Research Article Chaos and Control of Game Model Based on Heterogeneous Expectations in Electric Power Triopoly

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1 Discrete Dnamics in Nature and Societ Volume 29, Article ID , 8 pages doi:.55/29/ Research Article Chaos and Control of Game Model Based on Heterogeneous Epectations in Electric Power Triopol Weihuo Ji Department of Economics, Dalian Universit of Technolog, Liaoning Province 624, China Correspondence should be addressed to Weihuo Ji, jiweihuo668@ahoo.com.cn Received Ma 29; Accepted 2 November 29 Recommended b Xue-Zhong He A dnamic repeated game model has been established based on heterogeneous epectations in electric power triopol. Theoretical analsis and numerical simulation show the compleit of this model; suppose that the producers make decisions with naive epectation and bounded rationalit. The straight-line stabiliation chaos control method was successfull applied to the dnamic repeated game model. The results have important practical value for the producers in the electric power oligopol. Copright q 29 Weihuo Ji. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in an medium, provided the original work is properl cited.. Introduction In economic sstem research on chaos theor, bifurcation theor based on differential equations is the earliest chaotic dnamics branch with the most widespread application and the deepest advancement, 2. Whether the economic sstem is chaos or not has been a new hot topic in economic studies. A series of research on the dnamic game model of the output decision has been conducted. In references 3 7, the outputs would take on dnamic characters including chaos when the feedback rates of producers changed. In 23, Agia and Elsadan 7 studied the duopol game model based on heterogeneous epectations, that is, one plaer applied naive epectation rule and the other used bounded rationalit rule. In this paper, the model of duopol 8 was modified to triopol in the electric power market. Supposing two producers determined on the optimal outputs with naive epectation rule while the other with bounded rationalit rule for the maimal profits. Theoretical analsis and numerical simulations were performed in detail and the parameters leading to the chaos were obtained in the definite range. Then the straight-line stabiliation method was used to control the period-doubling bifurcation and unstable periodic orbits in chaotic attractor of the discrete nonlinear dnamic sstem 9. The research results help the electric power producers to decide appropriate outputs.

2 2 Discrete Dnamics in Nature and Societ 2. The Model in Electric Power Triopol Suppose that there are three electric power producers in the given region, and the provide electric power to consumers through the electric utilities. Each electric power producer makes the optimal output decision and q i t is the t-output of producer i. At each period t,thepricep is determined b the total outputs Q t q t q 2 t q 3 t. Suppose that the inverse demand function p p Q m nq 2, then m is the maimal price corresponding to Q and n is parameter of the price. The variable cost function VC i VC i q i b i q i c i q 2 i i, 2, 3 is nonlinear, too. Thereinto b i and c i are cost parameters. So the profit of the producer i in period t would be [ ] ] π i t q i t m nq 2 t [b i q i t c i q 2 i t rq i t, i, 2, 3, 2. where r is the wheeling rate. In this game, the producers make the optimal output decisions for the maimal profits; one of the methods to find out the Nash equilibrium is to let the partial differentiation of the profit equal ero as π i q i m nq 2 t 2nq i Q t b i 2c i q i r, i, 2, According to the Rational Epectation Theor, epectation rules include naive epectation, adaptive epectation, and bounded rationalit. Naive epectation is the simplest and the most common epectation form. Suppose that the plaers decide the future outputs according to the past 2 and take the output epectation in electric power market as an eample, then the epectant output of t equals the optimal output of t: q i t q i t. 2.3 The boundedl rational compan makes its output decision on the basis of a local estimate of the marginal income ratio. The compan decides to increase its production if it has a positive marginal income ratio or to decrease its production if the marginal income ratio is negative. According to the rules above, the dnamical equation of producer i has the form as following 3 : q i t q i t αq i t π i q i, <α<, 2.4 where α is the relative speed of adjustment; it is a positive parameter. We suppose that two producers adopt naive epectation rule and the other bounded rationalit. Let q t, q 2 t, q 3 t, q t X, q 2 t Y, andq 3 t Z,

3 Discrete Dnamics in Nature and Societ 3 then the nonlinear dnamic game model based on heterogeneous epectations in the electric power market can be epressed as X 6n Y 6n 4n 4n 2c 2 n 2 2 n 2 2 2n 2 4nc 4nc c 2 3mn 3nb 3nr, 4n 4n 2c 2 2 n 2 2 n 2 2 2n 2 4nc 2 4nc 2 c 2 2 3mn 3nb 2 3nr, Z α 3n 2 n 2 n 2 2n 4n 4n 2c 3 m b 3 r Numerical Simulations of the Sstem Now onl a few of simple dnamic sstems can be concluded as the chaotic characters b the analtical method, so the numerical analsis is the prime method in the research on the chaotic phenomenon. In order to deduce some numerical chaotic evidences of sstem 2.5, the inverse demand function and three cost functions will be taken as constants. The parameters take the values of m 5, n, b, c.3, b 2, c 2, b 3.7, and c 3. So the inverse function is p Q 5 Q 2, and variable cost functions are VC q q.3q 2, VC 2 q 2 q 2 q 2 2,andVC 3 q 3.7q 3 q 2 3, respectivel. Several numerical results have been concluded, such as bifurcation diagrams, strange attractors, Lapunov eponents, and the influence on the sstem with the changing parameters. 3.. The Chaotic Attractors of the Sstem When the parameters take the values of r, α 3 and the initial outputs are.2,,, the sstem 2.5 has the chaotic attractor as shown in Figure. The Lapunov eponents of the sstem 2.5 are λ.25, λ 2 467, as well as λ 3 656, and the Lapunov dimension is d λ /λ 2.25/ The fractal dimension is less than 2, so the occupied space is small and the structure is sparse, which can be seen in the chaotic attractor diagrams The Influence on the Sstem of the Parameters Changing Figure 2 shows the outputs bifurcation diagram when the parameters were set as r, α, 3. It is obvious that the outputs change from the initial status.2,, to the stable point then ehibit period-doubling bifurcations till chaos with the increase of output modification speed parameter. It is concluded 3 that the market presents the out-of-order phenomenon more easil when the modification speed parameter is larger. That is to sa that the feedback rate of producers to market can cause the outputs game model to show complicated characteristics. The output modification speed parameter has limited value and α ma 326 in sstem 2.5. In this stud, two of the producers adopt the same decision rules and the variable cost functions are similar quadratic function, so the evolvements of

4 4 Discrete Dnamics in Nature and Societ b.3 a.7.3 c.7 d Figure : a Three-dimensional view of the chaotic attractor, b - phase plane strange attractor, c - phase plane strange attractor, d - phase plane strange attractor with r, α 3. their outputs are similar as shown in the bifurcation figures. The range of α emerging chaos in the sstem 2.5 is 85, 326. When r, the diagram of the largest Lapunov eponent is changing with the parameter α in Figure 3. Different parameters α are accompanied b different Lapunov eponents. In addition, Figure 3 is in the same essence as that of Figure 2. When α < α, the sstem is regular periodic, so the largest Lapunov eponent is less than ero ecept at the critical point λ. When α is larger than α, λ > which means that the sstem is chaos while λ < at limited range of α corresponds to periodic windows in the chaotic region. 4. The Chaos Control of the Sstem The chaos control methods include perturbation feedback chaos control method and nonfeedback control method. The feedback objects are sstem parameters in the perturbation feedback chaos control method, sstem variable, and eternal parameters. Small continuous perturbation is regarded as control signal in feedback chaos control method. Even if the perturbation is ver small, the stable control of the specific goal can be realied. Recentl, Yang and Xu, and so forth, proposed a new control method called the straight-line stabiliation method 4, 5. So we adopt this method to control the chaos

5 Discrete Dnamics in Nature and Societ α.2 α a b.2.2 α c Figure 2: The state of the outputs with r, α 3..2 α λ α Figure 3: The largest Lapunov eponent of the sstem with r, α, 3.

6 6 Discrete Dnamics in Nature and Societ in the triopol game model. Impose the eternal control signal e n on the primar sstem, therefore, we obtain e t e n e 2 t ηi J q t q q 2 t q 2, e 3 t q 3 t q 3 4. where η < is the feedback control parameter, and q, q 2, q 3 are the outputs at the fied points. Suppose that q, q 2,andq 3, then the controlled game model is attained as X r ηi J, Y r ηi J, Z α r ηi J, 4.2 where J is the Jacobian matri of the primar sstem with r and α 3 given as 7 7 J The fied point of the sstem is 83, 848, 84, which means that q 83, q 2 848, and q 3 84, so the eternal control signal is η e n 96 η η The primar sstem is chaotic with r and α 3. Figure 4 is the third output of controlled sstem with the initial outputs.2,, and the eternal control signal η,. As can be seen from the figure, the controlling strateg is applied on the sstem b the eternal control signal, and the controlled sstem is stead with η,.442. The range of η corresponding to the stable region is larger than that corresponding to the unstable region. Figure 5 a is the 4-period time series of the third output with r, α 3, and η. Figure 5 b is the chaotic attractor of the controlled sstem with r, α 3, and η.

7 Discrete Dnamics in Nature and Societ η Figure 4: The output of the controlled sstem with r, α 3, and η, t a b Figure 5: a The time series of the output with r, α 3, and η ; b the chaotic attractor of the controlled sstem with r, α 3, and η. 5. Conclusions The chaotic dnamic theor is successfull applied to the electric power market. The discrete game model is obtained based on heterogeneous epectation rules. Suppose that the inverse demand function and cost functions are all nonlinear and take the wheeling rate into account, then theoretic analsis and numerical simulation results show that the sstem is chaotic. The influences on the market of the outputs modification speed parameters and the range of the parameters leading to the chaos have been discussed. Then the straight-line stabiliation method has been used to control the period-doubling bifurcation and unstable periodic orbits in chaotic attractor. Once a small perturbation is applied in the chaos region, the equilibrium point arrived quickl, so the producer s anticipated target will be achieved. The model is closer to the economic realit and worth spreading in oligopol though the dnamic analsis is more complicated.

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