Similarly to how recurrent networks are specialised for sequential input data, convolutional networks are specialised for grids of values, such as 1-dimensional time series or 2-dimensional images (Goodfellow et al., 2016). Compared to fully-connected layers that learn global patterns, convolutional layers learn local patterns (Chollet, 2018). Rather than having every input unit interact with every output unit, convolutional layers use fewer operations. In addition to reduced memory usage and computational cost, convolution layers also have other desirable properties:
they learn translation invariant patterns, meaning it learns to recognise a pattern in an image no matter where it appears; they learn spatial hierarchies of patterns, from initially small, local patterns to larger, more abstract and increasingly complex patterns made up of the underlying ones (Goodfellow et al., 2016).
An image consists of two spatial dimensions (height and width) and a depth of some channels;
a grey-scale image has one channel, whereas a colour image has three: red, green, and blue. A convolution operation slides somekernel, also called a receptive field, across the input grid, and computes some transformations at every possible location. Typical kernel dimensions are 3×3 or 5×5 with a stride — the number of steps the convolution takes when extracting the next patch — of 1127 (Chollet, 2018). The key is that these transformations are the same for each
127With no stride (stride of 1), successive receptive fields will overlap somewhat, which is usually desirable when searching for local patterns.
extracted patch, i.e. that the convolution looks for the same learned patterns in each spot. The resulting vectors are applied a non-linear activation function, then spatially reassembled into a feature map where the ”pixel” values are now these activations (Chollet, 2018).
While the original image has some given channel axis depth, the resulting feature maps from each convolutional layer can be arbitrarily deep. The width and height can shrink depending on thepadding and kernel dimensions, but the depth depends on the number offilters specified.
Filters encode different aspects of the input map or image, the number of which is specified as one of the layer’s parameters alongside the kernel dimensions, type of padding, and activation function — among others. As such, each filter learns to look for a certain pattern, and outputs a map with its activations at all possible locations in the input map (Chollet, 2018). The stack of these maps constitute the layer’s output feature map.
After a non-linear activation function has been applied to the output of the convolutions, it is typically followed bypooling to further refine the map to a form of summary statistic (Goodfellow et al., 2016). A popular method is max pooling, which outputs the maximum value found in each channel of rectangular grids in the input map. It is typically used with a 2×2 kernel with stride 2, to effectively downsample the feature map in both dimensions by a factor of two (Chollet, 2018). This reduces the number of feature map coefficients to compute, and more directly promotes hierarchical filters by making successive convolutions span increasingly larger spatial areas (Chollet, 2018).
A.3.1 Convolutional Pyramids
New record-setting convolutional architectures are announced with such frequency that defin-ing a generally ”best” architecture is impractical (Goodfellow et al., 2016). Instead, we design convolutional networks following the guiding principle of convolutional pyramids: large recep-tive fields are often necessary for the network’s structured understanding of low-level image processing tasks, but stacking sufficient layers or using adequately wide kernels can become cost-prohibitive (Shen, Chen, Tao, & Jia, 2017). Starting with the spatially wide (and high) input images with few channels, we stack successive convolution and pooling layers that progres-sively downsample the spatial dimensions while adding filters, thereby resulting in a pyramid of decreasing spatial footprint and increasing depth. This enables a spatial-filter hierarchy of increasingly wide fields, without sacrificing computational efficiency (Shen et al., 2017).
B Additional Information about Elbas
B.1 Prices in the Regulating Market
Figure 23 shows how the regulating price is derived. If there is a need for more power to be delivered in a given hour, the TSOs set the price so that up-regulation occurs (price above the market/Elspot price), which is the situation shown in the figure. If, on the other hand, there is a need for less power to be delivered in a given hour, the price is set so that down-regulation occurs (price below the market/Elspot price). Thus, the regulating prices are used to incentivise more or less power being produced in each hour of delivery. If consumption exceeds planned production, a higher price incentivises more producers to deliver power in a given hour.
If production exceeds consumption, a lower price incentivises more producers to buy cheap power from other producers instead of generating their own, and as such, reducing overall power production.
Figure 23: Up- and down-regulation in the regulating market. The illustration is based on:
Nord Pool (n.d.).
Table 7, which is based on Table 4 in Scharff and Amelin (2016), shows how production and consumption imbalances are settled.128 For consumption imbalances, one price is used for
set-128Production imbalances are calculated as the difference between the latest submitted production plan after settlements in the Elspot market and the observed production. Consumption imbalances are calculated as the latest submitted production/consumption plan after Elspot, adjusted for trading in Elbas, and observed consump-tion (Scharff & Amelin, 2016). The binding plan may be updated until 45 minutes before power delivery. Any changes to the production plan that is not corrected through Elbas trading, goes into the consumption imbalance, which implies that the production imbalance only reflects the deviations during the delivery hour.
tlement and it reflects the dominating direction of regulating power in a given hour. That is, it equals the up-regulating price if the dominating direction is up, and the down-regulating price if the dominating direction is down. For production imbalances, on the other hand, a two-price system is applied in settlements. Hence, if the dominating direction is up, producers with a pro-duction deficit pay the up-regulation price, while producers with a propro-duction surplus receive the day-ahead price. Conversely, if the dominating direction is down, the producer with a deficit pay the day-ahead price while the producers with a surplus receive the down-regulation price.
The purpose of the two-price system is to ensure that no market participants may profit from balancing in real-time compared to trading the corresponding power in the day-ahead market (Scharff & Amelin, 2016).
Own position in
regulating market Up-regulation No regulation Down-regulation Production deficit
(= prod. purchase)
Pay:
Pup×Edef icit
Pay:
Pda×Edef icit
Pay:
Pda×Edef icit
Production excess (= prod. sale)
Receive:
Pda×Eexcess
Receive:
Pda×Eexcess
Receive:
Pdown×Eexcess
Consumption deficit Pay:
Pup×Edef icit
Pay:
Pda×Edef icit
Pay:
Pdown×Edef icit
Consumption excess Receive:
Pup×Eexcess
Receive:
Pda×Eexcess
Receive:
Pdown×Eexcess
Table 7: Imbalance price settlement for production imbalances (two-price system) and consump-tion imbalances (one-price system). Source: Scharff and Amelin (2016).