Stability performance for primary frequency regulation of hydro-turbine governing system with surge tank

This paper aims to study the stability for primary frequency regulation of hydro-turbine governing system with surge tank. Firstly, a novel nonlinear mathematical model of hydro-turbine governing system considering the nonlinear characteristic of penstock head loss is introduced. The nonlinear state equations under opening control mode and power control mode are derived. Then, the nonlinear dynamic performance of nonlinear hydro-turbine governing system is investigated based on the stable domain for primary frequency regulation. New feature of the nonlinear hydro-turbine governing system caused by the nonlinear characteristic of penstock head loss is described by comparing with a linear model, and the effect mechanism of nonlinear characteristic of penstock head loss is revealed. Finally, the concept of critical stable sectional area of surge tank for primary frequency regulation is proposed and the analytical solution is derived. The combined tuning and optimization method of governor parameters and sectional area of surge tank is proposed. The results indicate that for the primary frequency regulation under opening control mode and power control mode, the nonlinear hydro-turbine governing system is absolutely stable and conditionally stable, respectively. The stability of the nonlinear hydro-turbine governing system and linear hydro-turbine governing system is the same under opening control model and different under power control model. The nonlinear characteristic of penstock head loss mainly affects the initial stage of dynamic response process of power output, and then changes the stability of the nonlinear system. The critical stable sectional area of surge tank makes the system reach critical stable state. The governor parameters and critical stable sectional area of surge tank jointly determine the distributions of stability states.     

Nonlinear dynamical analysis of hydro-turbine governing system with a surge tank

This paper brings attention to a new nonlinear mathematical model of a hydro-turbine governing system with a surge tank. The nonlinear mathematical model, which is described by state-space equations, is composed of Francis turbine system, electrical generator system, conduit system and governor system. Furthermore, the nonlinear dynamical behaviors of the system with different parameters are studied exhaustively including bifurcation diagrams, time waveforms, phase orbits, Poincare maps, spectrograms and power spectrums. Fortunately, some interesting phenomenons are found from numerical simulation results. More important, all of the above analyses supply some theory bases for designing and running of a hydro-turbine governing system.     

Nuclear power plant steam turbine—Modeling for model based control purposes

The nature of the processes taking place in a nuclear power plant (NPP) steam turbine is the reason why their modeling is very difficult, especially when the model is intended to be used for on-line optimal model based process control over a wide range of operating conditions, caused by changing electrical power demand e.g. when combined heat and power mode of work is utilized. The paper presents three nonlinear models of NPP steam turbine, which are: the static model, and two dynamic versions, detailed and simplified. As the input variables, the models use the valve opening degree and the steam flow properties: mass flow rate, pressure and temperature. The models enable to get access to many internal variables describing process within the turbine. They can be treated as the output or state variables. In order to verify and validate the models, data from the WWER-440/213 reactor and the 4 CK 465 turbine were utilized as the benchmark. The performed simulations have shown good accordance of the static and dynamic models with the benchmark data in steady state conditions. The dynamic models also demonstrated good behavior in transient conditions. The models were analyzed in terms of computational load and accuracy over a wide range of varying inputs and for different numerical calculation parameters, especially time step values. It was found that the detailed dynamic model, due to its complexity and the resultant long calculation time, is not applicable in advanced control methods, e.g. model predictive control. However, the introduced simplifications significantly decreased the computational load, which enables to use the simplified model for on-line control.     

Optimal control of backward stochastic heat equation with Neumann boundary control and noise

This paper considers a stochastic control problem in which the dynamic system is a controlled backward stochastic heat equation with Neumann boundary control and boundary noise and the state must coincide with a given random vector at terminal time. Through defining a proper form of the mild solution for the state equation, the existence and uniqueness of the mild solution is given. As a main result, a global maximum principle for our control problem is presented. The main result is also applied to a backward linear-quadratic control problem in which an optimal control is obtained explicitly as a feedback of the solution to a forward–backward stochastic partial differential equation.     

Generalized solution in singular stochastic control: The nondegenerate problem

This paper is concerned with singular stochastic control for non-degenerate problems. It generalizes the previous work in that the model equation is nonlinear and the cost function need not be convex. The associated dynamic programming equation takes the form of variational inequalities. By combining the principle of dynamic programming and the method of penalization, we show that the value function is characterized as a unique generalized (Sobolev) solution which satisfies the dynamic programming variational inequality in the almost everywhere sense. The approximation for our singular control problem is given in terms of a family of penalized control problems. As a result of such a penalization, we obtain that the value function is also the minimum cost available when only the admissible pairs with uniformly Lipschitz controls are admitted in our cost criterion.     

Evolution of Nonlinear Sloshing in a Tank Near Half the Fundamental Resonance

This article describes the evolution of shallow water waves in a tank that is closed at one end and is periodically forced at half the fundamental frequency at the other end. The nonlinear response occurs at the same order as the linear response and is governed by a forced Korteweg–de Vries ( K dV ) equation. Unlike the corresponding problem for a gas (or the hydraulic limit), there may be nonperiodic (beating) solutions and multiple steady solutions for the same frequency. The addition of a component at the fundamental frequency to the piston input can be used to cancel the nonlinear effects and leave only the linear response in the steady state.     

Nonlinear and bilinear models for chemical reactor control

The dynamic behavior of a continuously stirred tank reactor (CSTR) with an exothermic reversible reaction is studied. The balance equations of the reaction lead to a set of highly nonlinear differential equations. For system analysis and control synthesis the dynamic equation are rewritten as state space model. From this nonlinear model a bilinear model is derived. Then, two optimization problems are solved: The time optimal problem for the nonlinear model and the quadratic problem for the bilinear model. In case of the finite time bilinear-quadratic problem a modified Riccati approximation algorithm for a stabilizing feedback controller is presented.     

A dynamic wind farm aggregate model for the simulation of power fluctuations due to wind turbulence

An important aspect related to wind energy integration into the electrical power system is the fluctuation of the generated power due to the stochastic variations of the wind speed across the area where wind turbines are installed. Simulation models are useful tools to evaluate the impact of the wind power on the power system stability and on the power quality. Aggregate models reduce the simulation time required by detailed dynamic models of multiturbine systems.In this paper, a new behavioral model representing the aggregate contribution of several variable-speed-pitch-controlled wind turbines is introduced. It is particularly suitable for the simulation of short term power fluctuations due to wind turbulence, where steady-state models are not applicable.The model relies on the output rescaling of a single turbine dynamic model. The single turbine output is divided into its steady state and dynamic components, which are then multiplied by different scaling factors. The smoothing effect due to wind incoherence at different locations inside a wind farm is taken into account by filtering the steady state power curve by means of a Gaussian filter as well as applying a proper damping on the dynamic part.The model has been developed to be one of the building-blocks of a model of a large electrical system, therefore a significant reduction of simulation time has been pursued. Comparison against a full model obtained by repeating a detailed single turbine model, shows that a proper trade-off between accuracy and computational speed has been achieved.     

Non‐linear Control of a Hydraulic Piston Actuator without Velocity Information

The mathematical models of hydraulic actuators are known to be non‐linear. Therefore, in order to increase the dynamic performance of the closed‐loop system, we have to take into account the significant non‐linearities of the hydraulic plant in the controller design. In this contribution, we deal with so‐called valve‐controlled translational piston actuators. In general, they have the pleasing property to be exact input‐to‐state linearizable in the sense of the differential geometric control synthesis approach. However, in practical applications it often turns out that those controllers, which have to rely on the knowledge of the piston velocity, have problems in the case of noisy measurements. This is why, we propose an approach where the non‐linear controller is designed in such a way that the control law is independent of the piston velocity. It can be even proven that the closed‐loop system is globally asymptotically stable.     

A finite volume simulation model for saturated–unsaturated flow and application to Gooburrum,Bundaberg, Queensland,Australia

In this paper, a two-dimensional control volume finite-element computational model is developed for simulating saltwater intrusion in a heterogeneous coastal alluvial aquifer system at Gooburrum located near Bundaberg in Queensland, Australia. The model consists of a coupled system of two non-linear partial differential equations. The first equation describes the flow of a variable-density fluid, and the second equation describes the transport of dissolved salt via a form of the Fokker–Planck equation. The outcomes of the work demonstrate that transport simulation techniques provide excellent tools for hydraulic investigations even when complex transition zones are involved.     

Phase Field Approximation of Dynamic Brittle Fracture

The numerical assessment of fracture has gained importance in fields like the safety analysis of technical structures or the hydraulic fracturing process. The modelling technique discussed in this work is the phase field method which introduces an additional scalar field. The smooth phase field distinguishes broken from undamaged material and thus describes cracks in a continuum. The model consists of two coupled partial differential equations - the equation of motion including the constitutive behaviour of the material and a phase field evolution equation. The crack growth follows implicitly from the solution of this system of PDEs. The numerical solution with finite elements can be accelerated with an algorithm that performs computationally extensive tasks on a graphic processing unit (GPU). A numerical example illustrates the capability of the model to reproduce realistic features of dynamic brittle fracture. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Scaling limits for the Ruijgrok–Wu model of the Boltzmann equation

The Ruijgrok–Wu model of the kinetic theory of rarefied gases is investigated both in the fluid‐dynamic and hydro‐dynamic scalings. It is shown that the first limit equation is a first order quasilinear conservation law, whereas the limit equation in the diffusive scaling is the viscous Burgers equation. The main difficulties came from initial layers that we handle here. Copyright © 2001 John Wiley & Sons, Ltd.     

Symbolic computation of normal form for Hopf bifurcation in a retarded functional differential equation with unknown parameters

Based on the normal form theory for retarded functional differential equations by Faria and Magalhães, a symbolic computation scheme together with the Maple program implementation is developed to compute the normal form of a Hopf bifurcation for retarded functional differential equations with unknown parameters. Not operating as the usual way of computing the center manifold first and normal form later, the scheme features computing them simultaneously. Great efforts are made to package this task into one Maple program with an input interface provided for defining different systems. The applicability of the Maple program is demonstrated via three kinds of delayed dynamic systems such as a delayed Liénard equation, a simplified drilling model and a delayed three-neuron model. The effectiveness of Maple program is also validated through the numerical simulations of those three systems.     

A Maximum Principle for SDEs of Mean-Field Type

We study the optimal control of a stochastic differential equation (SDE) of mean-field type, where the coefficients are allowed to depend on some functional of the law as well as the state of the process. Moreover the cost functional is also of mean-field type, which makes the control problem time inconsistent in the sense that the Bellman optimality principle does not hold. Under the assumption of a convex action space a maximum principle of local form is derived, specifying the necessary conditions for optimality. These are also shown to be sufficient under additional assumptions. This maximum principle differs from the classical one, where the adjoint equation is a linear backward SDE, since here the adjoint equation turns out to be a linear mean-field backward SDE. As an illustration, we apply the result to the mean-variance portfolio selection problem.     

Radial basis function networks for internal model control

In this paper radial basis function (RBF) networks are used in the framework of nonlinear internal model control (IMC). A nonlinear IMC strategy is proposed that includes explicit input weighting. This strategy yields a control law in form of an analytical expression if a control-linear RBF model is used. Important implementation issues such as disturbance modeling and state estimation are discussed and an extended Kalman filter approach is proposed for combined state and model parameter estimation. Simulation studies for a continuous stirred tank reactor with multiple steady states indicate that the proposed control strategy is well suited for the control of unstable nonlinear systems.     

Optimal control of water level oscillations in surge tank of hydropower station with long headrace tunnel under combined operating conditions

This paper aims to realize the optimal control of water level oscillations in surge tank of hydropower station with long headrace tunnel under combined operating conditions. Firstly, the definitions of four typical combined operating conditions are presented, and the optimal control objectives of water level oscillations in surge tank under combined operating conditions are proposed. Then, the concept of the most favorable superimposition time of water level oscillation is proposed, and the analytical solutions of the most favorable superimposition time are derived and verified. Finally, for the superimposition operating condition, the attenuation of the water level oscillation under the most favorable superimposition time is analyzed, a novel control strategy of the load adjustments of the units is proposed, and an application example for the proposed load adjustment mode is presented. The results indicate that the optimal control of the water level oscillations in surge tank under combined operating conditions depends on the selection of the superimposition time and the optimization of the mode of the load adjustments under the superimposition operating condition. The most favorable superimposition time can minimize the extreme water levels in surge tank under the superimposition operating condition, and the analytical solutions under the four typical combined operating conditions are highly accurate. The cascaded load adjustment mode proposed in this paper can realize the ideal attenuation of water level oscillations under superimposition operating conditions, and can obviously improve the flexibility of the unit operations.     

Risk-averse profit-based optimal scheduling of a hydro-chain in the day-ahead electricity market

This paper presents a profit-based model for short-term hydro scheduling adapted to pool-based electricity markets. The objective is to determine a feasible and realistic operation of a set of coupled hydro units belonging to a small or medium-size hydroelectric company in order to build the generation bids for the next 24 hourly periods. The company is assumed to be price-taker, and therefore, market prices are considered exogenous variables and modeled via scenarios generated by an Input/Output Hidden Markov Model (IOHMM). In order to be protected against low prices scenarios, two different risk-aversion criteria are introduced in the model: a minimum profit constraint and a minimum conditional Value-at-Risk (CVaR) requirement, which can be formulated linearly in the context of the optimization problem. In order to ensure a feasible operation, the model takes into account a very detailed representation of the generating units, which includes forbidden discharge intervals, spatial–temporal constraints among cascaded reservoirs, etc. The non-linear relationship among the electrical power, the net-head and the turbine water discharge is treated by means of an under-relaxed iterative procedure where net-heads are successively update until convergence is reached. During each algorithm stage, previous iterations’ information is used to build the input–output curves. This way, the hydro scheduling problem can be formulated as a MILP optimization problem, where unit-commitment decisions are modeled by means of binary variables. The model has been successfully applied to a real-size example case, which is also presented in this paper.     

Reduced order dynamic model of the flat-plate solar collector

Padé approximation techniques are applied to obtain a reduced order dynamic model for the partial differential system representing the dynamic behaviour of the flat-plate solar collector. According to a prespecified accuracy, the collector is divided into n sections. At any position along each section the reduced order dynamic model is decoupled second order state equation, the input of which is the output of the preceding section. Numerical solutions obtained from the reduced order dynamic model are in very close agreement with the exact solution. Moreover, the computational efforts as well as the computer storage requirements are considerably reduced in comparison with other methods.The results obtained from the dynamic model are compared with those based on a simple steady-state model. The comparison reveals that the steady-state expression may only be used for collectors having a low thermal inertia and a high fluid-stream heat capacity.     

Dynamic modelling of power plant turbines for controller design

A nonlinear dynamic model of a steam turbine is formulated from approximation of fundamental equations, and does not rely on empirical relations. The model can be used as a part of an integrated power system model for dynamic simulation and control system design.