*2.2. Synthetic Time History Production*

For the dispatch model to perform the stochastic optimization, a method of producing synthetic data was required. An autoregressive moving average (ARMA) model coupled with a Fourier series detrending model, known as a FARMA model, was used to produce synthetic electricity-price time histories. RAVEN's capabilities allow users to train a FARMA on historical, time-dependent datasets, then sample that model to create synthetic time histories. The FARMA model takes an input time history and uses a Fourier series decomposition to extract the trends that occur due to temporal variations. For example, in a load profile, there are variations that occur due to seasonal weather variations or weekly use patterns. These trends are not noise and should be quantified and extracted prior to the ARMA producing synthetic noise. A Fourier fast transform was used to extract the time scales of these correlations. This process is detailed in depth in [16]. The Fourier detrending equation, used on the time scales found from the fast Fourier transform, is given in Equation (1), as taken from [16]:

$$\mathbf{x}\_t = y\_t - \sum\_{i=1}^p \left[ a\_m \sin(2\pi f\_m t) + b\_m \cos(2\pi f\_m t) \right] \tag{1}$$

After the Fourier detrending, the ARMA statistically quantifies the noise and allows for stochastic reproduction in future samples. The Fourier detrend pulls out all the strong, time-dependent trends in the dataset, leaving the noise. An ARMA algorithm can be used to model that noise. Equation (2) describes the ARMA process:

$$\mathbf{x}\_{t} = \sum\_{i=1}^{p} \phi\_{i} \mathbf{x}\_{t-1} + \alpha\_{t} + \sum\_{j=1}^{q} \theta\_{j} \alpha\_{j-1} \tag{2}$$

where *x* is the output vector for a given dimension *n*, the input vectors, *θ* and *φ*, are *n* by *n* matrices, and *α* is the error term. The variables *p* and *q* are the autoregressive and moving average terms, respectively. When parameter *p* is zero, only the moving average is used. When *q* is zero, the process is exclusively autoregressive.

The dispatch model can then sample the FARMA model and produce large numbers of synthetic time histories for stochastic optimization purposes. As a price-taker model, its dispatch is based on economic decisions dependent on the electricity prices from trained FARMA.

Historical electricity prices for the HTTR operating region (i.e., the Tokyo region of Japan's electricity system) were used for training the FARMA model. The HTTR-GT/H2 was assumed ineligible for the non-fossil or baseload markets due to its status as a small-scale test reactor. Larger commercial reactors could likely participate in the spot and intraday markets in addition to the baseload and non-fossil markets. The input data were separated into 30 min increments covering a 1-year period. Prices reflect 2018 historical prices.

RAVEN's advanced clustering methods were leveraged to improve the accuracy of the synthetic price data [17]. While Fourier detrending is useful for capturing seasonal effects, the clustering takes it a step further by isolating those segments with major differences.

RAVEN clustered the data set into representative four-day periods. Each representative four-day period is known as a cluster, and each cluster was trained as an individual FARMA. By training these individual four-day clusters—as opposed to a single FARMA over the year, RAVEN can achieve improved accuracy by further isolating the effects of long-term seasonal trends. Each specific four-day window, or segment, is then assigned to the cluster that best represents it. The clustering algorithm offers improved accuracy compared to overtraining the FARMA over the entire year.

Figure 3 shows each cluster and the time at which it occurred in the year. Each panel shows the four-day periods that are similar to each other and are thus representable by a single FARMA model. Note that the four-day periods in the shoulder months (usually in the spring and fall) tend to be similar. Additionally, the summer or winter peaks may have only a few four-day segments in their cluster. This is a feature of the clustering algorithm: by training a different FARMA for each representative window, the peak price events will not impact the production of synthetic data for the more typical shoulder months.

In Figure 4, a complete synthetic time history is plotted against the original data. This synthetic history reflects a possible time history of electricity prices that is statistically similar to the original input data. The FARMA can be sampled many times over to produce a broad range of synthetic time histories statistically similar to the input price profile.

For each model run, the FARMA was sampled 100 times to reduce the modeling uncertainties in the input electricity prices. The price duration curve (PDC) is shown in Figure 5.

The historical PDC is largely identical to the average synthetic PDC, except when comparing the 100 or so highest electricity price hours.

This dispatch model will be used to investigate the impact of the PDC discrepancy found in the 100 or so highest electricity price hours. Therefore, the model was run in two modes: one using 100 synthetic price histories and returning the expected breakeven sale price of hydrogen, and the other using the historical PDC to determine the expected breakeven sale price of hydrogen.

**Figure 3.** Four-day segments plotted by cluster, as produced by RAVEN when training the FARMA.

**Figure 4.** Historical 2018 Tokyo region electricity prices plotted against the synthetic time history produced by sampling the RAVEN FARMA.

**Figure 5.** PDC comparison between the synthetic and historical data.
