Energy Forecasting Methodologies

Estimating energy demand, generation, capacity expansion, and other parameters accurately is a prerequisite for energy system planning. The literature on forecasting includes many different approaches. These approaches include traditional statistical time series models, machine learning and deep learning models, and hybrid physics-informed models. An extensive examination is made in this section on forecasting techniques relating to long-term (multi-decade) energy capacity estimates, along with comparative performance as well as associated assumptions, and specifically related to Turkey's energy transition.

Traditional Statistical Time Series Methods

Based on econometrics and statistics, statistical time series forecasting dominated the energy forecasting literature through the 1990s and is still widely used for benchmark comparisons. These techniques model temporal dependencies using parametric assumptions related to data-generating processes, providing strongly indicates (consistency, asymptotic normality) and interpretable parameters. Nonetheless, because they rely on linearity and stationarity and make use of only a few exogenous variables, they cannot perform well in long-horizon, high-dimensional, non-linear energy forecasting problems.

#### ARIMA and SARIMA Models

The ARIMA (Auto-Regressive Integrated Moving Average) model, developed by Box and Jenkins in 1970, is the most widely used method of univariate time series forecasting. An ARIMA(p, d, q) model is formed by combining the following: Auto-Regression (AR) using $p$ past values, differencing $d$ times to make it stationary and Moving Average (MA) of $q$ past forecast errors. Mathematically:

$$\varphi (B)(1-B{)}^d{y}_t=\theta (B){ϵ}_t$$

The above equation can be explicitly stated by the following notation. Here, $y_t$ is the time series, $B$ is the backshift operator (that is, $B{y}_t=y_{t-1}$), $\varphi (B)=1-{\varphi }_{1}B-\cdot s-{\varphi }_p{B}^p$ is the AR polynomial, $\theta (B)=1+{\theta }_{1}B+\cdot s+{\theta }_q{B}^q$ is the MA polynomial, whereas ${ϵ}_t\sim N(0,{\sigma }^2)$ represents white noise.

The ARIMA model is popular because it has few parameters, has been made easy to use by Box-Jenkins methodology, and performs reasonable in the short run. Nonetheless, ARIMA operates under the assumption of linear relationships and cannot identify the non-linear behaviours that are typical of technology adoption (S-curves, exponential growth phases, saturation). Hong et al. [hong2016forecasting] perform benchmark ARIMA for energy forecasting across 25 studies. There were a mean absolute percentage error (MAPE) of 5 to 15% for forecasts over 1 year but deteriorated to 20-40% for more than 5 years. Apart from that, there is systematic under-forecasting when the variable rapidly increases.

SARIMA, or Seasonal ARIMA, is an extension of ARIMA that accounts for seasonal patterns in time series data. SARIMA includes periodic autoregressive (AR) and moving average (MA) terms at lag $s$. For instance, for monthly data with annual seasonality, $s$ would be set to 12. A SARIMA(p,d,q)(P,D,Q)_s model is:

$$\varphi (B)\Phi (B^s)(1-B{)}^d(1-B^s{)}^D{y}_t=\theta (B)\Theta (B^s){ϵ}_t$$

Where seasonal AR and MA polynomials are $\Phi (B^s)$ and $\Theta (B^s)$. According to De Oliveira and Oliveira [oliveira2018forecasting], SARIMA was applied to the Brazilian electricity demand. The MAPE was 3.2% for 1-month ahead. The 12-months ahead forecast MAPE was 18.5% which indicates the accuracy deteriorates as the forecast horizon increases. For Turkey, Akay and Atak [akay2007grey] used SARIMA(1,1,1)(0,1,1)_{12} for electricity consumption forecasting. Their results suggest that seasonal patterns dominate short-term dynamics. However, long-term consumption trends require the introduction of additional explanatory variables other than lagged or historical consumption.

#### Vector Autoregression (VAR)

VAR models extend ARIMA functionality from single series to multiple series. A VAR(p) model containing $k$ variables is given by:

$${\mathbf{y}}_t=\mathbf{c}+{\mathbf{A}}_1{\mathbf{y}}_{t-1}+\cdots +{\mathbf{A}}_p{\mathbf{y}}_{t-p}+{\mathbf{ϵ}}_t$$

${\mathbf{y}}_t$ is a $k * 1$ vector of time series, ${\mathbf{A}}_i$ are $k * k$ coefficient matrices, and ${\mathbf{ϵ}}_t\sim N(0,\mathbf{\Sigma })$ is multivariate white noise. The VAR facilitates the impulse response analysis (how the shocks from one variable impact the others) and Granger causality test (whether one variable predicts the other).

For energy-related applications, VAR captures the relationships that exist between electricity prices, demand and actual capacity. Ziel and Weron [ziel2018day] applied VAR to forecasting electricity prices in the European markets taking the generation mix, demand, and fuel prices. Nonetheless, the curse of dimensionality makes VAR difficult to apply to high-dimensional settings. That is, estimating the $k^{2}p$ parameters will require large samples. In this context, "high-dimensional energy systems" refers to energy systems with 26 or more technologies. The Bayesian VAR with shrinkage priors is somewhat a solution to this [banbura2010large], but essentially stays under 20 variables in practice.

Çetin and Egrican [cetin2011cointegration] develop a VAR model for Turkey to analyze the relationship between electricity demand, GDP, and prices. They show that electricity demand has an elastic response to price hike (elasticity $-0.5$) but has an inelastic response to GDP growth (elasticity $+0.8$). The exponential rise and saturation patterns of technology adoption cannot be captured because of linearity assumption of VAR.

#### Exponential Smoothing and State Space Models

Exponential smoothing methods use past observations with weights declining exponentially to forecast future values. The SES (simple exponential smoothing) technique:

$${\hat{y}}_{t+1}=\alpha y_t+(1-\alpha ){\hat{y}}_t$$

where $\alpha$ is the smoothing parameter in the interval $[0,1]$. Extensions include trends and seasonality, Holt's linear trend method, and Holt-Winters method. Error-Trend-Seasonality (ETS) state space models can give maximum likelihood estimates and prediction intervals [hyndman2008forecasting] and these are equivalent to exponential smoothing.

ETS represents a model with additive errors, additive trend, and additive seasonality is represented as ETS(A,A,A):

$$y_t={\ell }_{t-1}+b_{t-1}+s_{t-m}+{ϵ}_t{\ell }_t={\ell }_{t-1}+b_{t-1}+\alpha {ϵ}_t{b}_t=b_{t-1}+\beta {ϵ}_t{s}_t=s_{t-m}+\gamma {ϵ}_t$$

where ${\ell }_t$, $b_t$, $s_t$ respectively denote level, trend, and seasonal states. As per the findings made by Makridakis et al. (2020), exponential smoothing performs well in comparison to ARIMA in short term forecasting. However, its application in long term series with non-stationary trends is an issue. Hyndman and Athanasopoulos (2021) found that energy demand forecasts for 10 countries using ETS had a MAPE of 2-5% for 1-year ahead forecasts, but 15-30% for 5+ years. This suggests systematic over-smoothing during ramp-up phases.

Taylor [taylor2010triple] suggested the application of double-seasonal exponential smoothing to electricity demand as a two-dimensional time series with daily and weekly cycles. He achieved a MAPE of 2.1% on a day-ahead forecasting of hourly electricity demand. Yet, this micro level seasonality has no relevance for strategic capacity planning which is of annual aggregates over 26 year horizons and macro level growth dynamics dominates.