Modelling and Internal Fuzzy Model Power Control of a Francis Water Turbine

Abstract

This paper presents dynamic modelling of a Francis turbine with a surge tank and the control of a hydro power plant (HPP). Non-linear and linear models include technical parameters and show high similarity to measurement data. Turbine power control with an internal model control (IMC) is proposed, based on a turbine fuzzy model. Considering appropriate control responses in the entire area of turbine power, the model parameters of the process are determined from a fuzzy model, which are further included in the internal model controller. The results are compared to a proportional-integral (PI) controller tuned with an integral absolute error (IAE) objective function, and show an improved response of internal model control.

1. Introduction

In HPPs, a digital turbine governor is an indispensable part of the control system. The stability of frequency, active power control, water flow control, turbine start-up procedure and emergency shut-down implemented with algorithms are major functions of digital turbine governor. Connected to Supervisory Control and Data Acquisition (SCADA), the unit controller, the excitation system and the digital voltage controller, it enables the operator to change the operating states of generator, i.e., the rotational speed, active power, reactive power and voltage.

With reference to achieving suitable control results and exploring the dynamic responses of a hydro power plant unit, it is necessary to obtain a mathematical hydraulic model of the hydro turbine and other parts of water system. Different types of hydraulic non-linear models were proposed in [1]: a non-linear model assuming a non-elastic water column, a model including travelling waves, and a model including a surge tank effect. In [2–7], the authors applied a non-linear or linearised model with a non-elastic water column to study the transient behaviour of a Francis turbine. Chen et al. provided a nonlinear dynamical model with surge tank presented with state space equations, analysed with bifurcation diagrams, phase orbits and spectrograms [8], however a comparison to the real system was neglected. In [9] authors developed interesting non-linear and linearised hydro turbine models with transient stability analysis in the free and open source Power System Analysis Toolbox (PSAT) software, although a conventional controller without a sophisticated tuning method was used.

Many recommendations of modelling, design and testing control systems for hydraulic turbines are described in international standards [10,11] and a few modern methods in [12,13], nevertheless in most cases a proportional-integral-derivative (PID) or proportional-integral (PI) controller is used for speed, active power, water level, flow and wicket gate opening control. However, the parameter determining process is usually taken on site acceptance tests; some recommendation limits are proposed in [11].

In [14] a particle swarm method was applied as a strategy in an improved gravitational search algorithm for identification of an optimal water turbine model and controller parameters. Fang et al. introduced the particle swarm optimization method of controller tuning in water turbine frequency control [15]. In [16] an internal model control (IMC) tuned PID controller in comparison with Ziegler-Nichols and singular frequency based tuning was studied for frequency control of hydropower system with a water hammer effect. The design of a PID controller based on sensitivity margin specifications was introduced in [17], where the author proposed the controller with respect to sensitivity peak specification. As a result in these cases, authors obtained optimal tuned PID parameters for water turbine control that could not necessarily result in optimal transient behaviour due to simplifications of the models and nonlinearity of real processes at different operating points.

In this paper, the focus is on mathematical modelling and the fuzzy power control approach of a Francis turbine of Aggregate 1 at hydro power plant (HPP) Moste. In the second section, the paper includes a comparison of non-linear and linearised models, with and without the surge tank of the Francis Turbine 1 in HPP Moste. In the third section, a fuzzy model of the Francis turbine has been applied based on a first order linearised model. Section 4 presents a fuzzy IMC controller with tuning parameter T to ensure a suitable dynamic response in working points and middle sections. At the end of Section 4, the comparison to the PI controller has been applied, tuned by the integral absolute error (IAE) objective function.

2. Non-Linear and Linearised Model of Turbine Dynamics

2.1. Description of the Hydro Power Plant

Slovenian hydro power production consists of three HPP chains placed on the Sava, Soca and Drava Rivers. As the first HPP on the Sava River chain with a unique accumulative operation type, HPP Moste plays an important role in the Slovenian hydro power plant generation portfolio.

HPP Moste is part of the HPP Moste and HPP Zavrsnica water system depicted in Figure 1. Besides 2,940,000 m³ of useful volume of water storage, a −6.25 m water level deviation is allowed. After the water intake object, there is a surge tank connected by a tunnel. The pressure penstock connects the surge tank with a powerhouse, comprised of two Francis turbines (HPP Moste) and one Francis turbine (HPP Zavrsnica) with a joint tail-water conduit.

2.2. Turbine Model with Water Conduit

Turbine dynamics with the penstock are derived from second Newton's law [1,18] described with Equation (1):

$$\frac{d\overline{q}}{dt}=({\overline{h}}_0-\overline{h}-{\overline{h}}_f-{\overline{h}}_{sp})\cdot g\cdot \frac{A}{l}$$

(1)

where q̄ is turbine flow rate in m³/s, with the penstock area section A in m² and length l in m. In Equation (1), h̄₀ is the static head of the water column; h̄ is the head at the turbine admission; h̄_f is defined as friction head loss; h̄_sp is the tail water head loss in m units and g is the gravity acceleration.

With reference to water inertia time [10]:

$$T_w=\frac{Q_r}{g\cdot H_r}\cdot \text{∑}_{i=1}^n\frac{L_i}{A_i}$$

(2)

Equation (1) becomes expressed in per unit:

$$\frac{dq}{dt}=\frac{1-h-h_f-h_{sp}}{T_w}$$

(3)

In Equation (2), the water inertia time T_w includes the characteristics of all water passages with area sections A_i; corresponding lengths L_i with rated discharge Q_r; rated head H_r and gravity g. The change of water flow in the conduit is expressed in Equation (3), where h is the head at turbine; h_f is the friction head loss in the penstock and h_sp is the tail water head loss per unit. Water flow through the turbine [5] is given by:

$$q=G\cdot \sqrt{h}$$

(4)

where G is the wicket gate opening per unit of the Francis turbine and h is the head at the turbine admission per unit.

Turbine power P_m in p.u. is expressed by:

$$P_m=A_t\cdot h\cdot (q-q_{nl})-D\cdot G\cdot \mathrm{\Delta }w$$

(5)

where Δw is the speed deviation; D is the damping factor and q_nl is defined as no-load turbine flow. Proportionality factor A_t as described in [1] is given by Equation (6):

$$A_t=\frac{P_{m,r}}{P_{g,r}\cdot h_r\cdot (q_r-q_{nl})}$$

(6)

where P_m,r is the maximum turbine rated power; P_g,r is the generator rated power; h_r is the rated head and q_r is the rated turbine flow in p.u.

With purpose of analysing dynamic responses, linearisation in the surrounding area of operating point R was made with a Taylor series procedure (considering first two elements), e.g., linearisation of the non-linear head at turbine admission h:

$$h=\frac{q^2}{G^2}=\overline{h}+{(\frac{\partial h}{\partial q})}_R\cdot (q-\overline{q})+{(\frac{\partial h}{\partial G})}_R\cdot (G-Ḡ)$$

(7)

with calculation of partial derivation and substitution of:

$$\mathrm{\Delta }h=h-\overline{h};\mathrm{\Delta }q=q-\overline{q};\mathrm{\Delta }G=G-Ḡ$$

(8)

We obtain linearised Equation (9):

$$\mathrm{\Delta }h=\frac{2\cdot \overline{h}}{\overline{q}}\cdot \mathrm{\Delta }q-\frac{2\cdot \overline{h}}{Ḡ}\cdot \mathrm{\Delta }G$$

(9)

The linearisation process for all relations between Equations (3) and (5) was made with the real parameters of the Francis turbine 1 on HPP Moste. According to the synchronised generator and the fixed network frequency Δw = 0 assumption, only the first part of Equation (5) will be concerned in further analysis. The transfer function between turbine power P_m and wicket gate opening G is proposed in Equation (10):

$$\frac{\mathrm{\Delta }Pm}{\mathrm{\Delta }G}=\frac{b_0\cdot s+b_1}{a_0\cdot s+a_1}$$

(10)

with calculated numerator coefficients:

$$b_0=-\sqrt{\overline{h}}\cdot (\overline{q}-q_{nl})\cdot A_t\cdot T_w$$

(11)

$$b_1=\sqrt{\overline{h}}\cdot \overline{h}\cdot A_t-\sqrt{\overline{h}}\cdot A_t\cdot (\overline{q}-q_{nl})\cdot (\frac{2\cdot \overline{q}\cdot Q_r^2\cdot (f_{p1}+f_{p4})+2\cdot \overline{q}\cdot a_0\cdot Q_r^2+a_1\cdot Q_r}{H_b})$$

(12)

and denominator coefficients in Equations (13) and (14):

$$a_0=\frac{1}{2}\cdot \frac{\overline{q}}{\overline{h}}\cdot T_w$$

(13)

$$a_1=1+\frac{1}{2}\cdot \frac{\overline{q}}{\overline{h}}\cdot (\frac{2\cdot \overline{q}\cdot Q_r^2\cdot (f_{p1}+f_{p4})+2\cdot \overline{q}\cdot a_0\cdot Q_r^2+a_1\cdot Q_r}{H_b})$$

(14)

In connection to negative numerator coefficient b₀ in Equation (11), the non-minimal phase type of dynamic system is observed. In Figure 2, the matching of the nonlinear and calculated linear models is presented with simulation results at operating point P_m = 0.3422, G = 0.4416, q = 0.431, h = 0.9524, h_f = 0.0076 and h_sp = 0.015, all p.u. with real ramp reference of the wicket gate opening, in comparison to real system measurement.

2.3. Surge Tank Modelling

In high-head HPPs, the surge tank is often included in the water system for providing the limits of water pressure fluctuation and water speed decreases in water tunnels and penstocks. There are some other important functions of surge tanks that influence the water system [18,19]: hydraulic separation of water tunnel and pressure penstock, decreasing of water hammer effect evaluation, improved results of turbine control, and acceleration of water level stabilisation.

With reference to model improvement, the surge tank with Equations (15) and (16) were included in the previously described Francis turbine model with conduit:

$$Q_s=Q_w+Q_T$$

(15)

where Q_s is the water flow in the tunnel; Q_w is the surge tank water flow (positive or negative) and Q_T is the turbine water flow. Water level of surge tank [1,18] is described in Equation (16):

$$\frac{dz}{dt}=\frac{Q_s-Q_T}{C_S}$$

(16)

with C_s expressed in Equation (17), where A_w is surge tank section area:

$$C_S=\frac{A_w\cdot h_{\mathit{\text{base}}}}{q_{\mathit{\text{base}}}}$$

(17)

The surge tank implemented on HPP Moste has two extended horizontal chambers with enlarged upper and bottom surface areas, A_{1_kom} and A_{3_kom}. Therefore, three different areas A_w (A_{1_kom}, A_{2_kom} and A_{3_kom}) were added to the model, depending on absolute surge tank water level calculated with relation:

$$A_w=f(z_{\mathit{\text{abs}}});z_{\mathit{\text{abs}}}=H_{\mathit{\text{abs}}}-H_{b,max}-(z\cdot H_{b,max})$$

(18)

where H_abs is the nominal level of HPP water storage; H_b,max is the maximum available head and z_abs is the absolute level of surge tank.

The friction loss of head h_s and h_fr with friction factors f_p2 and f_p3 in the water tunnel are presented in Equation (19):

$$h_s=Q_s\cdot \left|Q_s\right|\cdot f_{p2}{h}_{fr}=f_{p3}\cdot {Q_s}^2$$

(19)

and the loss of head due to narrowed intake of the surge tank:

$$h_w=Q_w\cdot \left|Q_w\right|\cdot f_0$$

(20)

where f₀ is the intake loss coefficient. The net head of surge tank is expressed in p.u. with Equation (21):

$$H_{\mathit{\text{vod}}}=z-h_w$$

(21)

which is also presented in differential Equation (22), describing flow in the water tunnel:

$$\frac{d{Q}_s}{dt}=\frac{H_0-h_s-h_{fr}-H_{\mathit{\text{vod}}}}{T_{w1}}$$

(22)

and Equation (3) is modified in Equation (23):

$$\frac{d{Q}_T}{dt}=\frac{H_{\mathit{\text{vod}}}-h_s-h_r-h_{sp}}{T_{w2}}$$

(23)

where H₀ is static head of water column; T_w1 is the water inertia time of the water passages before the surge tank intake and T_w2 is the water inertia time between the surge tank and the Francis turbine. A dynamic system with differential Equations (16), (22) and (23) was linearised with the same Taylor series procedure as in the model without the surge tank. The third order transfer function of the HPP Moste turbine model with a conduit and surge tank was derived for three different turbine power working points: P_m = 0.3408 p.u., P_m = 0.6657 p.u. and P_m = 0.9684 p.u., with the transfer function shown in Equation (24):

$$\frac{\mathrm{\Delta }{P}_m}{\mathrm{\Delta }G}=\frac{s^3\cdot d_0+s^2\cdot d_1+s\cdot d_2+d_3}{s^3\cdot c_0+s^2\cdot c_1+s\cdot c_2+c_3}$$

(24)

Coefficients of the numerator, denominator and all other parameters in simulation are related to the real system (Francis turbine 1 HPP Moste with penstock and surge tank) and are described in Appendixes A.1. and A.2.

With the real ramp reference (slower opening rate) obtained from the HPP measurement (from 5 to 6 MW), the transient response is practically without undershot (Figure 3). With 1260 s of the simulation time, approximately ±1% of fluctuating power is presented after the reference was changed (ΔG = 0.0903).

Surge tank level H_vod and water flow Q_w were also observed (Figure 4) during the transient response in the nonlinear model (ramp reference). The maximum difference of the water level in the surge tank was 0.0096 p.u. (0.68 m).

3. Fuzzy Model

In connection with the high similarity of transient responses (models with/without surge tank) depicted in Figure 3, a first order model (without surge tank) is used for fuzzy modelling. A transfer function derived from Equation (10) with time constants T₁, T₂ and gain K is presented in Equation (25):

$$G_s(s)=\frac{K\cdot (1-T_1\cdot s)}{1+T_2(s)}$$

(25)

Due to the nonlinearity of the process, a fuzzy model was realised to determine the parameters in the middle sections of the working points. Membership functions, describing degrees of membership for input parameter P_m and output parameters K, T₁ and T₂ are presented in Figure 5.

Membership functions with triangular and trapezoidal shapes were determined based on the knowledge acquired from operating experiences and intuitiveness. Fuzzy rules added to the fuzzy model are presented with rule list interpretation:

IF turbine power Pm = small, THEN:

− K = medium

− T1 = small

− T2 = small

IF turbine power Pm = medium, THEN:

− K = high

− T1 = medium

− T2 = medium

IF turbine power Pm = high, THEN:

− K = small

− T1 = high

− T2 = high

Determination of output fuzzy set was made with an aggregation method (maximum) and with a defuzzification process, where the centroid method (also called centre of area, centre of gravity) was used to obtain crisp values of parameters [20]. Turbine model parameters in working points and middle sections obtained with fuzzy model are listed in

Table A1. Model parameters.

Parameter	Symbol	Value
Generator rated power	Pg, r	9 MVA
Maximum turbine rated power	Pm, r	7.5 MW
Rated turbine power	Pr	6.692 MW
Maximum turbine flow	Qmax	14 m3/s
Rated turbine flow	Qr	13 m3/s
No-load flow	qnl	0.19 p.u.
Rated turbine head p.u.	hr	0.824 p.u.
Static head p.u.	h0	0.9740 p.u.
Static head	hb	68.62 m
Gravity constant	g	9.80665 m/s2
Water inertia time	Tw1	2.2361 s
Water inertia time	Tw2	0.7267 s
Water inertia time	Tw3	2.9628 s
Surge tank intake loss	f0	0.0005
Friction loss penstock	fp1	0.0136
Friction loss water tunnel	fp2	0.0136
Additional friction loss of the water tunnel	fp3	0.01
Additional friction loss of the penstock	fp4	0.003
Surge tank parameter	Cs	225.5150 s
Turbine damping	D	0.5 p.u./p.u.
Surge tank upper chamber area	A1_kom	316.6 m2
Surge tank main chamber area	A2_kom	44.2 m2
Surge tank bottom chamber area	A3_kom	276.4 m2

.

4. Fuzzy IMC Control and Comparison to PI Control

In connection to IMC calculation, the fuzzy model of the Francis turbine proposed in Section 3 is used. With IMC, a model is parallel-added to the process (Figure 6) and the difference between the output of the process and model is used for the feedback signal.

In an ideal case, when disturbance d = 0 and assuming the model is identical to the process, the feedback is inactive. Conversion to classic control form is made with Equation (26):

$$G_R(s)=\frac{G_{\mathit{\text{IMC}}}(S)}{1-G_{\mathit{\text{IMC}}}(s)\cdot G_M(s)}$$

(26)

In the first step, the minimal phase and all-pass part of the process transfer function are expressed with Equation (27):

$$G_s(s)=\frac{K\cdot (1-T_1\cdot s)}{1+T_2(s)}=\frac{K\cdot (1+T_1(s))}{1+T_2(s)}\cdot \frac{1-T_1(s)}{1+T_1(s)}$$

(27)

The ideal IMC controller is an inversed minimal phase part that results in mirror-transformed zeros over the ordinate axis:

$$G_{\mathit{\text{IMC}}}^{(\mathit{\text{opt}},\mathit{\text{ideal}})}(s)=\frac{1+T_2(s)}{K\cdot (1+T_1\cdot s)}$$

(28)

Due to easier realisation and increased robustness, a PT1 (first order) filter with tuning parameter T is added to the ideal IMC controller:

$$G_{\mathit{\text{IMC}}}^{(\mathit{\text{opt}},\mathit{\text{real}})}(s)=\frac{1+T_2(s)}{K\cdot (1+T_1\cdot s)\cdot (1+T\cdot s)}$$

(29)

Considering Equation (26) and Figure 6, conversion to a classic controller G_R(s) is expressed with Equation (30):

$$G_R(s)=\frac{1+T_2\cdot s}{K\cdot T\cdot T_1\cdot s+(K\cdot T+2\cdot K\cdot T_1)}\cdot \frac{1}{s}$$

(30)

Referring to different parameters in working points from the fuzzy model, the IMC controller is presented with a matrix of transfer functions M_R(s), with different tuning parameter T:

$$M_R(s)=\left[\begin{array}{c}{G}_{R1}(s);P_m=0.34081;T=0.3\\ G_{R2}(s);P_m=0.6657;T=0.2\\ G_{R3}(s);P_m=0.9684;T=0.08\end{array}\right]$$

(31)

The results of IMC with a nonlinear model and surge tank are presented in Figure 7, with 0.05 p.u. step response of turbine power and output of controller signal (wicket gate opening) in the first (G = 0.4416, P_m = 0.3408) and second working point (G = 0.6367, P_m = 0.6657). In the third working point (G = 0.9, P_m = 0.9684), a −0.05 p.u. step response was realised. The results shows a small undershoot (first and second working point) and a small overshoot at the third working point.

The fuzzy IMC controller algorithm was additionally formed with a fuzzy system to determine parameter T depending on turbine power P_m; while other parameters K, T₁ and T₂ are observed from a turbine fuzzy model. Results after the defuzzification process (centroid method) of tuning parameter T are presented in

Table 1. Francis turbine model parameters in working points and middle sections, calculated with fuzzy model.

Turbine power	K	T1	T2
Pm = 0.34081	1.3583	0.7610	0.6614
Pm = 0.484	1.3565	0.8922	0.7150
Pm = 0.542	1.3935	1.2863	0.9139
Pm = 0.6657	1.4470	1.3918	0.9506
Pm = 0.7859	1.3537	1.6694	1.0515
Pm = 0.881	1.2929	2.0928	1.2593
Pm = 0.9684	1.2671	2.3073	1.3348

.

If a hydraulic model of turbine and water passages are known, a PID or PI controller could be used for turbine power control, tuned with IAE expressed with Equation (32) or integral time absolute error (ITAE) objective function (33) with a simulation procedure [11].

$$\mathit{\int }\left|P-P_r\right|\cdot dt=min$$

(32)

$$\mathit{\int }t\cdot \left|P-P_r\right|\cdot dt=min$$

(33)

In Equations (32) and (33), P_r is the reference power and P is the actual power output. In connection to the IMC result validation, a PI controller with the same model was proposed. The tuning process was made with the IAE objective function, expressed with Equation (32), and tested in the same working points as the fuzzy IMC controller. A similar fuzzy system (with input—P_m, output K_p and K_I) was realised with calculated PI parameters, listed in Table 2.

Figure 8 presents turbine wicket gate opening G, turbine flow Q, head at turbine admission h; friction head loss h_f; and tail water h_sp for both controllers in P_m = 0.484 middle section. The influence of the PI controller proportional part fast response results in a 6% higher variation of head h at turbine admission, which consequently causes a higher under-control effect of turbine power, compared to fuzzy IMC control.

The fuzzy IMC controller in middle section P_m = 0.484 provides 0.200 p.u. undershoot M_p in comparison to 0.915 p.u. with PI controller (Figure 9). Comparison results for the four middle sections of working points are presented in Table 2 (bolded values).

The smallest under-control power effect with 0.200 p.u. undershoot is obtained with IMC in P_m = 0.484 (

Table 2. Tuning parameter T for the internal model control (IMC) and proportional-integral (PI) parameters (K_p and K_I).

Turbine power	T (IMC)	KP (PI)	KI (PI)
Pm = 0.34081	0.3000	0.2983	0.3154
Pm = 0.484	0.2786	0.2912	0.3016
Pm = 0.542	0.2178	0.2709	0.2594
Pm = 0.6657	0.2000	0.2764	0.2437
Pm = 0.7859	0.1671	0.2520	0.2318
Pm = 0.881	0.1064	0.2329	0.1932
Pm = 0.9684	0.0800	0.2225	0.1759

); however, the highest under-control effect 1.088 p.u. is observed at P_m = 0.7859 with PI controller. Turbine head drop Δh has the smallest value 0.020 p.u. with the IMC controller in P_m = 0.881; and the highest value 0.086 p.u. with the PI controller in P_m = 0.484 middle section.

In comparison to the PI controller, fuzzy IMC presents better results, with lower undershoots M_p in all four middle sections and with approximately the same settle time on the step reference of turbine power P_m.

Fuzzy IMC also provides smaller variations of head Δh at turbine admission, which ensures fewer oscillations of the HPP water system.

5. Conclusions

In this paper, the dynamic modelling of a Francis turbine 1 (HPP Moste) with and without a surge tank with linearisation and IMC power control was proposed. With the actual HPP technical data (water tunnels and conduits), the presented models show a good match to the real data measurement.

With linearisation, a first-order transfer function with non-minimal phase was used for fuzzy modelling. Concerning appropriately defined input and output membership functions, a fuzzy rule list and the defuzzification process (centroid method), the crisp parameters of the Francis turbine model were calculated.

Based on the fuzzy model, a fuzzy IMC controller was proposed with calculation of tuning parameter T, depending on turbine power. For IMC validation, the PI controller with IAE objective function and fuzzy inference system (similar to IMC) was proposed. A comparison of the fuzzy IMC and PI control with maximum undershoot M_p and head variations Δh criterion in the middle sections of working points shows that IMC has better results due to the smaller undershoot on step response, smaller head variations Δh and easier tuning process (one tuning parameter T) in case of acquired model.

The applied methodology with the extension of working points with different static heads can be used as algorithm in digital turbine governor in order to ensure better results in water turbine power control.