Optimal operation of a single unit in a pumping station based on a combination of orthogonal experiment and 0-1 integer programming algorithm Open Access

c

Start-stop consumption parameter of pumping station

C

Investment expenses for pumping station, including construction engineering expenses, mechanical and electrical equipment and installation expenses, metal structure equipment and installation expenses, temporary works expenses, independent expenses and reserve expenses (RMB, which is the legal tender of the People's Republic of China)

f

Total water-lifting expense during operation process (RMB)

F

Fitness function

H_i

Head for period i (m)

k_i

Start-stop state of pumping station in period i, set the value to 1 when it is on and 0 when it is off

KT

The number of pumping station's start-stop during operation process

N

Divided time periods during operation process

N

Investment years of pumping station to date

Actual power of period i

P_i

Electricity price of period i (RMB/(kW·h))

Unit water-lifting expense (RMB/10⁴ m³)

Flow of pumping station in period i (m³/s)

t

Reduced service life of pumping station caused by frequent start-stop (h)

T

Actual life of pumping station (h)

Time increment of period i (h)

W_e

Target water extraction of pumping station issued by the administrative department (m³)

Z

Barrier function

Discount rate (%)

Unit efficiency of pumping station (%)

Motor efficiency (%). When the load is greater than 60%, the motor efficiency can be considered basically unchanged and calculated as 94%

Transmission efficiency (%), which refers to the ratio between the power transmitted to the pump and the output power of the motor. The transmission efficiency of direct connection is 100%.

Blade angle of pumping station in period i (°)

Penalty factor

Recently with the construction of the South-to-North Water Transfer Project, a number of studies have investigated optimal schemes for the operation of pumping stations, including single unit stations (Negharchi et al. 2016; Gong et al. 2021) and pumping station groups (Hashemi et al. 2014; Gong & Cheng 2018). In the process of model construction, objective function included minimum energy consumption (Fernández García et al. 2014; Negharchi et al. 2016; Olszewski 2016), maximum benefit (Cheng et al. 2013), or highest efficiency (Lamaddalena & Khila. 2013). The frequent start-stops of a pumping station is common and unavoidable in such optimization schemes. All transient events induced changes in most operating parameters (e.g., discharge, head, rotational speed, and voltage) in pumping stations. The impacts of these changes on the pumping station should be assessed at an early stage of any hydroelectric project. Arce (2001) emphasized the effect of generator unit start-stops on efficiency. Gagnon et al. (2010) indicated that runners were subject to complex dynamic forces which might lead to eventual blade cracking and generate unexpected down time and high repair cost. Trivedi et al. (2013) proposed that transients created both steady and unsteady pressure loading on the runner blade, resulting in cyclic stresses and fatigue development in the runner. In addition, pump stoppage could instantaneously increase the pressure within a pipeline, which was an extreme condition and poses a severe threat to the safety of long-distance water transmission projects. Zhang et al. (2019) provided information for the design of protection devices to alleviate the water hammer effect caused by sudden pump shutdown, and effectively prevented extreme pressure waves caused by rapid valve closure. Stoenescu et al. (2019) proposed that maintenance costs could double the energy production variable cost if wrong operational decisions are applied. These papers have laid a solid foundation for this study, from different fields and views; however, they only considered the effect on the unit caused by frequent start-stops and neglected the optimization scheme from the management perspective.

In the process of model solving, it mainly included dynamic programming methods (Arce 2001; Cheng et al. 2010; Zheng et al. 2016), genetic algorithms (Abkenara et al. 2015), and ant colony optimizations (Ostfeld & Tubaltzev 2008; Mehzad et al. 2019). When the models were solved using modern intelligent algorithms, it was not always possible to satisfy the constraints in a short time. Most intelligent algorithms adopted random sampling, which also had difficulty in solving the equality constraint (Birhanu et al. 2014; SaberChenari et al. 2016). However, the water demand constraint was an equality constraint that was used to minimize waste.

The improved model construction and algorithm were proposed to solve these problems. The blade angle and start-stop scheme were available parameters that could be controlled by end users. Therefore, they were taken into account as the double–decision variables. Meanwhile, a combination of an orthogonal experiment and a 0-1 integer programming algorithm was proposed to explore the optimal solution of the model with equality constraint.

• (1)
  Water demand constraint:

  

  (2)
• (2)
  Start-stop scheme constraint:

  

  (3)
• (3)
  Power constraint:

  

  (4)

The adverse effects on start-stops of a large plant have been studied by many scientists (Barán et al. 2005; Pulido et al. 2006). Yang et al. (2016) studied the wear and tear on hydro power turbines due to primary frequency control. Meniconi et al. (2014) derived an expression for the head envelope damping for turbulent flows in smooth and rough pipes, and indicated that the energy dissipation in unsteady-state pressurized pipe flow was quantified through a key parameter governing the dominance of unsteady friction in transient flows. Bi et al. (2010) proposed that it might induce a serious water hammer event in the pipes as the pump rapidly shut off. Thus, the transients in the pipes could be effectively alleviated by prolonging the valve closing time and reducing the valve opening. Nilsson & Sjelvgren (1997) pointed out that the factors affecting the start-stop consumption of a pumping station included wear of mechanical equipment and failure of control equipment during start-up. The largest cost was maintenance and personnel cost due to wear caused by equipment failure. The reviewed literature showed that frequent start-stops lead to load change, wear, and even fatigue damage, which affected the service life of the unit. However, the start-stop of the plant is inevitable and adjusting the scheme to minimize losses and prolong the service life of the plant is necessary. Therefore, quantification of the start-stop consumption is crucial.

Each day was divided into nine periods and considered the operation management of the pumping station, the peak-valley electricity price, and the impact of model accuracy. The flow rate, unit efficiency, electricity price and average head of different blade angles in each period are shown in Table 2.

The target water extraction (W_e) was 2.00 × 10⁶ m³ and the number of start-stop (KT) was limited to three. Taking periods as the experimental factor, the blade angle of each period was the experimental level. Each factor was separated into five experimental levels within the corresponding range (−4, −2, 0, +2, +4).

An orthogonal experiment method was adopted to construct the L₆₄(5⁹) orthogonal table, and the experimental combination scheme selected by the orthogonal table obtained the theoretical optimal blade angle corresponding to all combinations. The experimental combination of the L₆₄ (5⁹) orthogonal table are shown in Table 3.

Physical constraints:

• (1)
  Water demand constraint:

  

  (7)
• (2)
  Start-stop scheme constraint:

  

  (8)

Equations (7)–(9) were 0-1 integer programming models, which could be solved with a branch-and-bound method. The target water extraction (W_e) was 2.00 × 10⁶ m³ and the number of start-stops (KT) was limited to three. Sixty-four combination schemes selected according to the L₆₄ (5⁹) orthogonal table were successively substituted into equations (7)–(9) to obtain 64 target values of corresponding schemes– expenses, as shown in Table 3.

Through orthogonal analysis, as shown in Table 3, the theoretical optimal solution was obtained (period 1–9, 0°, −4°, 2°, 0°, 4°, −4°, −2°, −2°, −2°). Based on the theoretical optimal solution of blade angle, the 0-1 integer programming model was used to obtain the optimal combination of start-stop state for each period (period 1–9, which is 0, 0, 1, 1, 1, 0, 0, 1, 1). The optimization operation processes are shown in Table 4.

Table 1

Calculation table of the loss

Number .	Name .	Unit .	Amount .
1	The total investment/C	104 RMB	6,903.38
(1)	Construction engineering expenses	104 RMB	253.1
(2)	Mechanical and electrical equipment and installation expenses	104 RMB	4,942.46
(3)	Metal structure equipment expenses and installation expenses	104 RMB	275.09
(4)	Temporary works expenses	104 RMB	83.46
(5)	Independent expenses	104 RMB	782.35
(6)	Reserve expenses	104 RMB	506.92
(7)	Other expenses	104 RMB	60
2	Reduced use time for each start-up and shutdown/t	h	15
3	Total running time/T	year	30
4	Discount rate/	%	8
5	Year of investment to date/n	year	16

Number .	Name .	Unit .	Amount .
1	The total investment/C	104 RMB	6,903.38
(1)	Construction engineering expenses	104 RMB	253.1
(2)	Mechanical and electrical equipment and installation expenses	104 RMB	4,942.46
(3)	Metal structure equipment expenses and installation expenses	104 RMB	275.09
(4)	Temporary works expenses	104 RMB	83.46
(5)	Independent expenses	104 RMB	782.35
(6)	Reserve expenses	104 RMB	506.92
(7)	Other expenses	104 RMB	60
2	Reduced use time for each start-up and shutdown/t	h	15
3	Total running time/T	year	30
4	Discount rate/	%	8
5	Year of investment to date/n	year	16

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Table 4 indicates that the cost of water lifting was 3.29 × 10⁴ RMB, and water-lifting cost was 164.49 RMB/10⁴ m³ when the target water extraction was 2.00 × 10⁶ m³/d. The consumption of the machine's start-stop was considered, which was missing in previous papers (Cheng et al. 2010; Gong et al. 2021). Meanwhile, the start-stops were limited to fewer than three within the operation process. The consumption caused by the machine's start-stop was reduced artificially through management, including economic and security aspects, which was consistent with our expectations (Trivedi et al. 2013; Stoenescu et al. 2019).

With the development of computer technology, genetic algorithms, ant colony algorithms, and other modern optimization algorithms emerge one after another. The characteristics of these algorithms are to find the optimal solution by various random sampling and search approximation methods after the decision variables were discretized in the feasible domain. A genetic algorithm was selected for comparative analysis in this paper and a penalty function method was adopted to process it. The solution steps of the penalty function method are as follows: after selecting the target water extraction W_e, head H_i, electricity price P_i, flow rate Q_i, and efficiency ⁠, the start-stop state k_i and blade angle θ_i of each period were generated randomly within the feasible region. The problem was transformed into an unconstrained problem by using the penalty function method to deal with constraints.

The algorithm terminated after satisfying the condition. The running time and results of the two algorithms were compared, as shown in Table 5.

Table 5 showed that the running time of the proposed algorithm was within 10 seconds while the running time of the genetic algorithm was more than 40 seconds. In addition, the results calculated by the genetic algorithm failed to meet the constraints after several iterations. The optimization efficiency of the proposed algorithm was 4%–10% higher than that of genetic algorithm. The results obtained in a short time did not fully meet the constraints because there was equality constraint in the model and it was difficult to converge through the genetic algorithm. The comparison showed that the results calculated by the proposed algorithm was more rapid and effective than the genetic algorithm.

The results showed that: (1) when KT reached three times or more, the unit water-lifting expense was related to the target water extraction. The larger the target water extraction, the larger the water extraction cost and the smaller the optimization benefit. Therefore, when the target water extraction is large, it is recommended to give instructions and arrange the water-lifting task in advance, which is beneficial to reduce energy consumption. (2) when the target water extraction was 2.5 × 10⁴ m³, the optimization benefit was at a maximum when there were two start-stops; when the target water extractions were 2.0 × 10⁴ m³ and 1.5 × 10⁴ m³, the optimization benefit was at a maximum when there were three start-stops. Therefore, the maximum optimization benefit was obtained when there were fewer than three stop-starts during the operation process at the pumping station.

In view of the large pumping stations, a novel mathematical model was constructed to minimize the water-lifting expenses of pumping stations, and the consumption of start-stops was considered in the model. This complex nonlinear model with double-decision variables could be solved by a proposed combination of an orthogonal experiment and a 0-1 integer programming algorithm. The optimization results of the actual case (Jiangdu No. 4 Station) showed that the proposed algorithm improved the operation efficiency and optimization benefit compared with the modern intelligent algorithm, and solved the complex nonlinear model with equality constraint. This paper researched a single pumping station, but the optimal operation of multi-unit pumping stations will be studied in depth in future research.

This work was supported by the National Natural Science Foundation of China (NSFC) [grant number 52079119].

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.