Chaotic analysis of electrical energy unit prices in Turkey

The words "chaos" or "chaotic" are some commonly used terms in daily life to define the ambiguous and disordered situations. But in the contrary case, from scientific approach point of view, this definition can not express the concept of chaos since chaos theory actually is a mathematical method to analyse the systems ruled by laws of physics. The existence of chaos can not be the point in question in random systems. Looking from outside to a nonlinear time series, one can think the points are scattered randomly in phase space, but in chaotic systems those random-like points are actually there based on some critical events and parameters. Linear systems can not contain chaos but only nonlinear deterministic dynamical systems. The purpose of chaotic analysis is to identify the chaos existence by using some mathematical analysis and methods. The main two characteristics of chaotic systems are an existence of an attractor with fractal dimension and the exponential divergence of trajectories in phase space, which also means sensitive dependence to initial conditions. An attractor can be defined as the place a linear or nonlinear system leans to in long term. If an anticonservative attractor shows sensitive dependence to initial conditions, it is called as a strange attractor and since this phenomenon covers main two characteristics of chaos, it is a strong evidence of chaos existence all by itself. Strange attractors show similarity with the whole system and their dimensions are not integer values but fractal. The sensitive dependence to initial conditions can be measured by Lyapunov exponents, in case of having a positive value this phenomenon physically states that trajectories diverge in phase space. Here the divergence means the neighbour points can arbitrarily get far apart due to the direction of their trajectories but they do not leave the attractor and in time might get close again. Today electrical energy is one of the biggest needs in every field. All countries in the world use their energy sources and electrical energy production facilities to meet the energy need. In case of lack of energy sources, they import short ones from other countries to mitigate the risk of energy shortage. Per capita consumption of electrical energy is considered as a sign of development level of countries. In order to manage a need carrying this level of importance, every country has to consitute their own internal market. Energy sector is one of the most significant sectors in Turkey as well due to the size of capacity and volume. Having an above world-average level on per capita consumpion of electrical energy, Turkey has still way to go to reach the level of European countries. In early 2017, Turkey's gross electrical energy consumption per capita was around 3350 kWh, whereas the world average was 2900 kWh. However, this number was close to 9000 kWh on average in Europe which shows the difference between Turkey and European countries. Electrical energy production in Turkey is based on different energy sources, as of July 2017 the distribution of the energy production is as follows: 34% natural gas, 31% coal, 24% hydro, 6% wind, 2% geothermal and the rest 3% from other sources. However, this order is not the same when it comes to installed power levels for different sources. In early 2017, the installed power capacity of Turkey was 79,834 MW and 33% of this power was hydro-electric, 28% natural gas, 21.5% coal, 7.7% wind, 1.1% geothermal and the rest 7.4% other sources. Like most of the European countries, a monopoly system in electricity market had existed till 1980s in Turkey, due to the public utility approach to electricity supplying. With the changes in economical conditions, this approach also changed after then and privatization and liberalization were considered as solution. Even though several changes were applied after 1980s, electricity market structure in Turkey started to become mature considerably after 2001 with the the establishment of EPDK and also EÜAŞ, TEİAŞ and TETAŞ. There has been several updates on the structure until 2015 but the most significant ones for this study are day-ahead planning, day ahead market, and intra-day market periods. Day-ahead planning had been designed as transitioning period to day-ahead market, and was active between 2009-2011. During this period, energy producers had been trained for the essentials of day-ahead market. Day-ahead market structure was initiated in 2011 and it has been active since then. The purpose of day-ahead structure was to give opportunity to energy producers to balance their liabilities agreed with bilateral agreements, which also enables more-balanced market and is more useful for market regulator. Intra-day market became active in 2015 as a subsidiary system to day-ahead market, since it gives the chance to players to change and play with their offers in shorter term which ends up with more stable system. In summary, energy producers in Turkey firstly arranges their electricity buying and selling by bilateral agreements with public or private institutions which gives them a long term stability. But due to some unexpected changes, those prices might need to change and be balanced. Therefore they use day-ahead market to balance the prices one day before. Players can use intra-day market as a last stop in case any unpredictable situation arises in the short term like a malfunction in production facility. Within the scope of this study, chaos existence in the time series of electrical energy unit prices in Turkey between 2009-2017 has been searched and followed by short term prediction of the prices. As different energy market structures have been under operation during this time period, analysis have been done seperately for each to gain a better understanding on the changes. Day-ahead planning period considered as first time series and day-ahead and intra-day market considered as the second one. Modelling the system based on some critical parameters is not the aim of this study since the only data at hand are the time series itself. As a first step, assuming there will be noise like all other dynamical nonlinear systems have, noise reduction has been done by using local projective method. For noise reduction, several iterations have been done in TISEAN 3.0.1 programme and the level of the noise reduction for all iterations have been analysed by checking the correlation integrals and correlation dimensions in different embedding dimensions. The time delay of the system determined by mutual information function; first minimum of the mutual information function has been selected as appropriate time delay. Sufficient embedding dimension has been determined by fraction of false nearest neighbours calculation. Derived time delay and embedding dimension data enabled the reconstruction of the phase space. Attractor dimension has been calculated from correlation integral graphs in logaritmic scale for all embedding dimensions. Correlation dimension plots from TISEAN 3.0.1 has also been used for visual inspection of attractor dimension. For both time series, fractal dimension of the attractor has been detected. To analyse the sensitive dependence on initial conditions Grassberger's Lyapunov exponent calculation has been used by TISEAN programme and maximum Lyapunov exponent has been found positive. Therefore, strange attractor with fractal dimension existence and positive Lyapunov exponent, the most effective ways to demonstrate chaos, have been shown. The short term prediction has been done by finding near neighbours to every point in the phase space and predicting the resulting points based on linear mapping. For day-ahead planning period, predictions have been completed with 85.2% R2 and 6.0% NRMSE and 92.1% R2, 5.7% NRMSE for day-ahead /intra-day market period.