**1. Introduction**

In mining operations, the primary energy consumer is the comminution system, responsible for more than half of the entire mine consumption [1]. From all pieces of equipment that integrate the comminution circuit, the semi-autogenous grinding mill (SAG) is perhaps the most important in the system. With an aspect ratio of 2:1 (diameter to length), these mills combine impact, attrition and abrasion to reduce the ore size. SAG mills are located at the beginning of the comminution circuits, after a primary crushing stage. Although there are small SAG mills, their size usually ranges from 9.8 × 4.3 to 12.8 × 7.6 m, with a nominal energy demand of 8.2 and 26 MW, respectively [2], which make SAG mills the most relevant energy consumer within the concentrator. Modelling their consumption behaviour supports the operational control and energy demand-side management [3].

Most theoretical and empirical models [4–6] demand input feed characteristics, such as hardness, size distribution and inflow rate, SAG characteristics, such as sizing and product size distribution, and operational variables such as bearing pressure, water addition and grinding charge level. Although they are suitable to provide adequate design guidelines, they lack accurate in-situ inference since most assume steady-state and isolation from up and downstream processes. In response, model predictive control, SAG MPC [7], combines those methods with real-time operational information. However, expert knowledge is required to model the SAG mill dynamics properly.

From a geometallurgical perspective, the integration of new predictive methods that account for space and time relationships over real-time attributes has been defined as a fundamental challenge [8,9] in mining operations, particularly in an integrated system such as comminution. In response, data-driven approaches have been proposed ranging from support vector machines [10]

and gene expression programming [11] to hybrid models that combine genetic algorithms and neural networks [12] and recurrent neural networks [13]. As data-driven methods are sensitive to the context (available information) and representation (information workflow), the authors have studied the use of several machine learning and deep learning methods in modelling the SAG energy consumption behaviour based only on operational variables [14].

The energy consumed by a SAG mill is related to several factors such as expert operator decisions, charge volume, charge specific gravity and the hardness of the feed material. Knowing the output hardness material becomes relevant for the downstream stage in the primary grinding circuit. Ore hardness can be characterized at sample support by combining the logged geological properties and the result of standardized comminution tests. They can be used to predict the hardness of each block sent to the process. However, these attributes are not always available. In response, a qualitative characterization of the ore hardness processed at time *t*, relative to the operational hardness of the ore processed at time *t* + 1 can be done using only operational variables rather than a set of mineralogical characterizations. This qualitative characterization is referred and here used as operational relative-hardness (ORH).

We take advantage of previous works [14] by knowing that the Long Short-Term Memory (LSTM) [15] outperforms other machine learning and deep learning techniques on inferring the SAG mill energy consumption. Therefore, Section 2 presents the ORH and LSTM models, Section 3 establishes the SAG mill experimental framework, the results of which are presented in Section 4, and conclusions are drawn in Section 5.
