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Procedía Computer Science 12 (2012) 361 - 366

Complex Adaptive Systems, Publication 2 Cihan H. Dagli, Editor in Chief Conference Organized by Missouri University of Science and Technology

2012- Washington D.C.

A Hybrid Intelligent System for Designing a Contract Model for

Weather Derivatives

Hajime Fujitaa, Hiroyuki Moria*

_aMeiji University,Tama-ku, Kawasaki,214-8571, Japan_

Abstract

In this paper, a hybrid intelligent system is proposed for designing a contract model of the weather derivatives between energy utilities. In recent years, the weather conditions often bring about uncertainties on profits to companies that are related to weather conditions. As a result, they require a strategy to equalize the profits and avoid unexpected deficits. To satisfy the requirements, weather derivatives are developed to the companies. They may be expressed as the function of the weather conditions such as the average, the maximum temperature, etc. So far, a lot of the derivatives have been developed, but it is not clear how to design them due to the nondisclosure on practical knowledge. In this paper, an efficient method is presented to determine a contract model of weather derivatives. As the first stage, this paper focuses on the normal data. A reasonable model is constructed by the learning process for the normal data. This paper formulates the contract model as a two-phased problem. Phase 1 carries out data clustering of curves to extract the normal data with DA (Deterministic Annealing) of global clustering technique. Phase 2 optimizes the parameters of the model that equalizes the payoffs between two companies in a sense that the mean and the variance of the payoff are equalized for both companies with EPSO (Evolutionary Particle Swarm Optimization) of meta-heuristics. The proposed method is successfully applied to the real data in Tokyo, Japan.

Keywords: Evolutionary Particle Swarm Optimization; Deterministic Annealing; Optimization; Clustrering: Weather Derivatives

1. Introduction

This paper presents a contract model for the weather derivatives between energy utilities with a hybrid intelligent system that consists of DA (Deterministic Annealing) [1]-[5] clustering and EPSO (Evolutionary Particle Swarm Optimization) [6], [7]. Under new environment of Smart Grid, the degree of uncertainties increases due to EV charging/discharging and Demand Response (DR) as well as the random-like generation output of renewable energy and the markets of electricity and fuels. As a result, the energy companies are inclined to be faced with various risks under the Smart Grid environment. The risk management is required to deal with various risks in sales, demands, currencies, etc. This paper focuses on the weather derivatives as a tool to hedge the risk of weather conditions. In the risks that energy utilities are faced with, the weather is one of the most considerable risks. Recently, the weather

* Corresponding author to provide phone: +81-44-9347353; fax: +81-44-9347909. E-mail address: hmori@isc.meiii.ac.ip

1877-0509 © 2012 Published by Elsevier B.V. Selection and/or peer-review under responsibility of Missouri University of Science and Technology. doi: 10.1016/j.procs.2012.09.085

becomes much more changeable significantly. The weather derivatives are an attractive tool that hedges the weather risks to equalize profits [8]. The conventional methods on the weather derivatives [9]-[13] may be divided into time series model for (a) temperature forecasting [9]-[11], (b) MCS (Monte Carlo Simulation) for temperature forecasting [12], (c) analysis of risk hedge [13], etc. Method (a) predicts temperature during the contract period by extracting the features of time series model of temperature with GARCH (Generalized Auto Regressive Conditional Heteroskedasticity) model [9], the stochastic time series model [10] and the Dishel D1 model [11]. Method (b) evaluates the predicted value and its variance by adding the random numbers to the time series models or arbitrary distributions. Method (c) deals with how to hedge the risk in minimizing the variance of profits.

This paper proposes a contract model of the collar option that combines the put option with the call one to deal with the temperature risk. The collar option implies that one company takes the payoffs from the other company if the contract index such as temperature exceeds a specified interval. In other words, two companies do not have to give the payoffs to each other if the index exists within the specified permissible interval. The proposed method consists of two phases. Phase 1 selects an appropriate set of data that corresponds to the typical years excluding the outliers. DA clustering is used to select the appropriate data set. DA clustering is one of global clustering methods that are not influenced by the initial solutions. Phase 2 optimizes the parameters of the contract model for the data set obtained at Phase 1 to equalize the payoffs between two companies for a certain period. EPSO is used to evaluate a globally optimal solution. It has better performance in meta-heuristics that repeatedly makes use of some rules or heuristics to evaluate better solutions. The proposed method is successfully applied to the real data in Tokyo, Japan.

2. Weather Derivatives Models

In this section, the weather derivatives are described. They partly have the features in common with the financial derivatives. The financial derivatives make someone to take the risk instead of paying the charge. Namely, they provide the amount of compensation in case the price of commodities goes up or down. Although the structure of the weather derivatives is the same as the financial derivatives, the indices are not commodities but the weather variables such as temperature, wind, storm, rainfalls, snowfalls, etc. The mainstream is related to temperature such as average temperature, HDD (Heating Degree Day), CDD (Cooling Degree Day), etc. Now, let us consider the weather derivatives between Companies A and B. It is assumed that Companies A and B correspond to an electric power and a gas companies, respectively. In case of swap, Company A pays to Company B in case the weather index is over the strike. On the other hand, Company B pays to Company A in case it is over the strike. In other words, there is the transfer of money in case it does not equal the strike. In the collar, there is not payoff in case the weather index is in the range between two strikes. The collar may be considered as the swap that the index has the range of dead band. The collar is mainly negotiated by two companies that have opposite risks.

3. Problem Formulation of Curve Clustering

In this section, a clustering method is outlined. As the clustering method, DA clustering is used to obtain better results that do not easily get stuck in a local minimum. Now, let us consider a clustering problem that data sets x1,., xn are divided into clusters C1 ,..., Cm with centre vectors y1 ,..., ym. The clustering problem may be written as follows:

n m II ||2

d = £Xv..||x7. -yj -» min (1)

¡=i j=i

where, d : cost function, Xj : data j, yi : centre vector in Cluster i Vij : association of data j to Cluster isuch that

|1 (if Xj belongs to Cluster Cl ) 0 ( otherwise)

The solution of clustering may be evaluated by minimizing the cost function of (1).

Deterministic annealing (DA) clustering [1]-[5] is a global clustering method that stems from the free energy of statistical mechanics. Compared with k-means [14], this method evaluates the center vectors with the association probability. Also, this method introduces the parameter called temperature into the algorithm. At high temperature, the cost function corresponds to a quadratic function while it approaches the original one as temperature cools down.

This method evaluates a globally optimal solution by cooling down temperature gradually. The algorithm may be written as follows:

Step 1: Set the initial conditions.

Step 2: Calculate the association probability by the following equation:

p(■ 1/ iL (3)

where, P('): association probability, Ci(t> : Cluster i of t-th iteration , fi = 1/T(T: temperature) Step 3: Calculate the center vector:

txjPj ( x^Cf-1))

y (t) = j-1 J J J_

i ÏPi ( x,. eCf"1') (4)

j=1 j j

Step 4: Calculate the cost function:

/.\ n m II || 2

d (0= ZZPiXjeC, )|Xj-y| (5)

Step 5: Go to the next step if the following convergence criterion is satisfied:

|d d(M)|/d (H)<e (6)

where, & convergence criterion Otherwise, return to Step 2. Step 6: Stop if the following equation holds:

/?>/?max (7)

where, fimaX: the predetermined upper bound of fi

Otherwise, update fi and return to Step 2, where fi is updated by the following equation:

p{t >=£<">+AP (8)

In this paper, temperature data for summer is classified for 40 years. This paper carries out the classification of the temperature curves rather than temperature points. The temperature curves may be classified into 3 zones, Zones 1, 2 and 3. Zone 2 corresponds to a set of the typical temperature curves whiles Zones 1 and 3 correspond to sets of the outliers. In this paper, DA clustering is used to extracts the temperature curves in Zone 2 to select the appropriate learning data for the contract model. To cluster the curves, the cost function of clustering may be rewritten as

d'=£ t,v. \\x. -y\ —>min

,=i >1 'Jll ij -^11 (9)

where, d': cost function for classifying curves, xiJ: j-th element of curve belonging to Cluster i yiJ: j-th element of centre curve at Cluster i Also, (3)-(5) may be rewritten as

exp(-Z? Ilx h- yI2)

P(x„sC«) = m 1 ■h 'J!1 (10)

Xexp(-j0 x„- y

j=1 II

(t-i) N

I x hPh (x heC

y = ^4o-— (ii)

I p (x i))

d (»=î± P(x )|| x h- y j|| ' (i2)

where, x„ = (Xh,i,xK2,■■■Xh.éi)T , ys = (y,i,yj,2,■■■,yj,61 )T, h: No. of curve elements 4. EPSO

In this section, EPSO (Evolutionary particle swarm optimization) [6], [7] is described. The conventional PSO (Particle Swarm Optimization) [15], [16] that makes use of multi-point search with the agents has a drawback that it

often gets stuck in a local minimum. EPSO extends PSO into the method that adjusts the parameters with the evolutionary strategy and replicates the agents to keep the diversity of solution candidates so that better solutions are obtained. The algorithm stems from the behavior of social creatures such as birds or fishes. The individuals of birds or fishes called agents find out better solutions while sharing useful information in the search process. Each agent stores the best solution with the best cost function (pbest) that is obtained through the search process. At the same time, each agent has information on the best solution with the best cost function (gbest) that is determined by all the agents. Namely, gbest is the best solution of pbest. The algorithm of EPSO may be written as follows:

Step 1: Set the initial conditions.

Step 2: Reproduce r particles for particle s,.

Step 3: Update the original the reproduced particles with the transition rule, where the reproduced particles make use of the mutation weights.

Step 4: Evaluate fitness and select the particle with the weight fitness.

Step 5: Update pbest if the selected particle is better than pbest.

Step 6: Update gbest if the selected particle is better than gbest.

Step 7: Stop if the termination conditions are satisfied. Otherwise, return to Step 2. The solution may be updated by the following equations [6], [7]:

V'+1= w'j; + w* (pbest, -S't) + w;2 (gbestS't) (13)

wl=wlk+rN (0,1) gbest * = gbestxw*3 N (0,1) w*3 = wi3 +7-' N (0,1) S';1 = s' + V" 1

5. Proposed Method

In this section, a new method that makes use of DA clustering and EPSO is proposed to design the contract model for weather derivatives. Fig. 1 shows the concept of the proposed method, where it consists of Phases 1 and 2. Phase 1 extracts the typical summer season temperature data from the past temperature data with DA clustering to make contract reasonable in a sense that the outliers are excluded. Phase 2 estimates the parameters of the contract model with EPSO. The contract model are two strikes (xj,x2), cap (x5), floor (x6) and ticks (x3,x4) illustrated as shown in Fig. 2 where the payoff is described from a standpoint of Tokyo Electric Power Company [17]. This paper defines the parameters of contract as variables xj-x6. In this paper, the sum of the payoffs of DAT obtained from Fig. 2 is equalized for the typical summer season temperature in the past. Therefore, this problem may be formulated as an optimization problem of parameters. Cost function:

f (t) = Nhprice(tl) —> 0 (18)

Constraints:

x3(x,-x5) = x4(x6 -x2) = 1200 (20)

where, f (■): payoff/payout functions. N : number of days, t : average temperature on i-th day (i=1,2,...,N) Aprice: payoff/payout price

Aprice(tl) =

x3(x1- x5) (ti~ x5 )

x3 (xl — ti) (x5 < t,< x1)

0 (x,< ^<x2)

xA(t^xz) (xz< t,<x6)

x4(x6- x2) (x6^ ti )

Extraction

Temperature Data

A Set of Typical Annual Temperature

Parameter

Optimization

Reasonable Contract

DAT (° C)

Fig. 1 Concept of Proposed Method

Fig. 2 Parameters of Contract Model

6. Simulation

The obtained data is classified into learning and test data. The learning data consists of summer data for 40 years from 1961 to 2000. On the other hand, the test data is summer data in 2001. The proposed method utilizes DA clustering for learning data selection. It classifies the past summer temperature curves for 40 years. Each curve consists of 61 elements that correspond to 61 days for two months of August and September.

This paper compares EPSO with PSO and GA that are well-known for meta-heuristics. A thousand of initial solutions are used to examine the influence of them on the final solutions. For convenience, the following methods are defined : Method A: GA, Method B: PSO, Method C: EPSO. A hundred of initial random solutions are used to examine the effectiveness of k-means and DA clustering. The best results are used as the final solution.

Table 1 gives the statistics of the cost functions of Cluster 2 of the normal data with k-means and DA. It can be seen that the best, worst, and average cost functions obtained by DA are much smaller than those obtained by k-means. For example, the best cost function with DA is 3.5% of that with k-means. Also, the standard deviation of the cost functions with DA is 20% of that with k-means. Therefore, it can be shown that DA is much better than k-means. Table 2 gives the results of statistical analysis for the cost functions of Methods A-C where the best, the

Table 1 Statistics of Cost Functions of Cluster 2

Methods Cost Functions

Best Worst Ave. Standard Deviation Computational Times (s)

k-means 0.478 0.49 0.481 0.005 0.572

DA Clustering 0.017 0.021 0.018 0.001 3.200

Table 2 Statistics of Cost Functions of Methods A-C

Methods Cost Functions

Best Worst Ave. Median Standard Deviation

A 0.000 1.563 0.233 0.147 0.225

B 0.000 5.371 0.249 0.098 0.480

C 0.000 0.000 0.000 0.000 0.000

Table 3 Parameters Optimized by Methods A-C

Methods Parameters

Xl X2 X3 X4 X5 X6

A 25.48 26.40 8.152 -8.031 23.98 27.77

B 25.58 26.29 8.021 -8.015 24.22 27.52

C 25.56 26.36 8.145 -8.065 24.09 27.85

worst, the average and the standard deviation of the cost functions for the thousand trials. It can be confirmed that Method C outperforms Methods A and B significantly. Method A has smaller average cost function than Method B. In particular, it is noteworthy that Method C has the average cost function of zero and the standard deviation of zero.

Table 3 gives the optimized parameters (x1~x6) for Methods A-C. It can be seen that Methods A-C evaluated almost the same parameters because of selecting the best solutions in the thousand trials. Now, let us apply the proposed to real data of August and September, 2001. The proposed method gives the results that Tokyo Gas Co. should pay ¥165,000,000 to TEPCO in spite of the fact that Tokyo Gas Co. had paid ¥523,000,000 to TEPCO [17]. In other words, Tokyo Gas Co. overpaid ¥358,000,000 that corresponds to 217% of the payoffs estimated by the proposed method.

7. Conclusion

This paper has proposed a new method for designing a collar contract model of weather derivatives. It aims at the equalization of the payoffs between two companies to make the weather derivative model more reasonable. As the first stage, this paper focused on the normal data on maximum temperature. DA clustering of global clustering was used to select the typical learning data excluding the outliers at Phase 1. The simulation results have shown that DA clustering is much better than k-means of the conventional clustering method. Also, EPSO was utilized to optimize the model parameters of the weather derivative at Phase 2. It can be observed that EPSO outperformed PSO and GA in terms of the parameter optimization of the contract model. The proposed method was successfully applied to real data. The simulation results have shown that the proposed method provided better results.

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