Stochastic control of a micro-dam irrigation scheme for dry season farming
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Abstract
Micro-dams are expected to be feasible options for water resources development in semi-arid regions such as the Guinea savanna agro-ecological zone of West Africa. An optimal water management strategy in a micro-dam irrigation scheme supplying water from an existing reservoir to a potential command area is discussed in this paper based on the framework of stochastic control. Water intake facilities are assumed to consist of photovoltaic pumping system units and hoses. The knowledge of current states of the storage volume of the reservoir and the soil moisture in the command area is fed-back to the intake flow rate. A system of two stochastic differential equations is proposed as a model for the dynamics of the micro-dam irrigation scheme, so that temporally backward solution of the Hamilton–Jacobi–Bellman equation determines an optimal control, which represents the optimal water management strategy. A computational procedure using the finite element method is successfully implemented to provide comprehensive information on the optimal control. The results indicate that the water initially stored in the reservoir can support full irrigation for about 80 days under the optimal water management strategy, which is predominantly based on the demand-side principle. However, the volatility of the soil moisture in the command area must be reasonably small.
Keywords
Micro-dam irrigation Dry season farming Stochastic control Hamilton–Jacobi–Bellman equation Finite element method1 Introduction
Irrigation in general involves taking water from natural or artificial sources and supplying it to command areas where crops are grown. Different scales and types of irrigation schemes are being practiced throughout the world. In some regions, small reservoirs or tanks are built to provide common artificial water sources for rural communities. Tank irrigation systems in Monsoon Asia have a long history, and discussions on better management of them are still ongoing. Arumugam and Mohan (1997) proposed a supply-side decision support system for the release strategy from Veeranam Lake, which is a large historic irrigation tank in South India with a capacity of 26 million m^{3}. Unami et al. (2005) investigated the optimal water management strategy for a system of three tanks irrigating paddy fields of 40 ha in Japan. Srivastava et al. (2009) examined a rainwater recycling system consisting of six tanks and five open wells for the full irrigation of the command area of 23 ha in Eastern India. Guerra et al. (1990) analyzed the hydrological processes in terraced rice fields with farm reservoirs extending in central Luzon, Philippines. A farm reservoir of their study with a capacity of 2,000 m^{3} can support dry season rice irrigation for an area of 0.4 ha. Panigrahi et al. (2007) carried out experimental studies on a rainfed rice-mustard cropping system consisting of a small on-farm reservoir with a capacity of 61 m^{3} and a farm area of 800 m^{2}. It was revealed that such a system is an economically feasible option for small-scale supplemental irrigation. Smallholder irrigation schemes are also being developed in semi-arid regions of Sub-Saharan Africa (SSA), where erratic rainfall and high evaporation are serious constraints on agricultural production. Micro-dams, dugouts, sub-surface runoff harvesting tanks, and rooftop rainwater harvesting systems are all different rainwater harvesting practices found in the Makanya catchment of rural Tanzania (Pachpute et al. 2009). Here, the smallholder farmers have locally established robust water allocation arrangements among the micro-dams interconnected with furrows (Mul et al. 2011). Makurira et al. (2007) analyzed the water balance of the Manoo micro-dam system, which is part of the Makanya catchment. The storage capacity of the micro-dams in the Makanya catchment ranges from 200 to 1,600 m^{3}. Ngigi et al. (2005 found that small on-farm reservoirs with capacities of 30–100 m^{3} are adequate for supplemental irrigation to produce cabbage in the Laikipia district of Kenya, when they are combined with plot-scale low-head drip irrigation systems, which were introduced in the late 1990s. However, Kulecho and Weatherhead (2005) observed that the drip irrigation system was often discontinued in different parts of eastern Kenya due to lack of maintenance, irrelevant cultural background, and unreliable water supply. The Guinea savanna agro-ecological zone of West Africa is drawing research attention recently because of the extreme contrast between rainy and dry seasons (ILRI, 1993). Thousands of micro-dams are distributed over the upper and central parts of the Volta Basin (Liebe et al. 2005; Liebe et al. 2009; Leemhuis et al. 2009). The command areas of Tono and Dorongo irrigation schemes in the Upper East Region of Ghana are 2,500 and 10 ha, respectively. Mdemu et al. (2009) noticed that the smaller Dorongo scheme, which alone experienced drought, had better water productivity than that in the larger Tono scheme. Faulkner et al. (2008) made comparison between another set of two small reservoir irrigation schemes, namely, Tanga and Weega schemes in the Upper East Region of Ghana, where water demand in the irrigated crop fields was estimated to be in the range of 3.7–5.9 mm/day on weekly basis. In both schemes, the reservoirs’ capacities are almost the same with command areas of 1.6 and 6.0 ha in the Tanga and the Weega schemes, respectively. This variance resulted in different irrigation water depths actually supplied during the experimental period, which were in the ranges of 22.0–37.9 and 8.9–14.4 mm/day in the Tanga and the Weega schemes, respectively. The above two comparative studies of the irrigation schemes in Ghana attributed the reason for better performance of the schemes with scarce water availability to efficient water management, which might reduce some of the water losses as Carter et al. (1999) considered.
From the above-mentioned literature review, it can be inferred that more research effort should be directed toward seeking the maximum performance of the minimum sized irrigation schemes, though the operation of micro-dams is not yet clearly understood in conjunction with small-scale subsistence farming managed at the community level in semi-arid SSA. In this paper, a comprehensive description of an optimal water management strategy for a micro-dam existing in the Northern Region of Ghana, having a capacity of 14,748 m^{3} and fully irrigating a potential command area of 840 m^{2}, is outlined. The large dam capacity relative to the small command area is in fact quite reasonable under the assumption that no inflow to the reservoir can be expected during the whole period of dry season irrigation from mid-November to mid-March, which may be 120 days. The water level of the reservoir monotonically decreases in the dry season, while it is multiply utilized for domestic use, livestock watering, and bricks molding, as commonly practiced in SSA (Senzanje et al. 2008). Importance of the micro-dam as an aquatic habitat is secondary (Unami et al. 2012). Therefore, water management should primarily aim neither to empty the reservoir nor dry up the command area during the irrigation period, regardless of economic performance. A minimum stochastic model is developed to represent the physical processes in the micro-dam irrigation scheme to be optimally managed. The storage volume of the reservoir, the volume of readily available water contained in the soil of the command area, and the intake flow rate from the reservoir for irrigation to the command area are all considered as stochastic processes. A rigorous stochastic control approach that solves the Hamilton–Jacobi–Bellman (HJB) equation is employed. This method has an advantage over the conventional scenario-based optimization with a simulation method that do not take all the events that may occur into account (Srivastava 1996) or with an inexact programming method of predetermining the bounds of uncertainties (Lu et al. 2009).
The analytical or numerical solutions of the HJB equations are being studied mostly in the field of financial engineering where economic indices such as discounted utilities are to be maximized (Morimoto and Kawaguchi 2002; Chaumont et al. 2006; Baten and Miah 2007; Yiu et al. 2010). However, the concept presented here does not explicitly involve the economic aspect of the micro-dam irrigation schemes, but the determination of the optimal water management strategy is mathematically reduced to a stochastic control problem maximizing the expectation of the first exit time from a spatio-temporal domain where the stochastic processes should stay within. A computational procedure based on the finite element method is implemented to give a comprehensive description of the optimal water management strategy as well as the values of expected first exit time.
In the next Sect. 2, the physical details of the micro-dam irrigation scheme as well as the surrounding environment are described. In Sect. 3, the stochastic model is developed to formulate the stochastic control problem. In Sect. 4, being the main objective of this paper, it is demonstrated that the stochastic control problem can be numerically solved to determine the optimal intake flow rate based on the knowledge of current states of the storage volume of the reservoir and the soil moisture in the command area. Section 5 presents the conclusions.
2 Description of the micro-dam irrigation scheme
Estimation of water loss during dry periods in Dam 5
Dry period | Starting date | Ending date | Duration (day) | Water level decrease in 24 h | |
---|---|---|---|---|---|
AM (mm) | SD (mm) | ||||
DP07/08 | November 1, 2007 | March 11, 2008 | 132 | 10.74 | 3.51 |
DP08/09 | October 24, 2008 | March 16, 2009 | 144 | 9.51 | 3.80 |
DP09/10 | October 30, 2009 | February 11, 2010 | 105 | 9.62 | 3.14 |
DP10/11 | November 12, 2010 | February 22, 2011 | 103 | 9.42 | 3.02 |
There is no evidence that water from Dam 5 has been used for dry season irrigation, which is also not common in the area (Yiridoe et al. 2006). One of the motivations for developing the micro-dam irrigation scheme with Dam 5 is that farmers from a nearby community carried out small-scale gardening using the water from another of the six micro-dams, particularly, Dam 1 located at coordinates 09°28′36″N 001°00′50″W, during the dry season of early 2010. An exceptional torrential rain of 110 mm for one night on February 12th recovered the water levels of the reservoirs and might trigger off the farmers’ dry season small-scale gardening. Dam 1 is located 4.8 km downstream from Dam 5 and is rather a simple dugout intended for livestock watering, equipped with an intake pipe to draw water from the reservoir to a cattle trough. The catchment area of Dam 1 is 1,226 ha, including those of Dam 2 through Dam 5. At the time of the small-scale gardening, water was diverted from the cattle trough to basins with beds for the cultivation of Hibiscus calyphyllus, which provides vegetable leaves for human consumption as well as fiber materials. The dimensions of Dam 1 are as large as h_{max} = 1.8 m, A_{max} = 91,733 m^{2}, and V = 79,320 m^{3}, while the total area of the command is 350 m^{2}.
The area immediately downstream side of Dam 5, which is triangular in shape as shown in Fig. 2, is proposed as the potential command area of the micro-dam irrigation scheme. One side of the triangular area is the dam embankment and the other two are planted with trees, so as to control entry of animals. The total area is a = 840 m^{2}, and the land surface slope is less than 1% around an elevation of EL166.35m. Located in the hydromorphic valley bottom, the soil of the area is moderately permeable Dystric Planosols, which is normally used for rice cultivation (CERSGIS 2005). However, rainfed farming system for upland crops such as maize, yam, and groundnut is currently practiced in the area, since Dam 5 bypasses surface water flows which originally covered the valley bottoms during the rainy seasons.
3 Stochastic model and optimal control
4 Computational method and application
Key parameters used for the optimal control of the micro-dam irrigation scheme
T | V | y_{min} = aDθ_{w} | y_{max} = aDθ_{max} | y_{s} = aDθ_{s} | \( \bar{q} = aE_{c} \) |
---|---|---|---|---|---|
120 days | 14,748 m^{3} | 8.40 m^{3} | 58.80 m^{3} | 87.36 m^{3} | 16.80 m^{3}/day |
The numerical solutions of Φ are totally oscillation free. The contour graphs of \( E^{s,x,y} \left[ {\hat{T}^{*} } \right] \) on the x–y plane quantitatively represent the performance of the optimal water management strategy. The minimum of \( E^{s,x,y} \left[ {\hat{T}^{*} } \right] \) is achieved on the boundary of Ω_{x} × Ω_{y} for every s. The sub-domain of Ω_{x} × Ω_{y} where the value of \( E^{s,x,y} \left[ {\hat{T}^{*} } \right] \) is large appears near the boundary x = V with moderate y, and then it expands to the whole Ω_{x} × Ω_{y} as the time s approaches to T. A critical time s_{c}, which is defined as the minimum s where there exists (x, y) such that the expected first exit time \( E^{s,x,y} \left[ {\hat{T}^{*} } \right] \) is greater than 119 days, is on the 38th, 39th, and 96th day for C_{v} = 0.20, 0.30, and 0.40, respectively. In other words, the micro-dam irrigation scheme can be expected to last for about 80 days if C_{v} = 0.20 or 0.30, provided that the reservoir is full and the soil moisture of the command area is moderate at the initial stage. While, the micro-dam irrigation scheme becomes short of water within 24 days if the value of C_{v} is larger as 0.40.
The computational procedure is implemented for different values of the command area a as well, though detailed results are not presented here. When a = 420 m^{2}, half of the original value, the critical time s_{c} is on the 33th, 34th, and 90th day for C_{v} = 0.20, 0.30, and 0.40, respectively. When a = 210 m^{2}, one quarter of the original value, the critical time s_{c} is on the 31th, 32th, and 90th day for C_{v} = 0.20, 0.30, and 0.40, respectively. Qualitative behavior of the numerical solutions of u^{*} and Φ for these values of a is almost the same as that for the original a = 840 m^{2}, except that switching θ shifts to the dry side and the gray zones increase as the value of a becomes small. In practice, soil moisture should be controlled on the dry side if the command area is small, but it has the least effect of prolonging the irrigation period. When a = 1,680 m^{2}, the double of the original value, the critical time s_{c} is on the 110th, 112th, and 115th day for C_{v} = 0.20, 0.30, and 0.40, respectively. In that case there is substantially no possibility of successful dry season farming in the enlarged micro-dam irrigation scheme.
The results obtained above indicate that the rigorous stochastic control approach with the minimum model leads to simple optimal management strategies, which is consistent with common conventional practices of small-scale subsistence farming. This methodology is advantageous over the conventional scenario-based optimization by not requiring many ambiguous model parameters as well as offering feasible solutions at the community level.
5 Conclusions
A stochastic control approach is applied to the optimal water management of an irrigation scheme consisting of a reservoir, a command area, and water intake facilities. Only the first exit time is considered as the performance index, targeting at small-scale subsistence farming where profit is not a primary concern. The computational procedure that solves the HJB equation for the stochastic processes provides comprehensive information on the optimal control. Demonstrations are given for the micro-dam irrigation scheme planned at a site within the Guinea savanna agro-ecological zone of West Africa. The model parameters are estimated by different observed data, with specified values of volatility. No inflow is expected into the reservoir and the evapotranspiration is extremely high during the dry seasons, when the irrigation scheme is operational. This is a very severe condition observed here, which is not encountered in neither the Monsoon Asia nor in East Africa. Though the capacity of the micro-dam is seemingly large relative to the small command area, the computational results indicated that the irrigation scheme cannot last long enough for the cultivation of dry season crops when the volatility is too large. When the volatility is small enough, the initial water stored in the reservoir can support full irrigation for about 80 days under the optimal water management strategy, which is predominantly based on the demand-side principle. The feed-back rule is so simple that water should be taken from the reservoir when the command area requires water, and it is considered feasible at the community level. These facts explain the actual farmers’ practice of small-scale gardening in the smaller command area using the water from the larger reservoir. However, introduction of water intake facilities such as the PV pumping system unit will be a constraint to realizing the potential of the irrigation scheme.
Notes
Acknowledgments
This research was funded by a grant-in-aid for scientific research No. 20255012, from the Japan Society for the Promotion of Science. The authors also thank Mr. F. K. Abagale, University for Development Studies, Ghana, as well as the communities in Tolon/Kumbungu District of Northern Region, Ghana, for their valuable support in the field studies.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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