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A new pattern recognition methodology for classification of load profiles for ships electric consumers

A new pattern recognition methodology for classification of load profiles for ships electric consumers 1

GJ Tsekouras1 , IK Hatzilau1 , JM Prousalidis1,2 Hellenic Naval Academy, Department of Electrical Engineering and Computer Science 2 National Technical University of Athens, School of Naval Architecture and Marine Engineering, Division of Marine Engineering

In this paper a new pattern recognition methodology is presented for the classification of the daily chronological load curves of ship electric consumers (equipment) and the determination of the respective typical load curves of each one of them. It is based on pattern recognition methods, such as k-means, adaptive vector quantisation, fuzzy kmeans, self-organising maps and hierarchical clustering, which are theoretically described and properly adapted. The parameters of each clustering method are properly selected by an optimisation process, which is separately applied for each one of six adequacy measures: the error function, the mean index adequacy, the clustering dispersion indicator, the similarity matrix indicator, the Davies-Bouldin indicator and the ratio of within cluster sum of squares to between cluster variation. As a study case, this methodology is applied to a set of consumers of Hellenic Navy MEKO type frigates.

AUTHORS’ BIOGRAPHIES Dr GJ Tsekouras (Electrical & Computer Engineer from NTUA/1999, Civil Engineer from NTUA/2004, PhD on Electrical Engineering from NTUA/2006). Adjunct lecturer at the Hellenic Naval Academy, dealing with power system analysis, forecasting and pattern recognition methodologies. Prof Dr-Ing IK Hatzilau (Electrical & Mechanical Engineer from NTUA/1965, Dr-Ing from University of Stuttgart/ 1969). After few years in the industry, he joined the Academic Staff of Dept of Electrical Engineering and Computer Science in Hellenic Naval Academy dealing with marine electrical and control engineering issues. He is representative of Hellenic Navy in NATO AC/141(MCG/6)SG/4 dealing with warship-electric systems. Dr J Prousalidis (Electrical Engineer from NTUA/1991, PhD from NTUA/1997). Assistant Professor at the School of

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Journal of Marine Engineering and Technology

Naval and Marine Engineering of National Technical University of Athens, dealing with electric energy systems and electric propulsion schemes on shipboard installations.

INTRODUCTION he ‘load demand profile’ ie, the chronological energy demand curve is a ‘characteristic parameter’ of an electric consumer (equipment) of a ship electric network, the value of which vary depending on several factors such as ship type, state, operating modes and mission. In Fig 1 an indicative segment of a daily chronological curve of the refrigeration plant in ship-condition at ‘SHORE’ of HN MEKO type frigate is shown, based on records taken, every 1 min.1 The load demand profile is of primary importance in performing several studies such as load estimation, power sources selection, power cable rating, short circuit analysis,

T

45


a methodology in terms of the exploitation of the results yielded to a series of applications and studies of ship systems. In brief, these results, ie, the typical chronological load curves of each consumer, can be used as input information in:

• • • Fig 1: Indicative chronological energy demand curve of an electric consumer of HN MEKO type frigate load shedding, harmonics, modulation etc. The representative load demand profiles of each consumer can be obtained from a classification procedure of its chronological load curves in typical time intervals (also called ‘typical days’). This classification can be conducted by pattern recognition techniques2-6 , such as: -

the the the the the the

the mean square error3,5 the mean index adequacy2-3 the clustering dispersion indicator2-4 the similarity matrix indicator3 the Davies-Bouldin indicator 3-5 the modified Dunn index4 the scatter index4 the ratio of within cluster sum of squares to between cluster variation.3

Alternatively, the load curves’ classification of consumers can be performed by data mining7 , wavelet packet transformation8 and Fourier analysis. The last ones are also used with simplified clustering models. Conventional tools, like statistical techniques,9 need the knowledge of the ‘typical days’, which can be defined by experienced ship’s personnel, eg, chief engineers. Evidently, one of the major issues of this classification is the definition of the ‘typical day’. In the continental power systems the load curve’s time interval is usually a day for a study time period from a few weeks2 to one year.3 In ships’ power systems the corresponding time interval is not defined by any standards, while the total study period can not but be fairly limited (varying from a few days to one month). In this paper the typical time interval is assumed to be a day. In a previous paper,11 the authors have thoroughly discussed and highlighted the significance of developing such

46

•

•

‘modified follow the leader’2,4 k-means3-4 adaptive vector quantisation3 fuzzy k-means3-5 self-organising map4 hierarchical methods3-4 .

Regarding adequacy measures commonly used, these can be: -

•

the formalisation of the consumer’s behaviour and the corresponding clustering using the representative load curve of each consumer; the design of the ship’s electric power system estimating the respective total load demand more accurately and, hence selecting the generators properly; the operation of the ship’s power system succeeding more precise short-term load forecasting, increasing the respective reliability and decreasing the respective operation cost; the improvement of power factor taking into consideration the respective reactive load curves; the load estimation after the application of demand side management programs for each specific consumer, as well as the feasibility studies of the energy efficiency which normalise the total load demand and improve the total load factor; the improvement of the operation of the automatic battle management and load shedding systems, because the automatic supply of the critical consumers in each operating mode is facilitated in case of power system’s fault based on the available generators, the healthy part of the power distribution system and the load demand of each consumer.

The objective of this paper is to present a new methodology for the classification of the daily chronological load curves of ship electric consumers. This method is based on the socalled unsupervised neural networks. More specifically, for each consumer the corresponding set of load curves is organised into well-defined and separate classes, in order to successfully describe the respective demand behaviour without any intervention by the user in the classification procedure. The proposed methodology compares the results obtained from certain clustering techniques (k-means, adaptive vector quantisation (AVQ), fuzzy k-means, selforganising maps (SOM) and hierarchical methods) using six adequacy measures (mean square error, mean index adequacy, clustering dispersion indicator, similarity matrix indicator, Davies-Bouldin indicator, ratio of within cluster sum of squares to between cluster variation). The main points of this methodology are:

• •

• • •

the estimation of the representative typical daily load profiles for each consumer; the modification of the clustering techniques for this kind of classification problem, such as the appropriate weights initialisation for the k-means and fuzzy kmeans; the proper parameters calibration, such as the amount of fuzziness for the fuzzy k-means, in order to fit the classification needs; the comparison of the clustering algorithms performance for each one of the adequacy measures; the selection of the proper adequacy measure.


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Finally, the results of the application of the developed methodology are thoroughly presented for the case study of six electric consumers of Hellenic Navy MEKO type frigates, the chronological load curves of which have been measured within the framework of a research project.1

2.

PROPOSED PATTERN RECOGNITION METHODOLOGY FOR LOAD CURVES CLASSIFICATION OF SHIPS ELECTRIC CONSUMERS The classification of daily chronological load curves of each ship electric consumer is achieved by applying the pattern recognition methodology shown in Fig 2. The main steps are: 1.

Data and features selection: Using electronic meters or ship energy management system for main consumers, the active and reactive power values are recorded (in

3.

kWh and kVArh) for each period in steps of 1min, 15min, etc. The chronological load curves for each individual consumer are determined for each study period (week, month). Consumers’ clustering using a priori indices: Consumers can be characterised by their voltage level (high, medium, low), installed power, power factor, load factor, criticality according to ship’s operating mode etc. These indices are not necessarily related to the load curves. They can be used however for the pre-classification of consumers. It is noted that the load curves of each consumer are normalised using the respective minimum and maximum loads of the period under study. Data preprocessing: The load curves of each consumer are examined for normality, in order to modify or delete the values that are obviously wrong (noise suppression). If necessary, a preliminary execution of a pattern recognition algorithm is carried out, in order to trace erroneous measurements or networks faults,

Fig 2: Flow diagram of pattern recognition methodology for the classification of daily chronological load curves of ships electric consumer

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47


4.

which, if uncorrected, will reduce the number of the useful typical time intervals for a constant number of clusters. Typical load curves clustering for each consumer: For each consumer, a number of clustering algorithms (kmeans, adaptive vector quantisation, fuzzy k-means, self-organising maps and hierarchical methods) is applied. Each algorithm is trained for the set of load diagrams and evaluated according to six adequacy measures. The parameters of the algorithms are optimised, if necessary. The developed methodology uses the clustering methods providing the most satisfactory results. This process is repeated for the total set of consumers under study. Special consumers, such as seasonal or emergency ones (eg, machine tools, firepumps, etc) are identified. These results can be combined with the ships’ operating mode. At this stage, the size of time interval (1, 2, 3, 4, 6 hours, half or one day) could be investigated, too.

The typical load curves of consumers used are selected by choosing the type of typical time interval (such as the most populated one, the time interval with the peak demand load, etc).

MATHEMATICAL MODELLING OF CLUSTERING METHODS AND CLUSTERING VALIDITY ASSESSMENT

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u d u1 X 2 x‘2 Þ ¼ t d ð~ x‘1 , ~ ð x‘ i x‘2 i Þ d i¼1 1 2.

1.

48

the Euclidean distance between ‘1 , ‘2 input vectors of the set X:

j

the infra-set mean distance of a set, defined as the geometric mean of the inter-distances between the members of the set, ie, for the subset j : sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 X 2 ^ (3) d ~ x‘ , j d ð j Þ ¼ 2N j ~x 2 j

‘

The basic characteristics of the five clustering methods being used are the following.

k-means model The k-means method is the simplest hard clustering method, which gives satisfactory results for compact clusters.11 The k-means clustering method groups the set of the N input vectors to M clusters using an iterative procedure. The respective steps of the algorithm are the following: 1.

General In the study case of the chronological typical load curves of ship electric consumers, a number of N analytical daily load curves is given. The main target is to determine the respective sets of days and load patterns. Generally, N is defined as the population of the input (or training) vectors, which are going to be clustered. ~ x‘ ¼ (x‘1 , x‘2 , . . . x‘i , . . . x‘d ) T symbolises the ‘-th input vector and d its dimension, which equals to 1440 (the load measurements are taken every minute). The corresponding set is given by X ¼ f~ x‘ : ‘ ¼ 1, . . . , N g. It is worth mentioning that x‘i are normalised using the higher and lower values of all elements of the original input patterns set, in order to obtain better results from the application of clustering methods. Each classification process makes a partition of the initial N input vectors to M clusters. The j-th cluster has a representative, which is the respective load profile and is represented by the vector ~ w j ¼ (w j1 , w j2 , . . . , w ji , . . . , w jd ) T of d dimension. Vector ~ w j expresses the cluster centre. The corresponding set is the classes set, which is defined by W ¼ f~ w k , k ¼ 1, . . . Mg. The subset of input vectors ~ x‘ , which belongs to the j-th cluster, is j and the respective population of load diagrams is N j . For the study and evaluation of classification algorithms the following distance forms are defined:

the distance between the representative vector ~ w j of j-th cluster and the subset j , calculated as the geometric mean of the Euclidean distances between ~ w j and each member of j : sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 X 2 (2) d ~ x‘ w j, ~ d ~ w j, j ¼ N j ~x 2 ‘

3.

(1)

Initialisation of the weights of the M clusters is determined. In the classic model a random choice among the input vectors is used4 . In the developed algorithm the w ji of the j-th centre is initialised as: w(0) ji ¼ a þ b ( j 1)=(M 1)

2.

(4)

where a and b are properly calibrated parameters. During training iteration t (called ‘epoch’ t, hereinafter) for each training vector ~ x‘ its Euclidean distances d(~ x‘ , ~ w j ) are calculated for all centres. The ‘-th input vector is put in the set (jt) , for which the distance between ~ x‘ and the respective centre is minimum, which means:

(5) w k Þ ¼ min d ~ x‘ , ~ wj d ð~ x‘ , ~ 8j

3.

When the entire training set is formed, the new weights of each centre are calculated as: 1 X ~ ~ w(jtþ1) ¼ ( t) (6) x‘ N j ~x 2( t) ‘

j

N (jt)

4.

where is the population of the respective set (jt) during epoch t. Next, the number of epochs is increased by one. This process is repeated (return to step b) until the maximum number of epochs is used or weights do not significantly change, ie, (j~ w(jt) ~ w(jtþ1) j , , where is the upper limit of weight change between sequential iterations). The algorithm’s main purpose is to minimise the error function J:


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N 1X d 2 ð~ w k:~x‘ 2 k Þ x‘ , ~ N ‘¼1

(7)

between sequential iterations and the respective criterion is activated after Tin epochs.

The main difference with the classic model is that the process is repeated for various pairs of (a,b). The best results for each adequacy measure are recorded for various pairs (a,b).

The algorithm core is executed for a specific number of neurons and the respective parameters 0 , min and T0 are optimised for each adequacy measure separately. This process is repeated from M1 to M2 neurons.

J¼

Kohonen adaptive vector quantisation (AVQ)

Fuzzy k-means

This algorithm is a variation of the k-means method, which belongs to the unsupervised competitive one-layer neural networks. It classifies input vectors into clusters by using a competitive layer with a constant number of neurons. Practically in each step all clusters compete with each other for the winning of a pattern. The winning cluster moves its centre to the direction of the pattern, while the rest of the clusters move their centres to the opposite direction (supervised classification) or remain stable (unsupervised classification). Here, we will use the last unsupervised classification algorithm. The respective steps are the following:

During the application of the k-mean or the adaptive vector quantization algorithm, each pattern is assumed to be in exactly one cluster (hard clustering). In many cases the areas of two neighbour clusters are overlapped, so that there are not any valid qualitative results. If the condition of exclusive partition of an input pattern to one cluster is to be relaxed, the fuzzy clustering techniques should be used. More specifically, each input vector ~ x‘ does not belong to only one cluster, but it participates in every j-th cluster by a membership factor u‘ j, where: M X

u‘ j ¼ 1 & 0 < u‘ j < 1, 8 j

(12)

j¼1

1.

2.

Initialisation of the weights of the M clusters is determined, where the weights of all clusters are equal to 0.5, that is w(0) ji ¼ 0:5, 8 j, i. During epoch t each input vector ~ x‘ is randomly presented and its respective Euclidean distances from every neuron are calculated. In the case of existence of bias factor º, the respective minimization function is:

f winner_ neuron ð~ x‘ , ~ w j þ º N j =N (8) x‘ Þ ¼ j: min d ~ 8j

where N j is the population of the respective set j during epoch t-1. The weights of the winning neuron (with the smallest distance) are updated as:

ð tÞ ð tÞ ð tÞ ~ (9) w j ð n þ 1Þ ¼ ~ w j ð n Þ þ ð t Þ ~ x‘ ~ w j ð nÞ

Theoretically, the membership factor gives more flexibility in the vector’s distribution. During the iterations the following objective function is minimised:

3.

where 9 is the upper limit of error function change

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(13)

The simplest algorithm is the fuzzy k-means clustering one, in which the respective steps are the following: 1.

2.

where n is the number of input vectors, which have been presented during the current epoch, and (t) is the learning rate according to: t ð tÞ ¼ 0 exp (10) . min T 0 where 0 , min and T0 are the initial value, the minimum value and the time parameter respectively. The remaining neurons are unchangeable for ~ x‘, as introduced by the Kohonen winner-take-all learning rule.13,14 Next, the number of epochs is increased by one. This process is repeated (return to step b) until either the maximum number of epochs is reached or the weights converge or the error function J does not improve, which means: J ð tÞ J ð tþ1Þ (11) , 9 for t > T in J ð tÞ

M X N

1X u‘ j d 2 ~ wj x‘ , ~ N j¼1 ‘¼1

J fuzzy ¼

Initialisation of the weights of the M clusters is determined. In the classic model a random choice among the input vectors is used.5 In the developed algorithm the w ji of the j-th centre is initialised by equation (4). During epoch t for each training vector ~ x‘ the membership factors are calculated for every cluster: ð

Þ

¼ u‘ tþ1 j

3.

1

ð tÞ M d ~ x‘ , ~ wj X

ð tÞ k¼1 d ~ x‘ , ~ wk

(14)

Afterwards the new weights of each centre are calculated as: q N

X ð Þ u‘ tþ1 ~ x‘ j ð

Þ

~ w j tþ1 ¼

‘¼1 N

X

ð Þ u‘ tþ1 j

q

(15)

‘¼1

4.

where q is the amount of fuzziness in the range (1, 1) which increases as fuzziness reduces. Next, the number of epochs is increased by one. This process is repeated (return to step b) until the maximum number of epochs is used or weights do not significantly change.

49


This process is repeated for different pairs of (a,b) and for different amounts of fuzziness. The best results for each adequacy measure are recorded for different pairs (a,b) and q.

during the fine tuning phase the respective values are f , T 0 . The h i9 j (t) is the neighbourhood symmetrical function, that will activate the j neurons that are topologically close to the winning neuron i9, according to their geometrical distance, who will learn from the same ~ x‘ (collaboration stage). In this case the Gauss function is proposed: " # d 2i9 j (17) h i9 j ð tÞ ¼ exp 2 2 ð tÞ

Self-organising maps (SOM) The Kohonen SOM 13-16 is a topologically unsupervised neural network that projects a d-dimensional input data set into a reduced dimension space (usually a mono-dimensional or bi-dimensional map). It is composed of a predefined grid containing M 3 1 or M 1 3 M 2 d-dimensional neurons ~ w k for mono-dimensional or bi-dimensional map respectively, which are calculated by a competitive learning algorithm updating not only the weights of the winning neuron, but also the weights of its neighbour units in inverse proportion of their distance. The neighbourhood size of each neuron shrinks progressively during the training process, starting with nearly the whole map and ending with the single neuron. The training of SOM is divided into two phases:

•

•

rough ordering, with high initial learning rate, large radius and small number of epochs, so that neurons are arranged into a structure which approximately displays the inherent characteristics of the input data, fine tuning, with small initial learning rate, small radius and higher number of training epochs, in order to tune the final structure of the SOM.

The transition of the rough ordering phase to fine tuning one takes place after T s0 epochs. It is mentioned that, in the case of the bi-dimensional map, the immediate exploitation of the respective clusters is not a simple problem. The map can be exploited either by inspection or applying a second simple clustering method, such as the simple k-means method.16 Here, only the case of the mono-dimensional map is examined. More specifically, the respective steps of the monodimensional SOM algorithm are the following: 1.

The number of neurons of the SOM’s grid are defined and the initialisation of the respective weights is determined. Thus, the weights can be given by (a) w ki ¼ 0:5, 8k, i, (b) the random initialisation of each neuron’s weight, (c) the random choice of the input vectors for each neuron. 2. The SOM training commences by first choosing an input vector ~ x‘, at t epoch, randomly from the input vectors’ set. The Euclidean distances between the n-th presented input pattern ~ x‘ and all ~ w k are calculated, so as to determine the winning neuron i9 that is closest to ~ x‘ (competition stage). The j-th reference vector is updated (weights’ adaptation stage) according to:

ð tÞ ð tÞ ð tÞ ~ w j ð n þ 1Þ ¼ ~ w j ð nÞ þ ð tÞ h i9 j ð tÞ ~ x‘ ~ w j ð nÞ (16) where (t) is the learning rate according to equation (10). During the rough ordering phase, r , T 0 are the initial value and the time parameter respectively, while

50

3.

where d i9 j ¼ k~ ri9 ~ r j k is the respective distance between i9 and j neurons, ~ r j ¼ (x j , y j ) are the respective co-ordinates in the grid, (t) ¼ 0 exp (t=T 0 ) is the decreasing neighbourhood radius function where 0 and T 0 are the respective initial value and time parameter of the radius respectively. Next, the number of the epochs is increased by one. This process is repeated (return to step b) until either the maximum number of epochs is reached or the index Is gets the minimum value:10 I s (t) ¼ J (t) þ ADM ð tÞ þ TE(t)

(18)

where the quality measures of the optimum SOM are based on the quantization error J – given by equation (7), the topographic error TE and the average distortion measure ADM. The topographic error measures the distortion of the map as the percentage of input vectors for which the first i91 and second i92 winning neuron are not neighbouring map units: TE ¼

N X

neighbð i91 , i92 Þ=N

(19)

‘¼1

where, for each input vector, neighb(i91 , i92 ) equals either 1, if i91 and i92 neurons are not neighbours, or 0. The average distortion measure is given for the t epoch by: ADM ð tÞ ¼

N X M X

h i9!~x‘ , j ð tÞ d 2 ~ w j =N x‘ , ~

(20)

‘¼1 j¼1

This process is repeated for different parameters of 0 , f , r , T0 , T 0 and T s0 . Alternatively, the multiplication factors and are introduced – without decreasing the generalisation ability of the parameters’ calibration: T s0 ¼ T0

(21)

T 0 ¼ T0 =ln 0

(22)

The best results for each adequacy measure are recorded for all parameters 0 , f , r , T0 , and .

Hierarchical agglomerative algorithms Agglomerative algorithms are based on matrix theory6. The input is the N 3 N dissimilarity matrix P0 . At each level t, when two clusters are merged into one, the size of the dissimilarity matrix Pt becomes (N t) 3 (N t). Matrix Pt is obtained from Pt-1 by deleting the two rows and columns that correspond to the merged clusters and adding


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a new row and a new column containing the distances between the newly formed cluster and the old ones. The distance between the newly formed cluster C q (the result of merging C i and C j ) and an old cluster C s is determined as: d ðC q , C s Þ ¼ ai d ðC i , C s Þ þ a j d ðC j , C s Þ þ b d ð C i , C j Þ þ c d ð C i , C s Þ d ð C j , C s Þ

d ð1Þ ð C q , C s Þ ¼

(29)

•

ð Þ ð Þ ð ni þ ns Þ d 2 ðC i , C s Þ þ ð n j þ ns Þ d 2 ðC j , C s Þ n s d ð2Þ ð C i , C j Þ ð ni þ n j þ ns Þ

where a i , a j , b and c correspond to different choices of the dissimilarity measure. It is noted that in each level t the respective representative vectors are calculated by (6). The basic algorithms, which are going to be used in our case, are:

•

1 1 d ðC i , C s Þ þ d ðC j , C s Þ 2 2 1 d ð C i , C s Þ d ð C j , C s Þ 2

•

2.

(26)

where n i and n j - are the respective members’ populations of clusters C i and C j . the weighted pair group method average algorithm (WPGMA): 1 d ðC q , C s Þ ¼ d ðC i , C s Þ þ d ðC j , C s Þ (27) 2 the unweighted pair group method centroid algorithm (UPGMC):

ð1Þ ð1Þ ð Þ n d C , C d C , C þ n ð j sÞ i i s j d ð1Þ ð C q , C s Þ ¼ n þ n ð i jÞ d ð1Þ ð C i , C j Þ ð n i þ n j Þ2 (28) 2

wq ~ w s k and ~ w q is the reprewhere d (C q , C s ) ¼ k~ sentative centre of the q-th cluster according to the equation (6). the weighted pair group method centroid algorithm (WPGMC):

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(31)

The respective steps of the respective algorithms are the following:

1 1 d ðC i , C s Þ þ d ðC j , C s Þ (25) 2 2 1 þ d ð C i , C s Þ d ð C j , C s Þ 2 the unweighted pair group method average algorithm (UPGMA):

(1)

ni n j d ð1Þ ð C i , C j Þ ni þ n j

It is noted that in each level t the respective representative vectors are calculated by equation (6).

1.

ni n j

•

d ð2Þ ð C i , C j Þ ¼

the complete link algorithm (CL) – it is obtained from (23) for a i ¼ a j ¼ 0:5, b ¼ 0 and c ¼ 0:5: d ð C q , C s Þ ¼ max d ð C i , C s Þ, d ð C j , C s Þ

ni d ðC i , C s Þ þ n j d ðC j , C s Þ d ðC q , C s Þ ¼ ni þ n j

•

where:

(24)

¼

•

(30)

the single link algorithm (SL) – it is obtained from (23) for a i ¼ a j ¼ 0:5, b ¼ 0 and c ¼ 0:5: d ð C q , C s Þ ¼ min d ð C i , C s Þ, d ð C j , C s Þ ¼

the Ward or minimum variance algorithm (WARD): d ð2Þ ð C q , C s Þ ¼

(23)

•

1 d ð1Þ ð C i , C s Þ þ d ð1Þ ð C j , C s Þ 2 1 d ð 1Þ ð C i , C j Þ 4


3.

4. 5.

Initialisation: The set of the remaining patterns R0 for zero level (t ¼ 0 ) is the set of the input vectors X. The similarity matrix P0 ¼ P(X ) is determined. Afterwards t increases by one (t ¼ t + 1). During level t clusters C i and C j are found, for which the minimisation criterion d(C i , C j ) ¼ min r,s¼1,..., N , r6¼ s d(C r , C s ) is satisfied. Then clusters C i and C j are merged into a single cluster C q and the set of the remaining patterns R t is formed as: R t ¼ (R t1 fC i , C j g) [ fC q g. The construction of the dissimilarity matrix Pt from Pt1 is realised by applying equation (23). Next, the number of the levels is increased by one. This process is repeated (return to step b) until the remaining patterns R N 1 is formed and all input vectors are in the same and unique cluster.

It is mentioned that the number of iterations is determined from the beginning and it equals to the number of input vectors decreased by 1 (N-1).

Adequacy measures In order to evaluate the performance of the clustering algorithms and to compare them with each other, six different adequacy measures are applied. Their purpose is to obtain well-separated and compact clusters, in order to make the load curves self explanatory. The definitions of these measures are the following: 1. 2.

Mean square error or error function (J) given by equation (7). Mean index adequacy (MIA)2-3 , which is defined as the average of the distances between each input vector assigned to the cluster and its centre:

51

A new pattern recognition methodology for classification of load profiles for ships electric consumers vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u M u1 X

MIA ¼ t d2 ~ w j, j M j¼1 3.

(32)

Clustering dispersion indicator (CDI),2-4 which depends on the mean infra-set distance between the input vectors in the same cluster and inversely on the infraset distance between the class representative load curves: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u M u1 X CDI ¼ t (33) d^ 2 ð k Þ= d^ ð W Þ M k¼1

tem, the maximum peak load of which ranges between 3.5kW and 60kW. The respective data are the 1 minute ON/ OFF normalised load values for each individual consumer for a period of eleven days during November 1997 and January 1998.1 The respective consumers are:

• • • • • •

chiller, HP compressor FWD/AFT, refrigeration plant, LP compressor FWD/AFT, sanitary plant, boiler.

4.

Similarity matrix indicator (SMI),3 which is defined as the maximum off-diagonal element of the symmetrical similarity matrix, whose terms are calculated by using a logarithmic function of the Euclidean distance between any kind of class representative load curves: n

1 o SMI ¼ max 1 1=ln d ~ w p, ~ wq : piq (34) p, q ¼ 1, . . . , M

Initially, the analysis of the ‘chiller’ is presented in detail, while additional consumers can be analysed in a similar way. It is supposed that the time interval is a day. The representative consumer’s typical day has been chosen to be the most populated one. The respective set of the daily chronological curves has 11 members. No curves are rejected through data pre-processing.

5.

Davies-Bouldin indicator (DBI)3,5 , which represents the system-wide average of the similarity measures of each cluster with its most similar cluster: ( ) M d^ ð p Þ þ d^ ð q Þ 1 X

DBI ¼ max : M k¼1 p6¼ q d ~ w p, ~ wq (35)

Application of k-means

p, q ¼ 1, . . . , M 6.

Ratio of within cluster sum of squares to between cluster variation (WCBCR)3 , which depends on the sum of the distance square between each input vector and its cluster representative vector, as well as the similarity of the clusters centres: X M X M X

2 WCBCR ¼ d ð~ x‘ Þ d2 ~ wq wk, ~ w p, ~ k¼1 ~ x‘ 2 k

11000 epochs). the proper calibration of (a) the initial value of the neighbourhood radius 0 , (b) the multiplicative factor between T s0 (epochs of the rough ordering phase) and T0 (time parameter of learning rate), (c) the

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•

multiplicative factor between T 0 (time parameter of neighbourhood radius) and T0 , (d) the proper initial values of the learning rate r and f during the rough ordering phase and the fine tuning phase respectively. The optimisation process for the monodimensional SOM parameters is similar to that one of the application of the adaptive vector quantisation for 0 and T0 . The best results for each adequacy measures are presented in Table 1 for the case of 4 clusters. the initialisation of the weights of the neurons, where the three cases of the theoretical analysis of selforganising maps are examined and the best training behaviour is presented in case (a) (for w ki ¼ 0:5, 8k, i ).

Application of hierarchical agglomerative algorithms In the case of the seven hierarchical models the best results are given for different models for two and three clusters, while for four clusters or more, all models give the same results, as it is shown in Fig 5. It should be mentioned that there are not any other parameters for calibration, such as maximum number of iterations etc.

Comparison of clustering models for the chiller The best results for each clustering method (modified kmeans, adaptive vector quantisation, fuzzy k-means, selforganised maps and hierarchical methods) are presented in Fig 6. The optimised AVQ and WARD methods present the best behaviour for the mean square error J, the optimised AVQ and the unweighted pair group method average algorithm (UPGMA) for the MIA, the optimised AVQ and the modified k-means for the CDI, DBI and WCBCR, the modified k-means for the SMI. All measures (except DBI) show an improved performance, as the number of clusters increases. The number of dead clusters for the WCBCR indicator for all clustering techniques and for all adequacy measures for AVQ method are shown in Fig 6g and 6h respectively. Here, it is not obvious which measure is the best one because of the extremely small set of training patterns. However, taking into consideration the results3,10 and noting that the basic theoretical advantage of the WCBCR measure is that it combines the distances of the input vectors from the representative clusters and the distances between clusters (covering also the J and CDI characteristics), the application of WCBCR is proposed. Observing Fig 6f, the improvement of the WCBCR is significant up to 4 clusters. After this value, the behaviour

Fig 5: Adequacy measures of the 7 hierarchical clustering algorithms for 2 to 10 clusters for the set of 11 training patterns of the chiller

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Fig 6: Adequacy measures and dead clusters of each clustering method for 2 to 10 clusters for the set of 11 training patterns of the chiller

Type

Number of clusters

a b c d e f

4 3 4 4 3 4

Population of Best the clustering representative technique cluster 7 9 7 6 8 8

AVQ CL - UPGMC k-means UPGMC All hierarchical k-means

Date November 1997 20 2nd 2nd 4th 4th 1st 1st

21 3rd 1st 3rd 3rd 2nd 2nd

22 3rd 1st 4th 4th 1st 1st

23 1st 1st 4th 4th 1st 1st

24 4th 1st 4th 1st 1st 1st

January 1998 25 4th 1st 4th 1st 1st 1st

26 4th 1st 4th 1st 1st 1st

27 4th 1st 4th 1st 3rd 1st

1 4th 3rd 2nd 1st 1st 4th

2 4th 1st 1st 2nd 2nd 3rd

3 4th 1st 1st 1st 1st 1st

Table 2: Results of the methodology for 6 electrical consumers of HN MEKO Type frigates, where type ‘a’ is the chiller, type ‘b’ the HP compressor FWD/AFT, type ‘c’ the refrigeration plant, type ‘d’ the LP compressor FWD/AFT, type ‘e’ the sanitary plant, type ‘f’ the boiler respectively

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of the adequacy measure is gradually stabilised. It can also be estimated graphically, using the ‘knee-rule’, of which gives values between 3 to 4 clusters (see Fig 7). In Fig 8 the typical daily chronological ON/OFF load curves for the

chiller are presented using the AVQ method with the best WCBCR measure for 4 clusters. The training time for the methods under study has the ratio 0.05:1:22:24:36 (hierarchical: proposed k-means: optimised adaptive vector quantisation: mono-dimensional selforganising map: fuzzy k-means for q ¼ 6). Therefore, the k-means, AVQ and hierarchical models have been selected.

Application of the clustering models for the other electric consumers

Fig 7: WCBCR measure of the AVQ model for 2 to 10 clusters for the set of 11 training patterns of the chiller and the use of the tangents for the estimation of the knee

The same methodology is applied to the other five electric consumers. In Table 2, the total number of clusters, the population of the representative cluster, the respective clustering technique and the clusters calendar are registered for each consumer. It is obvious that the modified k-means, the AVQ and the hierarchical (especially UPGMC) clustering techniques compete each other, while the fuzzy k-means and self-organising map techniques have poor results in this kind of classification. It is reminded that the representative

Fig 8: The typical daily chronological ON/OFF load curves for the chiller for four clusters using AVQ method

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cluster is the most populated one. In Fig 9, the respective representative typical load diagrams are presented.

CONCLUSIONS This paper presents an efficient pattern recognition methodology for the study of the load demand behaviour of electrical consumers of ships. Unsupervised clustering methods can be applied, such as the k-means, adaptive vector quantisation (AVQ), fuzzy k-means, mono-dimensional self-organising maps (SOM) and hierarchical methods. The performance of these methods is evaluated by six adequacy measures: mean square error, mean index adequacy, clustering dispersion indicator, similarity matrix indicator, DaviesBouldin indicator, the ratio of within cluster sum of squares to between cluster variation. Finally, the representative daily load diagrams along with the respective populations per each consumer are calculated. This information is valuable for ship engineers, because it facilitates the formalisation of

the electrical consumer’s load behaviour, the design, the operation and the reliability of the ship’s power system and the improvement of the operation of the automatic battle management system for a battleship. From the respective application for six consumers of HN MEKO type frigates electric system for a small dataset (only 11 training sets), it is concluded that three to four clusters suffice for the satisfactory description of the daily load curves of each consumer. It is also concluded that the optimal clustering technique is the modified k-means, the AVQ and the hierarchical clustering techniques, while the optimal adequacy measure is the ratio of within cluster sum of squares to between cluster variation. These results surely depend on the population of the data set, the ship operating mode (‘anchor’, ‘shore’, ‘at sea’, ‘general quarter’) and the load curve’s time interval (in this case study, it was a day). In the future, it should be investigated in larger datasets for longer study periods with different time intervals (from few hours to days).

Fig 9: The representative (most populated one) chronological ON/OFF load curves for each one of the six consumers

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