# Governor Design for a Hydropower Plant with an Upstream Surge Tank by GA-Based Fuzzy Reduced-Order Sliding Mode

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Dynamics of a Hydropower Plant with an Upstream Surge Tank

_{R}(m), tunnel length L

_{1}(m), tunnel cross-section area A

_{1}(m

^{2}), head of surge tank H

_{s}(m), cross-section area of the surge tank A

_{s}(m

^{2}), penstock length L

_{2}(m), penstock cross-section area A

_{2}(m

^{2}) and tail water head H

_{0}(m). Both H

_{R}and H

_{0}are assumed to be constant. The conduits between the turbine and tail water lake are assumed to be of negligible length. The water in the surge tank is considered as being in steady flow conditions.

#### 2.1. Water Hammer

^{−1}) is acoustic (water hammer) wave speed, H (m) is the piezometric head, A (m

^{2}) is the cross-sectional area of the pipe, Q (m

^{3}· s

^{−1}) is the cross-sectional average flow rate, g = 9.81 m· s

^{−2}is the gravitational acceleration, D (m) is the pipe diameter, ρ (kg· m

^{−3}) is the water density, τ

_{w}(N· m

^{−2}) is the shear stress at the pipe wall, l is the spatial coordinate along the pipeline and t is the temporal coordinate. According to the Darcy–Weisbach equation [25], we have:

_{r}(s) is the penstock water reflection time, T

_{w}(s) is the water inertia time, H

_{f}is the hydraulic loss, s is the complex variable in the Laplace domain and $\mathcal{L}$[·] denotes the Laplace transfer. T

_{w}(s) is defined as $\frac{L{Q}_{r}}{gA{H}_{r}}$, where A (m

^{2}) is the cross-section area of the pipe, H

_{r}(m) is the rated head, Q

_{r}(m

^{3}· s

^{−1}) is the rated flow rate and L (m) is the pipe length.

#### 2.2. Tunnel

_{1}, Equation (6) can be deduced from Equation (5) under the assumption of the inelastic water hammer effect.

_{1}(per unit) is the head deviation of the tunnel input and output, q

_{1}(per unit) is the flow rate deviation of the tunnel input and output, H

_{f1}is the hydraulic loss in the tunnel and T

_{w1}(s) is the water inertia time of the tunnel, defined as $\frac{{L}_{1}{Q}_{r}}{g{A}_{1}{H}_{r}}$.

#### 2.3. Penstock

_{2}(per unit) is the head deviation of the penstock input and output, q

_{2}(per unit) is the flow rate deviation of the penstock input and output, H

_{f2}is the hydraulic loss in the penstock and T

_{w2}(s) is the water inertia time of the penstock, defined as $\frac{{L}_{2}{Q}_{r}}{g{A}_{2}{H}_{r}}$.

#### 2.4. Surge Tank

_{s}(per unit) is the water head deviation of the surge tank, q

_{s}(per unit) is the flow deviation of the surge tank and ${T}_{s}=\frac{{A}_{s}{H}_{r}}{{Q}_{r}}$ (s) is the filling time of the surge tank.

#### 2.5. Wicket Gate and Servomechanism

_{y}(s) is the response time of the wicket gate servomotor.

#### 2.6. Hydro-Turbine

_{r}(kN· m) and X

_{r}(r/min) are the rated turbine torque and rated speed, and G

_{max}(mm) is the maximum equivalent gate position. The coefficients in Equation (10) can be calculated at each operating point.

#### 2.7. Generator and Grid

_{a}(s) is the generator unit mechanical time, and e

_{g}is the rotational loss coefficient. T

_{a}is determined by $\frac{{J}_{g}{X}_{r}^{2}}{3580{P}_{r}}\times {10}^{-3}$, where J

_{g}(kN· m

^{2}) is the generator unit inertia torque, P

_{r}(kW) is the generator-rated power output and X

_{r}(r/min) is the rated speed.

## 3. Control Design

#### 3.1. Design of the Reduced-Order Sliding Mode Controller

_{4}with a known gain K

_{E}[28]. The expression of the state is written as:

**x**= [x

_{1}x

_{2}x

_{3}x

_{4}]

^{T}, x

_{1}= x, x

_{2}= m, x

_{3}= y,

**c**= [c

_{1}c

_{2}c

_{3}c

_{4}]

^{T}and c

_{i}(i = 1, 2, 3, 4) is constant.

_{sw}is the switching control, and u

_{eq}is the equivalent control law. Their expressions are deduced below. In Equation (14), u

_{eq}and u

_{sw}are model-based, so that we have to obtain a simplified model described by the three independent and measurable state variables. This is also the reason that we name the controller “reduced-order sliding mode”.

_{3}; the generator and grid component is also depicted by a first-order equation with the state x

_{1}; and the turbine component is depicted by an algebraic equation. To get a reduced-order model depicted by the limited state variables, we have to analyze the remaining components, i.e., the tunnel, the surge tank and the penstock. In Figure 3, the transfer function of the three components can be written as:

_{w1}is usually larger or much larger than T

_{w2}in Equation (15). Thus, one method to reduce the order of Equation (15) is to use the steady value of the unit step response of the first term on the right of Equation (15) instead of its original expression, and other two terms are kept unchanged. Along the route, the reduced-order expression of the three components can be described by a first-order equation with the state x

_{2}as:

**x**= [x

_{1}x

_{2}x

_{3}x

_{4}]

^{T}is the state vector, and d(t) is the signal of the load disturbance; the state matrix 𝔸, input vector 𝔹, output vector ℂ and disturbance vector 𝔽 are shown in Appendix A.

_{eq}works when the system states keep sliding on the sliding surface. Differentiate S with respect to time t in Equation (13); let $\dot{S}=0$; and substitute the nominal model of Equation (17) into $\dot{S}=0$. Then, we can obtain:

_{sw}is obtained as:

_{M}is satisfied, we have $\frac{\mathrm{d}V}{\mathrm{d}t}\le -\kappa {S}^{2}-(\eta -{\mathbb{D}}_{M})\left|S\right|<0$ in Equation (23), so that the reduced-order governing system Equation (17) possesses the asymptotic stability under the control law Equation (22) in the sense of Lyapunov. However, the control law Equation (22) will be applied to the original system. Consequently, it is necessary to analyze whether the reduced-order sliding mode controller is able to stabilize the original system or not.

**Proof.**

#### 3.2. GA-Based Parameter Optimization

#### 3.3. Design of the Fuzzy Inference System

- If $S\dot{S}$ is PB, then Δκ is PB.
- If $S\dot{S}$ is PM, then Δκ is PM.
- If $S\dot{S}$ is Z, then Δκ is Z.
- If $S\dot{S}$ is NM, then Δκ is NM.
- If $S\dot{S}$ is NB, then Δκ is NB.

_{0}and m

_{1}are constant. Figure 4b shows the output surface of the designed fuzzy inference system using the input $s\dot{s}$ and the output Δκ. The final value of κ regulated by the fuzzy logic is determined by:

_{0}is a basic value.

**Figure 4.**(

**a**) Membership functions of the linguist labels; (

**b**) Output surface of the fuzzy inference system.

## 4. Simulation Results

_{r}= 153 MW, H

_{r}= 312.0 m, Q

_{r}= 53.5 m

^{3}, X

_{r}= 333.3 r/min, ${L}_{1}=9387$ m, A

_{1}= 49.6 m

^{2}, A

_{s}= 113.04 m

^{2}, L

_{2}= 470 m, A

_{2}= 16.61 m

^{2}, J

_{g}= 4.0 × 10

^{4}kN·m

^{2}. The mathematic model in Section 2 is able to depict the operating condition in which one turbine is fed by one penstock. Under such an operating condition, H

_{f1}= 0.036 and H

_{f2}= 0.027. The time constants T

_{a}, T

_{y}, T

_{w1}, T

_{w2}and T

_{s}are 8.113, 0.500, 3.312, 1.244 and 659.224 s, respectively. Turbine coefficients under the operating points of Case 1 and Case 2 are determined in Table 1. A value of 0.200 is picked for K

_{E}, the gain of the additional state. The parameter vector of the sliding surface on the operating point of Case 1 is optimized by the improved GA as

**c**= [500 35 63 600]

^{T}. The parameters of the fuzzy inference system are set as m

_{0}= 2 and ${m}_{1}=\frac{1}{3}$. Since min Δκ = −1, κ

_{0}and η are selected as one and 0.5 from the viewpoint of the system stability in Equation (23).

Coefficients | e_{x} | e_{y} | e_{h} | e_{qx} | e_{qy} | e_{qh} | e_{g} |
---|---|---|---|---|---|---|---|

Case 1 | −1.000 | 1.000 | 1.500 | 0.000 | 1.000 | 0.500 | 0.210 |

Case 2 | −0.260 | 0.322 | 0.722 | 0.000 | 0.573 | 0.325 | 0.210 |

#### 4.1. Load Rejection

_{p}= 4.3086, K

_{i}= 0.8938 and K

_{d}= 2.8925 [30]. Another set of gains K

_{p}= 0.89, K

_{i}= 0.30 and K

_{d}= 2.00 are selected by trial and error [31].

_{g0}once the disturbance is injected into the governing system. All of the system states coordinate each other to make the whole governing system respond faster and better under the action of the reduced-order SMC governor, as shown in Figure 6. Further, we can also explain the reason that the presented control method possesses better performance from the viewpoint of information. The reduced-order SMC governor makes full use of the measurable information of the governing system to decide the final control input, whereas the two PID governors just employ the proportional, integral and derivative of x to formulate the final control input, only utilizing a limited part of the system information.

**Figure 6.**Simulation results of 10% load rejection on the operating point of Case 1 by the reduced-order sliding mode control (SMC) governor, compared to two PID governors.

#### 4.2. Robustness Testing

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix

## Appendix A. Reduced-Order Model

## Appendix B. Proof of Theorem 3.1

**Proof.**

_{2}(s) − G

_{2r}(s) as the error between the original system and the reduced-order one. Then, the block diagram in Figure A1b is obtained. We can simplify the block diagram shown in Figure A1b by block diagram algebra. This precess is illustrated in Figure A2, where $\delta =\frac{\Delta}{{G}_{2r}\left(s\right)}$.

**Figure A1.**Block diagram. (

**a**) The original governing system; (

**b**) The reduced-order governing system with error.

_{1}and the output ε

_{2}. Equation (A1) can be obtained in the frequency domain.

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**MDPI and ACS Style**

Xu, C.; Qian, D.
Governor Design for a Hydropower Plant with an Upstream Surge Tank by GA-Based Fuzzy Reduced-Order Sliding Mode. *Energies* **2015**, *8*, 13442-13457.
https://doi.org/10.3390/en81212376

**AMA Style**

Xu C, Qian D.
Governor Design for a Hydropower Plant with an Upstream Surge Tank by GA-Based Fuzzy Reduced-Order Sliding Mode. *Energies*. 2015; 8(12):13442-13457.
https://doi.org/10.3390/en81212376

**Chicago/Turabian Style**

Xu, Chang, and Dianwei Qian.
2015. "Governor Design for a Hydropower Plant with an Upstream Surge Tank by GA-Based Fuzzy Reduced-Order Sliding Mode" *Energies* 8, no. 12: 13442-13457.
https://doi.org/10.3390/en81212376