Dynamic evolution of a hydraulic-mechanical-electric system with randomly fluctuating speed based on Chebyshev polynomial approximation method

Dynamic evolution of a hydraulic-mechanical-electric system with randomly fluctuating speed based on Chebyshev polynomial approximation method Beibei Xua, Diyi Chen*a,b, Caraballo, Tomasc aInstitute of Water Resources and Hydropower Research, Northwest A&F University, Shaanxi Yangling 712100, P. R. China bKey Laboratory of Agricultural Soil and Water Engineering in Arid and Semiarid Areas, Ministry of Education, Northwest A & F University, Shaanxi Yangling 712100, P. R. China cDepto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, c/ Tarfia s/n, E-41012 Sevilla, Spain Corresponding author: Diyi Chen Mailing Address: Institute of Water Resources and Hydropower Research, Northwest A&F University, Shaanxi Yangling 712100, China Telephones: 086-181-6198-0277 E-mail: [email protected] Abstract The research proposed in this paper focuses on the dynamic evolution of a hydraulic-mechanical-electric system under the effect of randomly fluctuating speed. The rapid growth of installed wind power capacity may potentially affect the stability of power grids, causing larger fluctuations of the generator speed to hydropower stations. In this work, a probabilistic component is associated to the generator speed of a deterministic hydraulic-mechanical-electric system providing a novel random model. This latter is analyzed to investigate the dynamic evolution of the system adopting the Chebyshev polynomial approximation method. A careful comparison of the numerical application results obtained by the deterministic and the probabilistic approaches is carried out. In addition, the influence of the fluctuation intensity (D) on the differential gain (kd) of the PID is investigated, proposing a law for kd as function of D. Finally, the operating ranges of the grid water hammer and of the elastic water hammer models are compared in order to validate the consistence of the law. The results of the study provide robust bases for the stable and safe operation of hydropower stations. Key words: vibration characteristics; hydraulic-mechanical-electrical system; randomly fluctuating generator speed; elastic water hammer; Chebyshev polynomial approximation; 1. Introduction Over the last decades, the hydropower sector has experienced a rapid development in China, with the constructions of power stations in every corner of the country, reaching an installed capacity of ten hundred million kilowatts [1-4]. Furthermore, according to the 2014 plan agreed by most international hydropower industries, the construction of new hydropower installations will be promoted in order to double the worldwide installed capacity within the next thirty years [5, 6]. Due to this rapid expansion, the reliability and safety of such installations have nourished the concerns of the public opinion as well as of the regulators [7-12]. In particular, the hydraulic-mechanical-electrical system is a crucial component which plays an essential role for the safety of individual installations and can hence affect the stability of the whole electric power grid. Therefore, the analysis of the stability of hydraulic-mechanical-electric systems is a high-priority topic of research. In spite of this, the scientific literature in this field is still quite limited. Studies focusing on hydraulic-mechanical-electric systems can be roughly classified in two groups. The first focuses on the grid water hammer theory, aiming at establishing deterministic models for the hydraulic-mechanical-electric system [13-16]. This approach is suitable to describe the dynamic evolution of hydropower stations with short penstocks characterized by uniform geometry. The works belonging to the second group adopt the elastic water hammer theory for the definition of the determinist model [17-25]. This approach is used to accurately analyze the dynamic characteristics of hydropower stations with long penstocks and allows to take into account eventual shape changes in the geometry of the penstock. In addition to water power, renewable energy includes solar, wind, geothermal or tidal power amongst others [26-31]. Moreover, as highlighted by the last Global Wind Energy Outlook, the wind power sector has grown rapidly over the last decades in terms of both technological and commercial competitiveness: this is expected to result in an installed capacity of nearly 2 TW by 2030, supplying between 16.7% and 18.8% of global electricity [32]. Obviously, this trend may potentially affect the stability of power grids, leading to larger random fluctuations of hydro-turbine generators speed [33-36]. High-intensity random fluctuations can lead to the breaking of the torque balance of hydro-turbine generator units and hence threaten its operating stability. Lots of studies have focused on the increasing randomness of power grid. Therefore, the increasing generation capacity of wind power installations introduces new important challenges to the stable operation of hydropower stations of any size. At light of this, our work proposes a novel approach to the stability analysis of the hydraulic-mechanical-electric system with respect to the existing literature. First, a probabilistic component u is integrated with the generator speed w , obtaining a novel probabilistic model of the hydraulic-mechanical-electric system. Second, by comparing the dynamic evolutions of the deterministic and probabilistic approaches, the advantages and drawbacks of each method are identified. Third, a mathematical definition of the differential gain kd for the PID in function of the fluctuation intensity D is proposed. Finally, the consistence of such definition is verified through a numerical application. The content of this paper is organized as follows: Section 2 presents the nonlinear probabilistic model of the hydraulic-mechanical-electric system. Numerical simulations along with detailed analysis of the results obtained are presented in Section 3. Conclusive remarks and discussions are included in Section 4. 2. Nonlinear random models of hydraulic-mechanical-electric systems The hydraulic-mechanical-electric system considering the elastic water hammer model adopted in this study is shown in Eq. (1): 12 23 3011223 '' 2 '' 4 4 1[ sin sin 2 ] 2 1(( ) ) s qs d q s tt ab ddq pid y xx xx xaxaxaxy EV x x V mD Txxx ykr kxky T xr dww wddw ww w SS SSS = ì ï= ï ï=- - - + ï = ï ï- í=- - - ï ï ï =-+-- ï ï ï=- î (1) Details about the parameters in Eq. (1) can be found in Ref. [24]. Fig. 1 Evolution in time of generator speed fluctuation According to the nature of the fluctuation affecting the generator speed, two different types can be identified: the first refers to a limited range, of about 49.5~50.5 r/min, as shown in Fig. 1 for values of time t lower than 120 s. This type of fluctuation is difficult to detect from the generator running sound due to the negligibility of the associated noise compared to that of the operating environment. The second type refers to fluctuations within the range 49.5~50.5 r/min, as shown in Fig. 1 for time values t higher than 120 s. Both the types of identified fluctuation have the potential to break the dynamic balance of torques between the hydro-turbine and the generator. In light of this, a probabilistic representation of the torque caused by the speed fluctuation is essential to obtain the state-space representation of the generator speed. Let Du w be the random torque, then the generator speed can be written as: '' 2 '' 1sin sin 2 2 qs d q s tt ab ddq EV x x V mDDu Txxx wddww SS SSS éù - =- - -- êú êú ëû (2) where u is the random variable introduced in this study. Its probability density function, shown in Fig. 2, can be described as [37] 2 21; 1. () 0; 1. uu pu u p ì-£ ï =í ï> î (3) -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 u p(u) Fig. 2 The diagram of the probability density function ()pu . Chebyshev polynomial approximation is used to simplify the probabilistic model of the hydraulic-mechanical-electric system. Such approximation can be presented as: 22 0 (1)( )! () (2) !( 2 )! n k nk n k nk Uu u kn k - = -- = - å . (4) Its recurrence relation is [ ] 11 1 () () () 2 nnn nU u U u U u -+ =+ , (5) and the approximation property is 12 1 1, 21()() 0, ij ij uU uU udu ij p - = ì -= í¹ î ò (6) According to the approximation theory of orthogonal polynomials and the above analysis, the random variables of the system can be written as ( ) ( ) ( ) ( ) ( ) ( ) 11 0 22 0 33 0 0 0 0 44 0 (, ) () ( ) (, ) () ( ) (, ) () ( ) (, ) () ( ) (, ) () ( ) (, ) () ( ) () () ( ) N i i i N i i i N i i i N i i i N i i i N i i i N iii i xtu x tUu xtu x tUu xtu x tUu tu tU u tu tU u ytu y tU u xt xtUu dd ww = = = = = = = ì= ï ï ï= ï ï ï= ï ï ï ï= í ï ï = ï ï ï = ï ï ï ï= ï î å å å å å å å , (7) where N is the maximum number of Chebyshev polynomials; ( ) 1 11 1 () ( ) (, ) ( ) i i xt puxtuUudu + - =ò ; ( ) 1 22 1 () ( ) (, ) ( ) i i xt puxtuUudu + - =ò ; ( ) 1 33 1 () ( ) (, ) ( ) i i xt puxtuUudu + - =ò ; 1 1 () ( ) (, ) ( ) ii tputuUudu dd + - =ò ; 1 1 () ( ) (, ) ( ) ii tputuUudu ww + - =ò ; 1 1 () ( ) (, ) ( ) ii yt puytuUudu + - =ò ; 1 44 1 () ( ) (, ) ( ) iii xt puxtuUudu + - =ò . At light of this, the random state-space equations of the hydraulic-mechanical-electric system can be rewritten as: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 12 00 23 00 3011223 00000 0 () ( ) () ( ) () ( ) () ( ) () ( ) () ( ) () ( ) () ( ) () ( ) () ( ) NN ii ii ii NN ii ii ii NNNNN iiiii iiiii iiiii N i i i dxtUu xtUu dt dxtUu xtUu dt dxtUu a xtUua xtUua xtUu ytUu dt dtU u dt d == == ===== = éù = êú ëû éù = êú ëû éù =- - - + êú ëû åå åå ååååå ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 ' 00 4 00000 () ( ) 1 () ( ) () ( ) 1 () ( ) ( ( () ( )) () ( ) () ( ) () ( )) N si i i NN itet i ii ii ab NNNNN ip iiid i i iiiii iiiii y tU u dtU u m P D Du tU u dt T dd ytUu kr tUu k x tUu k tUu ytUu dt T dt ww ww ww = == ===== éù = êú ëû éùé ù =--+ êúê ú ëûë û éù éù =-+- - êú êú ëû ëû åå åå ååååå ( ) ( ) 4 00 () ( ) () ( ) NN ii ii ii dxtUu r tUu dt w == ì ï ï ï ï ï ï ï ï ï ï í ï ï ï ï ï ï ï ïéù ï=- êú ïëû îåå (8) where ( ) ( ) '' 2 ' '' 00 sin ( ) ( ) sin 2 ( ) ( ) 2 NN qs d q s ei i ii ii ddq EV x x V PtUu tUu xxx dd åå == ååå - éù éù =- êú êú ëû ëû åå . From Eq. (5), the following relationship can be obtained: 2 00 01 1 12 3 1 () 22 22() Uu U UU U UU U U = = =+ (9) Combining the expressions obtained so far, ( ) 0 (t) ( ) ( ) N i i i Du t U u w = éù êú ëû å can be simplified as: Du(t) ω i ( ) (t)Ui(u) i=0 N ∑ " # $ $ % & ' ' =1 2 D ω i ( ) (t)Ui−1(u)+Ui+1(u) " #% & i=0 N ∑=1 2 D ω i−1 ( ) (t)+ ω i+1 ( ) (t) " # $% & 'Ui(u) i=0 N ∑ , (10) where 1 w - =0, and 1N w + =0. Replacing Eq. (10) into Eq. (8), the probabilistic model of the system can be written as: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 12 00 23 00 3011223 00000 0 () ( ) () ( ) () ( ) () ( ) () ( ) () ( ) () ( ) () ( ) () ( ) () ( ) NN ii ii ii NN ii ii ii NNNNN iiiii iiiii iiiii N i i i dxtUu xtUu dt dxtUu xtUu dt dxtUu a xtUua xtUua xtUu ytUu dt dtU u dt d == == ===== = éù = êú ëû éù = êú ëû éù =- - - + êú ëû åå åå ååååå ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 11 000 4 000 () ( ) 11 () ( ) () ( ) () () ( ) 2 1 () ( ) ( ( () ( )) () ( ) () ( N si i i NNN iteti i iiii iii ab NNN ip iiid i iiii iii y tU u dtU u m P D tU u D t t U u dt T dd ytUu kr tUu k x tUu k tUu dt T dt ww wwww ww = -+ === === éù = êú ëû éùé ù éù =-- - + êúê ú ëû ëûë û éù =-+- êú ëû åå ååå ååå ( ) ( ) ( ) 00 4 00 )()()) () ( ) () ( ) NN i i ii NN ii ii ii ytUu dxtUu r tUu dt w == == ì ï ï ï ï ï ï ï ï ï ï í ï ï ï ï ïéù ï- êú ïëû ïéù ï=- êú ïëû î åå åå (11) Let us multiply the above system of equations by () i Uu . Considering the mathematical expectation with regard to the random variable u on both sides of Eq. (11), and setting i=0, 1, 2, 3, and 4, the initial probabilistic model can be approximated by the system in Eq. (12): 10 20 20 30 30 0 10 1 20 2 30 0 00 '' 2 00 0 0 1 '' 00 0 0 40 11 ()sin () () ()sin2 () () 22 1 s NN qs d q s tii iit ii ab ddq pi d y xx xx xaxaxaxy EV x x V mE Uu tUu Uu tUu D D Txxx ykkxk T dww wd dww ww SS == SSS = = =- - - + = éù æö - æö æö =-+-- êú ç÷ ç÷ ç÷ ç÷ èø èø êú èø ëû =-+- åå ( ) 00 40 0 11 21 21 31 31 0 11 1 21 2 31 0 11 '' 2 11 1 1 0 '' 00 1 2 11 ()sin () () ()sin2 () () 22 s NN qs d q s tii iit ii ab ddq y xr xx xx xaxaxaxy EV x x V mE Uu tUu Uu tUu D D Txxx w dww wd dww SS == SSS ì ï ï ï ï ï ï í ï ï ï ï- ï ï=- î = = =- - - + = æö - æö æö =-+--+ ç÷ ç÷ ç÷ ç÷ èø èø èø åå ( ) 2 1 1 41 1 1 41 1 14 24 24 34 34 0 14 1 24 2 34 4 44 '' 2 44 4 4 '' 0 1 1 2 1()sin () () ()s 2 pi d y s N qs d q s tii i ab ddq d ykkxky T dt x xx xx xaxaxaxy EV x x V mE Uu tUu Uu Txxx w ww w dww wd SS = SSS ì ï ï ï ï ï ï ï íéù ïêú ïêú ëû ï ïæö =-+- - ç÷ ïèø ï ï=- î = = =- - - + = - æö =-+ ç÷ èø å3 0 4 4 44 4 4 43 4 1 in 2 ( ) ( ) 2 1 N ii t i pi d y tU u D D d ykkxky T dt x dww ww w = ì ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï í ï ï ï ï ï ï ï ï ïì ïï ïï ïï ïï ïï ïï ïï í ïéù æö æö ï ï-- êú ç÷ ç÷ ç÷ ï ïèø êú èø ëû ï ïïæö ï=-+- - ç÷ ï ïèø ï ïï ï=- î î å (12) The formula of the mathematical expectation of the electromagnetic torque ' e P can be then written as: ( ) ( ) ( ) ( ) '' 2 ' '' 00 '' 2 44 1 '' 100 sin ( ) ( ) sin 2 ( ) ( ) 2 sin ( ) ( ) sin 2 ( ) ( ) ( ) 2 NN qs d q s ei i ii ii ddq qs d q s ii ii ii ddq EV x x V EP E tUu tUu xxx EV x x V tU u tU u pudu xxx dd dd åå == ååå åå -== ååå æö - éù éù éù =- ç÷ ëû êú êú ç÷ ëû ëû èø æö - éù éù =- ç÷ êú êú ç÷ ëû ëû èø åå åå ò (13) Finally, the average response can be evaluated according to the Eq. (12): [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] 4 11 10 0 4 22 20 0 4 33 30 0 4 0 0 4 0 0 4 0 0 44 (, ) () ( ) () (, ) () ( ) () (, ) () ( ) () (, ) () ( ) () (, ) () ( ) () (, ) () ( ) () (, ) () i i i i i i i i i i i i i i i i i i ii Extu x tU u x t Ex tu x tU u x t Ex tu x tU u x t Etu tUu t Etu tUu t Eytu y tUu y t Ex tu x tU dd d ww w = = = = = = == == == == == == = å å å å å å 4 40 0 () () i i uxt = ì ï ï ï ï ï ï ï ï ï ï í ï ï ï ï ï ï ï ï ï= ï îå (14) 3. Numerical simulations The proposed model has been applied to a case-study in order to prove the efficiency and consistency of the approach. The values adopted for the parameters involved in the computation have been selected within realistic ranges: the rated generator speed is s w =314; the inertia time constant of the hydro-turbine generator unit is ab T =8.0; the damping factor of the generator is Dt=0.5; the transient internal voltage of the armature is ' q E =1.35; the direct axis transient reactance is ' d xå =1.15; the quadrature axis reactance is ' q xå =1.474; the major relay connecter response time is y T =0.1 and the bus voltage at infinity is s V =1.0. The first-order partial derivative value of flow rate with respect to water head is qh e =0.5; the first-order partial derivative value of torque with respect to wicket gate is ey=1.0; the intermediate variable is e=0.7; the length of the phase of the water hammer wave is Tr=1.0; the elastic time constant of the penstock is hw=2.0; the reference input is r=0; the proportional gain of the PID controller is kp=2 and the integral gain of the PID controller ki=1. The initial values of the deterministic model of the hydraulic-mechanical-electric system (1) are [x1(0), x2(0), x3(0), (0) d , (0) w , y(0), x4(0)]=[0.001, 0.001, 0.001, 0.001, 0.001, 0.001, 0.001]. Similarly, the initial values of the random hydraulic-mechanical-electric system (11) are different values of the intensity D and kd varying from 0 to 6. The trend associated with point 3, namely the value of kd for which the generator speed w pass from the non-tunable to the vibration state, is shown in Tab. 3. -8 -6 -4 -2 0 2 4 6 8 -0.05 0 0.05 0.1 kd ʍ -8 -6 -4 -2 0 2 4 6 8 -0.05 0 0.05 0.1 kd ʍ D=0.06 D=0.16 -8 -6 -4 -2 0 2 4 6 8 -0.05 0 0.05 0.1 kd ʍ -8 -6 -4 -2 0 2 4 6 8 -0.05 0 0.05 0.1 kd ʍ D=0.36 D=0.52 Fig. 6 Dynamic evolution of the generator speed w for the probabilistic model in Eq. (15) with different values of the random intensity D and kd varying from -8 to 8. (a) D=0.06; (b) D=0.16; (c) D=0.36; (d) D=0.52; Tab. 3 Location of point 3 according to increasing values of the random intensity D D 0.04 0.06 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.40 Point3 -5.854 -5.646 -5.542 -5.021 -4.604 -3.979 -3.25 -2.521 -1.479 -0.2292 0.083 As shown in Fig. 6 and Tab. 3, increasing levels of the intensity D cause the region of the domain associated with the non-tunable state shifting to the right, while the operating state on the right of point 3 remains unchanged. As expected, the trend registered for the probabilistic model in Eq. (15), considering the rigid water hammer model, matches perfectly the results obtained for the previous probabilistic model of Eq. (11). In addition to this, as shown in Fig. 5(a) and Fig. 6(a), it can be noticed that when the system operates in the regular vibration state, the amplitude of the fluctuations for the probabilistic model of Eq. (11) varies in the range of [0.0188, 0.02015], and the adjustable range of kd is [1.951, 2.563]. Regardless of the shape changes of the penstock wall, the amplitude of the fluctuations for the probabilistic model in Eq. (15) varies in the range [0.01754, 0.02281], which is similar to the previous case. Conversely, the adjustable range of kd, which is [0.083, 4.458], results much larger than that associated with the previous probabilistic model. Moreover, when the hydro-turbine generator unit operates in the quasi periodic vibration state, although the amplitude of the fluctuations associated with the model of Eq. (11) is similar to that of the model of Eq. (15), the range of kd is much larger than that associated with the latter probabilistic model. 4. Conclusions In this study, a random variable u is integrated with the generator speed of the deterministic model of a hydraulic-mechanical-electric system to establish the corresponding probabilistic model. Using this latter, the dynamic evolution of the system is analyzed, and three main conclusions can be achieved. First, although the dynamic evolution of the random system is broadly similar to that of the deterministic approach, the two sets of results show significant differences, carefully investigated in the paper. Second, the point 3, which highlights the transition from the not-tunable state to the random vibration state, shifts to the right with increasing values of the random intensity D, which leads to the decreasing adjustable range of kd. Third, the consistence of the trends highlighted by the probabilistic model implemented in this paper is compared and verified through the use of another model. Moreover, when the shape of the penstock is assumed not uniform, the adjustable range of kd is narrowed from left to right and the operating state of the system becomes less stable. 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