Modeling and Power Estimation of Continuously Varying Residential Loads Using a Quantized Continuous-State Hidden Markov Model
Misbah Aiad
Peng Hin Lee
School of Electrical and Electronic Engineering Nanyang Technological University 50 Nanyang Avenue, Singapore 639798 Email:
[email protected]
School of Electrical and Electronic Engineering Nanyang Technological University 50 Nanyang Avenue, Singapore 639798 Email:
[email protected]
system stability [1]. Several conducted studies showed that energy savings between 9% to 20% are achievable in residential homes if occupant are aware about detailed individual appliances consumptions rather than total aggregate consumption. This information was found to be influencing to users in a way that they use appliances less often (especially those with high power consumption) [2]. Non-intrusive load monitoring (NILM) or energy disaggregation aims to extract power consumption profiles of individual home devices only from the aggregate consumption signal measured by the smart meter. A very essential requirement of NILM is to use aggregate total measurements only without sub-metering or installation of sensors at device level. To simplify the problem of energy disaggregation, devices in households can be divided into four possible categories [3] - Always ON devices: these devices do not switch to the OFF state. Lumped together, always ON devices will determine the household baseload; the minimum level of total household power consumption. - ON/OFF devices: these devices can be either OFF or operating (ON) at a consistent power level. - Finite State Machines (FSM): these devices can be either OFF or operating (ON) at one of possible finite number of states. Each state is usually a specific consistent power level. Ceiling of stand fans are typical examples of devices under this category. - Continuously varying devices: these devices can be either OFF or operating (ON) at any power level within its operation characteristics. Since this type of devices vary their consumptions continuously, they can be considered as infinite state machines. Light dimmers, power tools and electronic devices are typical examples of devices that fall under this category.
Abstract— Hidden Markov Models (HMMs) and their extensions have broad useful applications in several fields. Energy disaggregation, or non-intrusive load monitoring (NILM), is the process of analyzing and decomposing the total aggregate energy consumption of a household into the individual consumptions by respective devices. These details were found informative and can influence occupants in a way that achieve noticeable energy savings. Hidden Markov Models (HMMs) were found efficient in modeling and detection of household devices. In this work, we propose a quantized continuous-state HMM so as to model continuously varying loads which is a challenging problem in the domain of energy disaggregation. Two core enhancements to the standard quantized continuous-state HMM are proposed. First, we propose a method that estimate the transition matrix considering potential probabilities to states neighboring that the model switches to. This method reduces the effect of domination of a state transition and achieve better simulation of switching cases in real variable loads. Second, the consumption of the variable load is estimated from the collective mean resulting from the Viterbi algorithm instead of the assigning the center value of the state with the maximum likelihood. In this way, the effect of quantization can be reduced. The proposed approach was tested on simulated and real variable loads from the REDD public data set. It was found that the proposed models outperform the reference HMM that applies standard estimation algorithms. Keywords-Hidden Markov Model (HMM), nonintrusive load monitoring (NILM), energy disaggregation, Viterbi algorithm.
I.INTRODUCTION Methods and techniques that can achieve energy savings have an increasing interest in the deployment of smart grids due to several impacts of electricity lifecycle on environment and other sectors. Electricity consumption in residential sector can be reduced in two principle ways: by using less appliances or using appliances less often, or by using appliances that consume less electricity. The overall reduction in the consumption of electrical energy will likely reduce peak load, reduce losses and improve the power
Since they represent the majority of appliances in households, research in the field of NILM was mainly focused on disaggregating ON/OFF appliances and
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transition matrix 𝐴 is an 𝑁×𝑁 matrix with elements 𝑎𝑖,𝑗 = 𝑃(𝑠𝑡+1 = 𝑗|𝑠𝑡 = 𝑖), where each element 𝑎𝑖𝑗 defines the transition probability from state 𝑖 at time 𝑡 to state 𝑗 at time 𝑡 + 1. The observation probability 𝑏𝑖 (𝑋) is essentially dependent on the current hidden state of the model. Beside modeling and disaggregation of devices in NILM approaches, HMM were applied broadly in areas such as speech recognition and bioinformatics [8].
those with finite number of states. This paper presents preliminary work that targets modeling and disaggregation of continuously varying loads. An enhanced quantized continuous-state Hidden Markov Model (HMM) is proposed to model and disaggregate a single variable load available from a public data set. II.
BACKGROUND AND RELATED WORK
The initial work in NILM by Hart [4] suggested to extract possible features (signatures) from the aggregate signal and then to group them in distinct clusters which represent the corresponding appliances. However, there were noticeable challenges such as overlapping devices clusters, dealing with varying loads, noisy signals, etc. Thereafter, progression in NILM research was achievable by focusing on either extracting new valuable features (which depends on the sampling time of measurements), or improving the clustering and detection techniques [5]. High frequency measurements can provide fine and distinguishing features such as harmonics, current and voltage waveform shapes, electromagnetic interference and frequency spectrums. Nonetheless, using low frequency measurements (e.g. 1 sample/second) is more practical and provide more applicable solutions since households’ smart meters are usually designed to monitor total consumption at low frequencies [3][5].
The Factorial HMM (FHMM) is an extension to the principle HMM where the observations are believed to be evolving due to several hidden and independent states chains [9]. Figure 1 shows a schematic of the FHMM structure where 𝑋0 , 𝑋1 , … 𝑋𝑇 are the observation sequence from time 𝑡 = 0 to 𝑡 = 𝑇 and the several hidden states chains are denoted by symbols 𝐴, 𝐵, 𝑒𝑡𝑐. The FHMM are functional in modeling of devices in a NILM technique since the observation sequence can model the total aggregate consumption measured by smart meters. The several hidden state chains represent the individual devices present in the household under study.
Looking at how these features were used for appliances detection; approaches can be classified to supervised and unsupervised approaches. There is a learning phase associated with the supervised learning approaches where devices features are first analyzed and recorded in a database. Thereafter, the disaggregation process is either performed by means of optimization or pattern recognition methods [3] [5]. It is noticeable that features of individual devices may not always be available or easy to obtain in real life applications. That is why supervised approaches have less interest in literature [6]. On the other hand, learning in the unsupervised approaches is performed online together with the disaggregation process. This is usually performed using proper statistical analysis and classification methods applied merely on the total aggregate signal [6] [7].
Fig. 1. Structure of FHMM Kolter and Jaakkola [7] proposed a new approximate inference method applied on a FHMM to tackle the problem of NILM. Two complementary models were used, the additive and the difference FHMM. The additive FHMM captures well the total aggregate output signal while the difference FHMM encodes the signal differences between subsequent levels (when a device switches ON or OFF). Kim et. al. [6] tested another three extensions of HMMs (beside FHMM) so as to accomplish better representation of devices and the aggregate signal. Parson [10] proposed a learning approach using HMM, where a one-time supervised learning process with already labeled data set was used to create general probabilistic models of appliances. Thereafter, these general models can be tuned to previously unseen households in an unsupervised manner. The tuning process resulted in specific models of devices that were present in the testing households. In an earlier
Hidden Markov Model (HMM) and its extensions were utilized extensively to model and detect appliances under the scope of NILM. The basic version of a HMM consists of an observable sequence (emissions) that is believed to be evolving depending on a hidden sequence of states. The first order HMM is the most common model where a transition to the next state depends only on the previous state. That is, given a HMM with 𝑁 possible hidden states, the
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this paper. The transition matrix is an 𝑁×𝑁 matrix, where 𝑁 represents the number of quantization levels. Each of these levels has a width ∆ that can be obtained from
work [11], we introduced an energy disaggregation model that considers mutual devices interactions and embeds the information on devices interactions into the FHMM representations of the aggregated data. Johnson and Willsky [12] introduced an explicitduration Hierarchical Dirichlet Process Hidden semiMarkov Model (HDP-HSMM) and developed a sampling algorithms for efficient posterior inference.
Where Pmax and Pmin represents the maximum and minimum power levels (i.e. the range) in which the variable load is consuming, respectively. Each quantization level has a corresponding center value denoted as 𝑐𝑖 .
For continuously varying loads, the amount of research tackling this problem is still limited. The main method was to extend the basic NILM algorithm by incorporation of harmonics. To save computational resources and improve performance, only transient signals were used for harmonic analysis [5]. An extension to harmonics analysis is the concept of a spectral envelope which is based on the analysis of the short-time fast Fourier transform (FFT) of the signal. The spectral envelopes can be characteristic features of appliances, including continuously varying appliances [5]. In a more recent work, Wichakool et. al. [13] proposed method that provides a systematic process to derive an estimator for any variable power load with structural features in the current waveforms without a full analysis of internal circuits. Nevertheless, to capture such fine features, these methods generally require measurements at high frequencies which often require additional metering equipment.
A. Estimation of transition matrix The transition matrix in the standard empirical HMM may be estimated using the time occupancy approach [7]. That is, transition probabilities are estimated from the total time occupied in each state and number of transitions between different states. We propose a three-steps approach to estimate the transition matrix. First, define all transition probabilities with uniform distribution across each row in the transition matrix. That is 𝑎𝑖,𝑗 =
1 , 𝑓𝑜𝑟 𝑒𝑎𝑐ℎ 𝑖 = 1,2, … 𝑁 𝑎𝑛𝑑 𝑗 = 1,2, … 𝑁 𝑁
Thereafter in the learning phase, whenever a specific transition 𝑎𝑖𝑗 is detected, increase the probability of 𝑎𝑖,𝑗 , 𝑎𝑖,𝑗+1 and 𝑎𝑖,𝑗−1 in a way that gives major increase of probability to 𝑎𝑖,𝑗 . We assign small probabilities to states neighboring 𝑎𝑖,𝑗 essentially to reduce the effect of domination of a state transition as illustrated in the next subsection. To achieve this goal, we may use a triangular probability distribution as shown in Figure 2.
A continuous-state (or infinite) HMM is theoretically capable of modeling cases where the number of hidden states is considerably uncountable. Beal et. al. [14] utilized the theory of Dirichlet Processes (DP) to implicitly integrate out the infinitely many transition parameters, leaving only three hyperparameters which were learned from data. III.
Pmax Pmin N
PROPOSED QUANTIZED CONTINUOUS-STATE HIDDEN MARKOV MODEL
However, discretizing the states domain via quantization would reduce computational complexity and can still provide desired performance levels. In the proposed HMM model, we focus on tackling two basic shortcomings in the standard continuous-state HMM. First, a method is proposed to estimate the transition matrix that reduces the effect of domination of a specific transition which appears when using the standard time occupancy estimation technique. Second, to mitigate the impact of quantization, a collective mean based on Viterbi algorithm outcomes is used to estimate power consumptions instead of assigning the center value of the corresponding quantization level.
Fig. 2. Learning new transition probabilities ∗ ∗ The new transition probabilities 𝑎𝑖,𝑗 , 𝑎𝑖,𝑗−1 and ∗ 𝑎𝑖,𝑗+1 then obtained by increasing the corresponding transition probabilities as follows ∗ 𝑎𝑖,𝑗 = 𝑎𝑖,𝑗 + 𝐴𝑗 ∗ 𝑎𝑖,𝑗−1 = 𝑎𝑖,𝑗−1 + 𝐴𝑗−1
To articulate the proposed methods, we firstly define notations that would be used subsequently in
∗ 𝑎𝑖,𝑗+1 = 𝑎𝑖,𝑗+1 + 𝐴𝑗+1
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Where,
In the standard Viterbi algorithm, whenever a state 𝑖 is inferred at time 𝑡, the quantization level center value 𝑐𝑖 can be considered as an estimate of the power consumption at that time 𝑡 . This means the estimated signal will look like steps going up and down according to variations of power consumption. The standard Viterbi algorithm will result in a vector 𝑉, where each element 𝑣𝑖 shows the likelihood of each hidden state as follows
𝐴𝑗−1 + 𝐴𝑗 + 𝐴𝑗+1 = 1 Finally, normalize the updated matrix row 𝑖 in which new transition probabilities were learnt. That is, ~ each normalized 𝑎𝑖,𝑗 is obtained from ~ 𝑎𝑖,𝑗 =
∗ 𝑎𝑖,𝑗 ∗ ∑𝑁 𝑗=1 𝑎𝑖,𝑗
𝑉 = [𝑣1 = 𝑃(𝑠 = 1), 𝑣2 = 𝑃(𝑠 = 2), … ]
The normalization step is important to recover transition probabilities that sum to 1 in each row of the transition matrix.
The proposed collective mean method is carried out in two steps
The above mentioned three steps are iteratively repeated in the learning phase, whenever a state transition is detected.
1. Normalization of elements in 𝑉 to obtain 𝑉̃ = 𝑉 . Hence, we have ∑𝑁 ̃𝑖 = 1.0, where 𝑣̃𝑖 is 𝑖=1 𝑣 ∑𝑁 𝑖=1 𝑣𝑖
the element 𝑖 in 𝑉̃ and 𝑖 = 1, 2, … 𝑁.
B. Effect of domination of a specific transition Approach in subsection III.A is mainly proposed to tackle possible domination of a transition in the transition matrix. When using standard time occupancy method to estimate transition probabilities, a specific transition may dominate others in the same row of the transition matrix. Reducing the impact of this domination is useful in the following two situations
2. The collective mean uses the convex combination from quantization levels to estimate power consumption. That is, the estimated power is 𝑝̂ = ∑𝑁 ̃𝑖 ∗ 𝑐𝑖 . The 𝑝̂ is analogous to the expectation 𝑖=1 𝑣 of a probability distribution function whose sample space items are 𝑐𝑖 ’s with corresponding probabilities of 𝑣̃, 𝑖 where 𝑖 = 1, 2, … 𝑁. The collective mean method will result in flexible power estimates that are not restricted to the center of quantization levels; which could contribute to reduce the gap between actual and estimated data.
1. When there is a smooth transition of one state to its direct neighbor, the observation sequence of the HMM could be not that indicative of such transition. Thus, the model could fail in detection of such smooth transition. 2. When there is a very frequent transition between two states (e.g. 𝑖 → 𝑗), but in some future occasions it happened to have transitions like (𝑖 → 𝑗 + 1) or (𝑖 → 𝑗 − 1). In such cases, the model is prone to mistake in inferring the new state if 𝑎𝑖,𝑗 is dominating other transitions in the same row of the transitions matrix. That is, the inference of the hidden state is more likely to stuck in state 𝑗 in cases where 𝑎𝑖,𝑗 is noticeably large.
IV.
RESULTS
We tested the proposed approach on generated data that simulate a light dimmer and actual data of loads from the REDD public data set [15]. We applied a 9sample median filter on both training and testing phases to remove noisy signals and outliers. The simulated light dimmer was assumed to have possible power consumptions from 0.0 W to 150.0 W. The transitions were generated to switch to both neighboring and far states. The model was learnt from a portion of the generated data and then validated on the remaining portion. To compare the performance of the proposed model, the following notation of HMM models is used
C. Collective mean from Viterbi algorithm The objective of the Viterbi algorithm is to infer the most probable hidden sequence of states, given some sequence of observations [10]. However, in energy disaggregation, we benefit from Viterbi algorithm in two ways
• Model A: a quantized continuous-state HMM with standard Viterbi algorithm and time occupancy method to learn the transition matrix. • Model B: a quantized continuous-state HMM with standard Viterbi algorithm and the proposed method to learn the transition matrix. • Model C: a quantized continuous-state HMM with collective mean from Viterbi algorithm and the proposed method to learn the transition matrix.
1. Inferring the hidden states typically the same as in the standard Viterbi. 2. Estimating the power consumption of the variable load.
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Table 1 shows the performance results obtained when applying different methods on a generated data that simulates a light dimmer. The performance metric used is the accuracy in retrieving the actual power consumption signal.
impact of quantization. The new models were tested on both generated and real data for continuous loads and showed improved performance when compared to standard HMM. REFERENCES
Table 1. Comparison of models performance applied on a generated data Number of quantization Model A Model B Model C levels 5 95.8% 95.9% 96.7% 10 97.6% 97.7% 98.6%
[1]
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[3]
We applied the proposed learning and testing methods on variable loads from the REDD public data set. We investigated the individual consumptions of available devices and found that mainly electronics may behave as continuously varying loads. Table 2 shows the performance results obtained when applying different methods on real loads from the REDD.
[4]
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Table 2. Comparison of models performance applied on real loads from REDD Load Model A Model B Model C 95.4% 95.4% 96.2% Load A 72.4% 72.8% 73.1% Load B
[6]
[7]
Load A represents electronic load from house 3 and load B represents electronics load from house 5 from the REDD public data set. V.
[8]
DISCUSSION
[9]
The proposed methods helped in improving the accuracy of retrieving the power consumed by continuously varying home devices. In general, increasing the number of quantization levels can result in better performance but will also increase the computation complexity due to the increase of the transition matrix size. The modified method to estimate the transition matrix (model B) showed better performance than the standard time occupancy method because it helped the HMM not to stuck in some states with relatively large transition probabilities. The collective mean method outperformed the standard Viterbi method in estimation of power consumption as it mitigates the impact of quantization especially when quantization levels are relatively far from each other. VI.
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CONCLUSION
Two techniques were proposed to enhance the performance of the reference quantized continuousstate HMM. To tackle possible dominations of state transitions probabilities, a modified method to estimate the transition matrix was presented. A collective mean method based on convex combination over Viterbi outcomes was proposed to mitigate the
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