Probabilistic Forecast Bands Across The Horizon

The probabilistic forecasts provide an additional perspective on how forecast uncertainty changes with the delivery horizon. As discussed in Section 5.c, for a D+1 forecast, the prediction intervals are generally tighter because the model has access to more recent information about the upcoming delivery period. As the horizon extends toward D+7, the information available at the forecast origin provides less certainty about the eventual market outcome, and the forecast distribution correspondingly expands, resulting in wider probabilistic bands or cone. This progression allows increasing uncertainty to be represented explicitly rather than reflected only through larger point-forecast errors.

The table presents the forward forecasts generated on September 22, 2026, covering delivery periods from D+1 (September 23) through D+7 (September 29). This provides a consistent forecast-vintage view of how modeled forecast uncertainty evolves across the delivery horizon. The probabilistic bands are relatively tight for D+1, with a (P95 − P5) width of $51.84, and become progressively wider towards D+7, where the width reaches $119.09. This illustrates the increase in modeled forecast uncertainty as the delivery period moves further into the future.

Forward Forecasts	Probabilistic Forecast Bands (P5, P10, P25, P50, P75, P90, and P95)	P95 – P5 at peak price (19:00)
D+1		51.84
D+2		56.42
D+3		57.32
D+4		66.26
D+5		71.83
D+6		79.02
D+7		119.09

Incremental widening between consecutive horizons

The table below examines the incremental change in probabilistic forecast uncertainty between consecutive delivery horizons, using the (P95 – P5) interval width at the 19:00 peak hour as the measure of predictive dispersion. The results show how the forecast bands expand from one horizon to the next and identify where the largest increases in modeled uncertainty occur as the delivery period moves further into the future.

Denoting the incremental change in the width of probabilistic bands across forecast days D by

∆W_D+1 = (P95_{D +2} – P5_{D +2}) – (P95_D+1 – P5_D+1)

The results show that forecast uncertainty generally expands as the delivery horizon increases. It should be noted the widening is not uniform across consecutive horizons and exhibits nonlinearity as demonstrated by the pronounced jump between D+6 and D+7, where the (P95 – P5) width increases by $40.07, from $79.02 to $119.09 in a single additional forecast day.

This demonstrates that Quantile forecasts and scenario forecasts represent uncertainty differently:

• Scenario forecasts typically provide discrete possible outcomes, such as low, average, and high scenarios. Each scenario represents a possible realization or trajectory under a particular set of assumptions and does not, by itself, assign a probability to that outcome. The spacing between scenarios and their progression across the forecast horizon are determined by the scenario-generation methodology.
• Quantile forecasts identify values at specified probability levels of a conditional forecast distribution. For example, P5, P50, and P95 represent the 5th, 50th, and 95th percentiles of the modeled distribution. This has several important consequences:
  • The forecast distribution is not necessarily symmetric around P50; the distances between P5, P50, and P95 can differ.
  • The expansion of probabilistic forecast bands is not deterministic or necessarily linear across the forecast horizon. While overall dispersion generally increases with lead time, the amount of widening between consecutive horizons can vary substantially, as demonstrated by the $40.07 increase from D+6 to D+7.
  • Forecast uncertainty can be price-level dependent, producing wider forecast distributions during higher-price periods and narrower distributions during lower-price periods. Thus, uncertainty is not represented as a constant absolute band around the point forecast.

6.4 Forecast Extreme Event

In a different example, an unexpected price spike occurred on September 2, 2026, at 19:00, in PJM. The peak deviated significantly from trailing historical averages, and its magnitude and timing were not consistent with the historical price patterns observed over the preceding month, making the event difficult to anticipate from historical seasonality alone.

The forecast-vintage error analysis shows how QR AMPS progressively adjusted its forecast as the delivery period approached. This adjustment occurs as additional and more current feature information becomes available closer to delivery. The AI engine automatically incorporates updated feature values as they become available, without manual intervention. As a result, successive forecast vintages benefit from an increasingly informative information set.

Forecasts generated as early as D-7 captured part of the developing deviation from the expected price pattern, while subsequent forecast vintages progressively refined the forecast trajectory and the magnitude of the event. This example demonstrates the model’s ability to adapt its forecast as new information becomes available, rather than relying solely on recurring historical patterns.

The evolution from the D-7 forecast through the subsequent vintages also illustrates the value of the forecast-vintage approach. It allows the adjustment process for an atypical market event to be observed directly, including how the forecast responds to information that was not available at the earliest forecast origin.

The table below summarizes the forecast errors for the extreme event across the D-7 to D-1 forecast vintages, providing a quantitative view of the adjustment of the peak magnitude and its contribution to overall forecast error.

The table below presents the pointwise forecast errors at the peak hour (19:00) with actual DAM value of 416.64$ across the D-7 to D-1 forecast vintages, showing progressive improvement as the delivery day approaches.

The relative error decreases from approximately 74.3% at D-7 to 13.2% at D-1, indicating that successive forecast vintages progressively adjust the expected peak toward the realized DAM price of $416.64. Although the improvement is not strictly monotonic at every intermediate vintage, the overall reduction in error demonstrates the model’s ability to incorporate increasingly informative, more recent information and substantially refine the forecast of the extreme peak as delivery approaches.

Other particularly important findings are:

• D-7 → D-1 absolute error: decreases from $309.21 to $54.97, a reduction of about 82.2%.
• Relative error: decreases from 74.25% to 13.20%, an improvement of 61.05 percentage points.
• In relative terms, the D-1 error is only about 17.8% of the D-7 error.
• The largest refinement occurs between D-2 and D-1, where the absolute error falls from $198.40 to $54.97, a reduction of about 72.3% in just one additional day.

The forecast may remain substantially biased at longer horizons, but the final pre-delivery vintage can incorporate information that materially changes the expected peak.

7. Conclusion

Multi-day-ahead DAM price forecasting presents a forecasting problem in which the information available to the model changes substantially with forecast lead time. The analysis in this case study demonstrates why evaluating such forecasts requires consideration of both forecast accuracy and the information state associated with each forecast. The forecast-vintage methodology preserves each model execution according to its original forecast timestamp, allowing each forecast to be evaluated using only the information that was available when it was generated.

The results show that forecast accuracy generally improves as the delivery period approaches, with longer-lead forecasts typically exhibiting higher MAE, RMSE, and MAPE. However, the progression is not necessarily monotonic at every individual horizon, reflecting the influence of changing market conditions and newly available information. Directional accuracy also varies across vintages and provides a complementary view of forecast performance.

The probabilistic analysis provides a second dimension of evaluation. The forward forecasts show that forecast uncertainty generally expands with the delivery horizon, with wider P95 – P5 intervals for more distant delivery periods. This allows the forecasting system to represent increasing uncertainty explicitly rather than relying solely on larger point-forecast errors.

The extreme-event example further demonstrates the value of the forecast-vintage perspective. As an atypical price event developed, successive forecast vintages incorporated newly available information and progressively adjusted the forecast trajectory. This illustrates that multi-day-ahead forecasting should not be viewed as a single static prediction, but as a sequence of forecasts that evolves as the information set changes.

Together, the point-forecast, probabilistic, and forecast-vintage analyses provide a more complete characterization of multi-day-ahead DA forecasting performance. They show not only how far forecasts are from realized market prices, but also how forecasts evolve as new information becomes available and how modeled uncertainty changes with the delivery horizon.

The forecast data and system screenshots are proprietary to QuantRisk. They may be reproduced or used only with clear attribution to QuantRisk.