Economics of reservoir sedimentation and sustainable management of dams

Journal of Environmental Management (2001) 61, 149–163 doi:10.1006/jema.2000.0392, available online at http://www.idealibrary.com on

Economics of reservoir sedimentation and sustainable management of dams A. Palmieri† , F. Shah‡ and A. Dinar†*

Accepted practice has been to design and operate reservoirs to fill with sediment, generating benefits from remaining storage over a finite period of time. The consequences of sedimentation and project abandonment are left to the future. This ‘future’ has already arrived for many existing reservoirs and most others will eventually experience a similar fate, thereby imposing substantial costs on society. Such costs could be avoided if sedimentation was minimized and dams were allowed to live forever. The fact that the world’s inventory of suitable reservoir sites is limited provides an additional reason for encouraging the sustainable management of dams. This paper provides a framework for assessing the economic feasibility of sediment management strategies that would allow the life of dams to be prolonged indefinitely. Even if reduced accumulation or removal of sediment is technically possible, its economic viability is likely to depend on physical, hydrological and financial parameters. The model presented incorporates such factors and allows a characterization of conditions under which sustainable management would be desirable. The empirical implementation of the model draws upon the substantial amount of technical information available. We analyze the sustainability of reservoirs, with a focus on the trade-off between such sustainability and the short to medium term benefits which a reservoir is expected to produce. The results show that, for a very wide range of realistic parameter values, sustainable management of reservoirs is economically more desirable than the prevailing practice of forcing a finite reservoir life through excessive sediment accumulation.  2001 Academic Press

Introduction Dams and reservoirs are developed by humans to cope with the variability of water supplies over time. As of today, there are nearly 45 000 large dams (over 15 m tall) in the world, the vast majority of which were constructed after 1950 (ICOLD, 1988; Schnitter, 1994). These dams produce several benefits including irrigation water, hydropower generation, flood control, recreation, fishing, and others. Until now the common engineering practice has been to design and operate reservoirs to fill with sediment slowly. This generates benefits from storage over a finite period of time. With such an approach, the consequences of sedimentation and project abandonment are left to be taken care of An earlier version of this paper was presented at the World Congress of Environmental and Resource Economists, Venice, Italy, June 23–27, 1998. The views in the paper are those of the authors and should not be attributed to the World Bank. Email of corresponding author: emadinar@worldbank. org 0301–4797/00/020149C15 $35.00/0

by future generations. For many dams this future has already arrived, and in some cases earlier than anticipated. Mahmood (1987:ix) summarized the situation of worldwide reservoir sedimentation as follows: ‘‘. . .the average age of man made storage reservoirs in the world is estimated to be around 22 years. The loss of capacity due to siltation is already being felt at a number of structures. It is entirely possible that, unless ingenious solutions are developed, we will lose the struggle to enhance the available water resources’’. For example, the Tarbela reservoir near Pakistan’s capital Islamabad impounds about 15% of the Indus river flows annually. After its first partial filling, in the summer of 1974, the reservoir has been in operation for the last 23 years. Sedimentation in the reservoir is rapidly reducing the economic benefits of the project. Surveys in 1995 indicated that the gross storage capacity of the reservoir has been reduced by about 20% and usable capacity by 15%. It is clear that an intervention is needed since the deposits are putting the operation of the intake works

Ł Corresponding author † Rural Development Department, World Bank, Storrs, CT 06269, USA ‡ Department of Agricultural and Resource Economics, University of Connecticut, 1818 St. NW, Washington DC 20817, USA Received 23 November 1999; accepted 29 September 2000  2001 Academic Press

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at risk. Intervention alternatives include modification of the reservoir operating rules, construction of an underwater dike to protect the intake works and massive tunneling to carry out annual sluicing (see definition in the next section) of incoming sediments. Preliminary estimates indicate cost in excess of $700 million.1 Although the Tarbela case and several others (see, for example, Morris and Fan, 1998; ICOLD, 1997) indicate that there is an increasing concern about this issue in several countries, on a global basis the trend has not changed. In fact most of the reservoirs continue to be designed and operated with the traditional concept of dead storage giving a finite reservoir life-span for granted. This is particularly regrettable since there are several techniques and management alternatives today to address the sediment problem, and their potential use needs to be evaluated. Now that many reservoirs have ‘matured’ in terms of sedimentation accumulation, it is crucial to also consider additional private and social costs of decomissioning (retiring) filled-up reservoirs. There are many environmental problems associated with retirement of dams, several of which are similar in nature to those of retirement of nuclear power plants (Nordhause, 1997). The limited number of remaining suitable dam sites for dam construction is another significant problem associated with the retirement of dams. Retiring a dam is a major environmental, engineering, and socioeconomic undertaking. Yet, few studies exist that document the cost of partial or full retirement of dams. The aging of the world’s stock of dams and reservoirs has prompted recent interest in retirement issues at a global level. ASCE (1997) provides a clear indication that the problem of aging dams is one of great relevance to developed countries. While several developing countries are also experiencing this problem, they tend to concentrate their efforts on developing additional resources, and satisfying the increasing national demands for water and energy. These countries cannot afford to address the problem of aging waterworks unless they constitute an immediate hazard to their populations. 1

This has been done in several cases at concrete dams (e.g. Shamenxia, China). It was done by drilling tunnels through valley abutments in the case of embankment dams (e.g. Tarbela, Pakistan).

The economic literature on dams and reservoirs addresses issues related to cost-benefit analyses of dam projects with and without social and environmental effects (e.g. Goldsmith and Hildyard, 1984; Paranjpye, 1988, 1992), and economic analyses of watershed resource management and soil erosion in the context of dams and reservoirs (e.g. Pearce, 1986; Easter et al., 1991; de Graff, 1996; Doolette and Magrath, 1990). The conceptual literature of the economics of dams is rather limited. A handful of papers address dam safety issues and develop a methodology for incorporating the risk of dam failure in benefit cost-analysis (Baecher et al., 1980; Pate-Cornell and Tagras, 1986; Cocharane, 1989). Krawczyk (1995) presents a framework for managing reservoir levels so as to reduce environmental damage. De Janvry et al. (1995) analyze watershed-management policies and their possible impact on a dam located downstream. In their study, they find that soil erosion control is desirable from the perspective of upstream users. It yields a side-benefit of increasing the life of the downstream dam by 23 years by reducing the sediments being deposited in it. However, no reservoir sediment management strategy is included in their framework. Our paper contributes to the existing literature by providing a new economic model of reservoir level strategy. We suggest in this paper to revisit the concept of ‘reservoir finite life’, which is terminated by sedimentation, and to evaluate a strategy of longterm storage maintenance through appropriate sediment management. We are aware of the fact that a comprehensive analysis must consider a system comprised of the reservoir and its catchment area in order to assess the influence of watershed management on the operation of the reservoir. However, recent literature includes opposite views on the effectiveness of watershed management vis a vis reservoir sedimentation. While some studies (Eakin and Brown, 1939; Easter et al., 1991) suggest direct and quantifiable relationship between watershed management and prevention of reservoir siltation, other (Doolete and Magrath, 1990; Pearce, 1986) suggest insignificant benefits to reservoirs, that result from watershed management. Therefore, at this stage of the research we exclude watershed optimization from the analysis and refer to the watershed

Economics of reservoir sedimentation and sustainable dam management

management as a given. This issue will be addressed in a separate investigation. Although reservoir sedimentation can be a major factor in limiting the productive life of a dam, the economic feasibility of the various sediment management strategies that may be used depends on physical, hydrological and financial parameters. The paper develops a stylized model that incorporates these considerations to help policy makers compare between sustainable and non-sustainable management options, and identifies situations in which sustainable management is economically desirable. The paper draws on the substantial amount of technical information available to analyze the sustainability of reservoirs and the tradeoff between such sustainability and the shortmedium term benefits which the reservoir is expected to produce. The next section reviews sediment management techniques, some of which can be considered for application at various dam sites. Then a mathematical model of dam management is developed and a general solution is presented, using optimal control framework. The model is modified to address certain management issues and demonstrate solution validity under given values of key parameters. The final section concludes the paper and provides several policy implementation suggestions.

Sedimentation and sediment management techniques

Annual sediment yield/reservoir storage

Sediments are generally considered an undesirable but unavoidable consequence of water storage. A correct approach to the design of reservoirs should consider rivers as sources of water and solid elements (sediments). When the natural water-sediment equilibrium is

substantially modified by the creation of a reservoir, the river banks downstream of the impoundment become affected by accelerated erosions. The process may in some cases extend as far as the sea coast. Minimizing alteration to the natural water-sediment equilibrium (in a dynamic sense) is not only desirable for the conservation of storage, but also from the standpoint of river stability downstream of the dam. A sediment balance will eventually be achieved at all sites as a result of either management of natural phenomena or complete siltation of the reservoir. If the reservoir is completely silted, the dam will be overtopped by water and sediments. Whenever possible, sediment management should be sought as a means of maximizing usable storage capacity. The range of specific sediment yields (SSY) into existing reservoirs varies worldwide over three orders of magnitude. Expressing SSY in [m3 /km2 . year] a range from 20 to 5000 was reported (ICOLD, 1997) with a world average in the order of 100. Data on relative silting rate, ratio between annual silting and original storage capacity of the reservoir, are presented in Figure 1 (ICOLD, 1997). They refer to seven major countries including China and the USA. Methods and techniques for achieving the sustainable use of reservoirs are described in several books and papers (Mahmood, 1987; ICOLD, 1989; Annandale, 1987; Doolette and Magrath, 1990; ICOLD, 1997; Morris and Fan, 1998). Each of these methods can be used under particular circumstances. Often a combination of methodologies may bring the most efficient results. A brief list of the sediment management methods includes: (1) reduction of sediment yield by measures in the catchment area (soil conservation measures, correction of landslides and accelerated erosions, reforestation

0.05 0.04 0.03 0.02 0.01 World average 0

China

USA

Romania

Slovakia

Japan

Spain

Algeria

Figure 1. Reservoir silting rates in selected countries. Note: minimum, average and maximum values do not exist for all countries. Minimum, ; average, ; maximum, . Source: ICOLD (1997, question 74).

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etc.), or by debris dams, which intercept coarse grained sediments, in mountainous streams); (2) sediment routing2 through construction of off-stream reservoirs, construction of sediment exclusion structures, and by sediment passing through the reservoir (e.g. sluicing); (3) sediment flushing, whereby the flow velocities in a reservoir are increased to such an extent that deposited sediments are re-mobilized and transported through bottom outlets; and (4) sediment removal by mechanical dredging, or by hydraulic removal (siphoning).3 Environmental aspects must not be overlooked. Sediment flushing must be carefully planned because the sluiced fluids could have unacceptably high concentrations of solids. The polluting potential of the sediments to be released must also be determined. A more complete description of the sediment management techniques can be found in Morris and Fan (1998). Each technique or method may be satisfactory by itself or in conjunction with others. The actual choice of the most convenient strategy is a complex process involving hydrology, hydrogeology, morphology and dam engineering. The results will always be site specific and no standardization is possible apart from the basic principles on which the solution is based. It is possible to illustrate our general ideas by modeling sluicing or flushing as the sediment management techniques. Prior to doing that, we should provide some definitions. Sluicing is an operational technique aimed at reducing the trap efficiency of the reservoir by releasing most of the sediment load with the flow through a dam before the sediment particles can settle. This is accomplished in most cases by operating the reservoir at a lower level during the flood season in order to maintain sufficient sediment transport capacity (turbulent and colloidal) through the reservoir. After the flood season, the pool level is raised to store relatively clear water. As a consequence, the storage capacity is 2

Routing is an operation that allows sediments to pass through a facility without deposition. 3 White et al. (2000) suggest several requirements for successful flushing, that include low level bottom outlets of adequate capacity; quantity of water theoretically available for flushing; quantity of water actually available for flushing; mobility of reservoir sediments (size, harmoring); and reservoir plan shape (long, relatively narrow reservoirs most suited for flushing).

limited to a fraction of the annual runoff and reservoir operation is limited to part of the year. Effectiveness of sluicing operations depends on reservoir morphology and on the availability of excess runoff. Flushing is a technique whereby the flow velocities in a reservoir are increased to such an extent that deposited sediments are remobilized and transported through bottom outlets.4 In the remainder of the paper we will use the term ‘flushing’ also in cases for which the term sluicing would be more appropriate. In other words, we will regard sluicing as a particular aspect of flushing in the physical sense explained in the definition above.

Reservoir operation selection criteria Basson (1997) has introduced two empirical indices to make a preliminary judgement on which reservoir operation mode should be selected for sedimentation control: Kw D S0 /MAR, which is the ratio of reservoir capacity to mean annual runoff and Kt D S0 /MSY, which is the ratio of reservoir capacity to mean annual sediment yield. The importance of these indices is illustrated in Figure 2 where the range of the 177 dam cases reported by Basson (1997) is shown. Approximate contours indicate the areas where most of the cases are concentrated. The diagram permits us to visualize the different modes in which a reservoir may be operated in order to manage the sedimentation process. The availability of excess water flows which can be used to flush sediments out of the reservoir is the single most important factor in assessing which mode of operation is preferable for that particular reservoir. Excess flows are available when the mean annual runoff (MAR) is large in relation to the storage capacity .S0 / of the reservoir or, in other terms, when Kw is less than one. Basson (1997) suggests that no flushing can be carried out for Kw >0Ð2 and that, for Kw <0Ð03 flushing is mandatory and no 4 Rivers with graded sediments and medium slopes are most suited for flushing. Coarse sediments are difficult to move, but usually not present in large quantities. Other techniques may be more effective than flushing when coarse sediments are conspicuous: debris dams in the catchment area, by-passing, tactical dredging, off-stream storage.

Economics of reservoir sedimentation and sustainable dam management 100 000 100%

Kt (storage/MSY)

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75%

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50% of cases within this boundary

Range of cases reported

1000 Storage Flushing 10 1 0.1 0.001

0.01

0.1

1

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Flushing Kw (storage/MAR)

Figure 2.

Basson’s diagram on reservoir operation modes. Modified from Basson (1997).

storage operation can be performed on a sustainable basis. The corresponding limits for Kt are relatively closer (100 and 30, respectively) and reflect the concept that small sediment yields (high values of Kt ) allow full storage operation modes. Most reservoirs have been designed to be large enough to accommodate 100 or more years of sediment accumulation, and are located in the upper right quadrant of the diagram. Since Kw >0Ð2 for these reservoirs, not enough excess water is available for a flushing operation and reservoir drawdown. In these cases density current venting5 is the only possible sediment management method when favorable conditions exist for the application of such a method. Conversely, flushing is mandatory for many reservoirs located in the lower quadrant .Kt <30/: excess runoff is available for flushing .Kw <0Ð03/ and sustained storage operation would result in a very fast depletion of the active capacity.

Salvage value of dams When a dam is operating, some of the sediments will unavoidably be trapped on the reservoir banks. After a few years of operation, if the amount of sediments that leaves the reservoir does not exceed6 the amount 5 A density current consists of the movement of a fluid, of higher density, underneath a fluid of lower density, through a reservoir. 6 Sediment accumulated in previous years can only be removed if the de-sedimentation rate increases during

that enters it (and if other aspects such as the structural integrity remain constant), then the active storage will be lost and the operation of the dam will no longer be economically viable. If a new investment in freeing storage volume will be economically justified, the dam may continue to operate. Otherwise, the dam should be considered for retirement. If the retirement cost of a dam is sufficiently high, its salvage value will be negative. In the majority of the cases, the salvage value of dams is a negative one.7 This fact can often be attributed to the sedimentation process and to the absence of adequate sediment management strategies. Other cases may occur in which the retirement of an old dam is recommended for safety reasons or for substantial modification of the demand (e.g. fishery could become prevalent with respect to an original hydropower purpose). Careful investments in the rehabilitation and refurbishment of existing dams should be aimed not only at increasing safety and productivity, but also at introducing efficient sediment management strategies. This is the life of the reservoir. Sometimes this happens as a consequence of the reduced trap-efficiency of the reservoir. 7 A remarkable example of positive salvage value are the dams being constructed in the Loess Plateau in China (Voegele, 1997). This region is characterized by an extremely high erosion rate which makes any kind of agricultural production unsustainable. The dams being constructed in this area are expected to lose their storage capacity in a very short time (nominally 5 to 20 years), but once silted, the reservoir may become a source of stable, cultivable land, thus creating a positive salvage value if retirement costs are low.

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exemplified by the case of the Tarbela dam in Pakistan (TAMS and Wallingford, 1998). New dam projects would benefit from a design approach similar to that used in China’s Three Gorges Project (Morris and Fan, 1998), which calls for a sediment equilibrium approach. To achieve intergenerational equity (vis a vis sedimentation), a finite useful life of the project would only be acceptable if adequate measures are taken to face, in due time, the retirement issue. Such measures may include setting up a retirement fund and/or investing in a suitable insurance policy.

Economic analysis Model The model presented below is intended to answer two related but distinct questions: (1) is the extra cost incurred in undertaking greater sediment flushing worthwhile in terms of extending the productive life of a dam; and (2) is it economical to extend the life of a dam indefinitely. The first question relates to determining the desirable sediment management strategies, regardless of the ultimate life of the dam, while the second question asks if sustainable management is the best practice (even if it is technically feasible). We assume that the planner has perfect information regarding all parameter values

and functional forms. Net benefits (B) in each period are taken to depend positively on remaining live storage capacity of the reservoir (S) and negatively on the amount of sediment removed (X) in that period. For the sake of simplification, we specify the following form for B: BDPÐ.S Y/ C1 where Y D f .X, Kw / is an inverse sediment flushing function that determines the stored water (Y) needed for flushing X units of sediment when the capacity-water inflow ratio is Kw . The ratio Kw is defined above and used in Figure 3. The rationale for including it in the flushing function is that higher values of Kw will make less surplus water available for flushing, thereby increasing the need for carrying out these operations with stored water. In other words, one should expect @f .X, Kw //@Kw ½0. It is also reasonable to assume that f .0, Kw / D 0, @f .X, Kw //@X ½ 0, and @2 .X, Kw //@X 2 ½ 0. The signs of the last two derivatives respectively imply nonnegative and non-increasing marginal productivity of water used in flushing. As for the other terms in the net benefits expression, P is the unit value of water (in irrigation and/or power generation) and C1 is the periodic O&M cost of the reservoir. Note that our simple expression for B does not allow water stored in one period to be used at a later date. This effectively prevents us

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D Area in which the model indicates non-negative benefits for unsustainable use of the reservoir 1

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Storage Kw (storage/MAR)

Figure 3. Base case superimposed on Basson’s diagram. The present solution features singularity in the vicinty of of Kw D1. This is not shown where the ABC line appears to end against the vertical line .Kw D1/. Analytically, a sharp variation in curvature occurs in the intermediate proximity of Kw D1, but this is not shown.

Economics of reservoir sedimentation and sustainable dam management

from considering reservoirs with Kw >1. Since such dams (with an excess storage capacity) would be serving the purpose of smoothing out multiyear seasonality in natural water supply, it is reasonable to expect that flushing would not be considered practical in these cases anyhow. Let the initial construction cost of the dam be denoted by C2 and its expected operating life-time is T. If T is finite, then the dam has a salvage value, V, which may be positive or non-positive. Using a continuous time formulation, with the subscript t denoting time and r the interest rate, the planner chooses Xt and T to: Z T BÐe rt dt C2 CV Ðe rT Maximize 0

subject to: dS/dtD MCXt

.1/

0Xt M

.2/

where St ½0, and S0 is given. The objective function is simply the discounted sum of net benefits less initial construction costs plus salvage value discounted back to the initial period. Equation (1) describes the behavior over time of St at each t, the remaining live storage goes down by the amount of incoming sediment (M) less the amount flushed .Xt /. This implies a non-increasing time-path of St under the constraint specified in Equation (2). The final conditions state that the starting value of S is known and S is always non-negative.

Analytical solution

associated with the problem, while satisfying a co-state equation, appropriate end-point conditions, and all other constraints specified in the statement of the problem. The relevant Hamiltonian function for our case is: Ht DfPÐ[St f .Xt , Kw /] C1 gÐe

rt

Cqt Ð. MCXt /

where qt is the co-state variable associated with St and has a shadow price interpretation (i.e. it measures the discounted value of a marginal increase in remaining storage capacity).9 It is possible to interpret Ht as a discounted net benefit function that accounts for the value of reduced reservoir capacity in the future due to current choice of Xt (note that Ht is equal to the usual discounted net benefits if Xt DM and it is lower if Xt
Case 1 f .Xt , Kw / is strictly convex in Xt This case allows an interior solution as the Hamiltonian is strictly concave in Xt . Assuming such a solution, the first order condition for maximization of H with respect to the control variable Xt is: @H/@Xt D0) PÐ[@f .Xt , Kw //@Xt ]Ðe

rt

D0)qt DPÐ[@f .Xt , Kw //@Xt ]Ðe The above formulation is quite similar to the exhaustible resources model due to Hotelling (1931). In that respect the dam site, or more precisely, the remaining reservoir capacity can be viewed as a stock of an exhaustible resource that is depleted as siltation occurs. The control variable is Xt (the amount flushed). The problem may be solved using optimal control theory (Seierstad and Sydsaeter, 1987). The standard procedure is to select a time path of the control variable to maximize the Hamiltonian function8 8 The Hamiltonian function is an expression equivalent to the Lagrangian function that is used for optimization

Cqt rt

.3/

The above equation simply says that sediment removal should continue until the discounted marginal benefit in terms of increased storage made available (i.e. qt ) is of an objective function subject to constraints in the case of a static problem. The Hamiltonian is used in solving dynamic optimization problems with the help of optimal control theory. 9 Units used in the empirical implementation of this model are as follows: B, $/year; C2 , $; V, $; P; $/m3 ; C1 , $/year, S, m3 /year; Y, m3 /year; M, m3 /year; X, m3 /year. The implied units for H and q would be $/year and $/m3 , respectively. 10 For a more detailed and general economic interpretation of optimal control theory, see Dorfman (1969).

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just equal to the discounted marginal cost in terms of the value of water needed to be sacrificed. The co-state equation for this problem is: dq/dtD @H/@SD PÐe

rt

.4/

Equations (3) and (4) imply that q is positive but continuously declining in the range of the interior solution. Using Equations (3) and (4), we also get: dXt /dtD[rÐ@f .Xt , Kw //@Xt 1]/@2 f .Xt , Kw //@Xt2 .5/ Given our assumptions on the flushing function f .Xt , Kw /, Equation (5) suggests that Xt may show a declining tendency for relatively small values of r, while it may possibly increase for some period with relatively high values of r. This result is consistent with intuition: for example, the damage done by sedimentation implies a progressive loss of reservoir capacity, so that with a low r, higher initial values of X may be selected, and consequently X would need to decline in the future to meet the conditions of a non-sustainable program; on the other hand, X may start low with a high value of r, and may rise for some time to lead to a sustainable outcome in the future. These two examples are, of course, only illustrative and do not cover the entire spectrum of possibilities. Indeed, a sufficiently low r could very well result in a sustainable outcome with a steady-state starting from tD0. Note that a sustainable program implies that dXt /dtD0 in the steady-state, but Equation (5) need not hold at that stage since it occurs when Xt D M, which is a corner solution. In a similar way, Equation (5) is not necessarily applicable when Xt D 0, which is another corner solution. Since the terminal time is to be determined optimally, we now proceed to discuss the procedure for doing so. The following endpoint condition is necessary for an interior solution with finite T: HT D @.V Ðe

rT

C1 gÐe

rT

//@T)fPÐ[ST f .XT , Kw /] CqT Ð. MCXT /DrÐV Ðe

rT

.6/

One would expect the minimal level of S to be reached by time T, so that the marginal economic value of preserving the remaining storage capacity is 0

at that instant; in other words, qT D 0. Given the assumption that @f .0, Kw //@X D0, this implies that XT D 0, which is reasonable since no advantage is to be gained from flushing at the terminal instant. Substituting this information in Equation (6) yields: PÐST .C1 CrÐV/D0

.7/

Equation (7) says that the terminal value of S should be such that it just covers general O&M costs plus the interest costs of deferring a positive salvage value (or minus the interest benefits of deferring a negative salvage value). Note that structural considerations may require ST to be greater than .C1 CrÐV//P, in which case this larger terminal value would be used. In the following discussion, we will assume that such structural considerations are not binding, but the analysis can be extended quite easily to accommodate them. Another point to note is that if V is negative and rÐV exceeds C1 in absolute value, then it will not be optimal to retire the dam. This makes perfect economic sense and may well explain why certain old dams continue to be operated even though their annual benefits do not fully cover O&M costs. Aside from these qualifications, Equation (7) helps determine ST , which can be used in equations (1) and (5) to solve for T with a given value of S0 . Once T is known, the time paths of Xt (and Yt ) can be specified completely, thereby allowing the optimized value of the objective functional to be calculated. The above discussion assumes a finite optimal T. If it is infinite, the program involves a steady-state with Xt DM for t½T Ł . We may have T Ł D0, in which case the net benefit calculations are straightforward. However, as the discussion of Equation (5) above indicates, the solution allows Xt to start low and rise to the value M by t D T Ł . Hence, this possibility must be permitted in any computer algorithm written to solve the problem. In a similar way, the other possibility which cannot be ruled out a priori is Xt D0 for 0tt0 , where t0 may be less than or equal to T (with T D .S0 ST //M, and ST is calculated using Equation (7)). In this case t0 DT would represent the situation in which sediment removal is never economically desirable.

Economics of reservoir sedimentation and sustainable dam management

Case 2 f .Xt , Kw / is linear in Xt , namely, f .Xt , Kw /DX Ðg.Kw /

In this case, the Hamiltonian Ht is linear in Xt , and maximization with respect to Xt requires investigation of the switching function st , where: st D@H/@Xt D PÐg.Kw /Ðe

rt

Cqt

.8/

Note that H Dst ÐXt CPÐ.St C1 /Ðe rt qt Ð M. Thus, if st > 0, then Xt D M maximizes H, whereas Xt D 0 maximizes H if st < 0. These two possibilities represent the corner solutions, while the singular solution (in which X is indeterminate in the range 0 Xt M) would occur if st D0. The co-state Equation (4) still applies. Integrating this equation between 0 and t, we have: qt D.P/r/Ð.e

rt

1/Cq0

.9/

Substituting the above in Equation (8) yields: st D[.P/r/ PÐg.Kw /]Ðe

rt

.P/r/Cq0

.10/

This expression indicates that st D 0 can generally hold for at most one value of t. Thus, the possibility of the singular solution occurring over any positive length of time is ruled out for most parameter values.11 Next, we need to check the stability of the corner solutions. This can be done by examining the sign of dst /dt. If dst /dt>0 for all t, then a switch from Xt DM to Xt D0 at a later date is impossible (since st cannot become negative after staring out as positive), but a switch from Xt D0 to Xt DM at a later date is allowed. In a similar way, if dst /dt<0 for all t, then the converse holds. The only case which definitely results in stability is when dst /dtD0 for all t. Differentiating both sides of Equation (8) with respect to time and using Equation (4) gives: dst /dtDPÐ[rÐg.Kw / 1]Ðe

rt

.11/

11 The only exception is when g.K /D1/r and q DP/r, w 0 but this case is too specific to be of general interest.

Equation (11) says that the discount rate and the capacity-water inflow ratio are fundamental in determining the stability properties of the corner solution since these parameters uniquely establish the sign of dst /dt for all t (with a given form of g(Ð)). If a corner solution happens to be stable, calculations of the time path of St and the value of the objective function are trivial. However, if a switch cannot be ruled out a priori, then the timing of the switch, or switches, will need to be determined to perform these calculations. Equation (11) implies that the sign of dst /dt does not change over time if r and Kw are fixed. This rules out multiple switches. As a consequence, a computer algorithm that allows for at most one switch may be written to solve the entire problem.

Empirical specification and simulation results Further characterization of the solution requires empirical specification of the flushing function. This function is likely to vary considerably from one dam site to another. Hydrological conditions, hydraulic characteristics of the water ways, and reservoir operating rules (of which there are several) will be the important determinants of the function for any specific case. Here we report on an analysis that was carried out applying a linear flushing function, which is calibrated using information from a published source (Basson, 1997). In flushing operations, the ratio of volume of water needed to volume of sediment sluiced—WD ratio—typically ranges from 7 to 50 (Basson, 1997). We use the value of WD D 20 as our base value12 for the case in which Kw D 1. This would happen when mean annual runoff (MAR) equals initial live storage .So /, so that all flushing must be done with stored water. As MAR increases relative to So , and Kw decreases, greater amounts of runoff (surplus) water become available for flushing, which would lower the ratio of 12

The WD ratio of 20 is based on data from Chinese sites, where the reservoir is sluiced for the whole 3 months summer period. For flood sluicing, the WD ratio should be 10. For flushing operations it is even lower. In that respect, our empirical estimate is rather conservative.

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stored water needed to remove sediments. We define the latter to be an adjusted WD ratio (AWD), and use the information provided in Basson (1997) to infer the following simple relationship: AWDD3C17Kw

.12/

The linear flushing function implied by Equation (12) is: Y D.3C17Kw /ÐX.

.13/

Substituting these results into Equations (10) and (11), respectively, yields: st D[.P/r/ PÐ.3C17Kw /]Ðe .P/r/Cq0 dst /dtDPÐ[rÐ.3C17Kw / 1]Ðe

rt

.14/ rt

.15/

These equations are helpful in describing the nature of the solution. For instance, when rD0Ð05 and Kw D1, we see that st D 20PCq0 and dst /dtD0. In this case, the solution is completely stable for given initial reservoir capacity, mean sediment yield, water price, and other parameters. If . 20PCq0 />0, then Xt DM for all t and we have the sustainable outcome, whereas if . 20P C q0 / < 0, then Xt D 0 until t D T. In the latter case, the terminal time T can be calculated by using the following equation: MÐTDS0 .C1 CrÐV//P

.16/

Equation (16) is obtained by integrating both sides of Equation (1) between 0 and T (assuming Xt D0), and using Equation (7) to substitute for ST . It simply says that the total incoming sediment over the time horizon T should just equal the effective available capacity of the storage reservoir to accommodate silt. Equation (15) also implies that if rD0Ð05 and Kw <1, then dst /dt<0. In this case, a switch from Xt D0 to Xt DM cannot occur, but the reverse is allowed. The opposite implication holds when r>0Ð05 and Kw D1, since now dst /dt> 0. Whether such switches actually occur for reasonable parameter values is an empirical question. In order to answer such a question, and more generally, to analyze the results for our model in relation to the classification system suggested by Basson (1997),

we used the software GAMS to solve our optimization problem.13 The objective function and constraints are as specified above (see Equations (1) and (2)), with BDPÐ.S Y/Ð C1 , and Y D .3 C 17Kw / Ð X. The base values for the parameters used in the program’s initial run are as follows: PD$0Ð03/m3 , rD0Ð05, V D $0, MAR D 1000Ð106 m3 , Kw D 1, and KT D 25 years. In addition, we assumed C2 D0Ð15ÐST Ð, and C1 D0Ð02ÐC2 . Also, note that the above parameter values of MAR, KT and Kw imply S0 D1000Ð106 m3 and MD40Ð106 m3 .14 This base case resulted in an optimal solution with T D23 years, ST D120Ð106 m3 , and aggregate net benefit of $8Ð119Ð107 . Recall that a finite value of T is synonymous with a non-sustainable outcome. Given r D 0Ð05 and Kw D1, no switching was expected and none happened. A sustainable outcome in this case would have resulted in an aggregate net benefit of $8Ð7Ð107 . Raising KT tends to narrow the gap between the aggregate net benefits of the two outcomes, but nonsustainability dominates even for such high values of KT D200 (and T D181), although the difference in monetary values is small by that point ($3Ð5401Ð108 vs. $3Ð5400Ð108 ). As may be expected, ceteris paribus lowering of Kw should improve the economic value of the sustainable outcome. Indeed, with Kw D0Ð5 and all other parameters held to their base values, we find that the value of the sustainable outcome is $6Ð36Ð107 and it dominates any non-sustainable outcome. Note that the maximized aggregate net benefits are substantially lower than those obtained when Kw D1. This is reasonable, since Kw D0Ð5 13

This software is capable of solving a wide range of linear and non-linear programming problems. In most cases, it is user friendly—one simply needs to specify the objective function, constraints, and parameter values before running the optimization program. However, our problem presents a unique complication, namely that T is to be determined optimally while also being used to discount V in the objective function. The help provided by Jerome Krueser in writing a suitable GAMS program to address this issue is gratefully acknowledged. The program is available from the authors upon request and the software may be purchased from GAMS Development Corporation (Email: [email protected]). 14 We tried to aim for consistency in selecting these base values—in addition to relying on broad consensus. For example, $0Ð15/m3 of stored water is generally considered a reasonable construction cost figure for relatively large reservoirs (say S0 >500 Mm3 ), and the assumption that annual O&M costs are about 2% of initial construction cost of a dam is also quite common in the economic analyses of dams.

Economics of reservoir sedimentation and sustainable dam management

implies that only half of the original MAR is now available for potential storage. The sustainable outcome with Kw D 0Ð5 tends to dominate as KT is raised above 25—and increasingly so, up to a point—which is consistent with the fact that a smaller sediment load requires a lower fraction of the stored water for flushing. Conversely, the advantage of the sustainable outcome declines with lower values of KT , until finally at KT D19, the nonsustainable outcome dominates (with an aggregate net benefit of $2Ð09Ð107 , vs. $1Ð784Ð107 for the sustainable outcome). Additional simulations show that, the critical value of KT (around which one strategy dominates over the other) goes down as Kw decreases. The solid line ABC in Figure 3 depicts the path of this critical KT for variations in Kw , holding the other parameters at their base values. Points located above the line ABC are characterized by sustainable outcomes, whereas points that are below the line ABD represent situations with negative net benefits. Therefore, the triangular zone CBD indicates cases where positive net benefits are associated with nonsustainable use of the reservoir. It is interesting to observe that this area starts at Kw D0Ð2, which is also the critical value suggested by Basson for a storage operation mode. Another point to note is that the line ABD roughly follows the lower limit of the envelope representing the cases reported by Basson (1997). The presence of a portion of the envelope area below the limit line should be interpreted as the present inability of our model to fully depict the behavior of

the system at very low Kw values. Actually, only about 5% of the total sample of dam cases fall below the limit line in that area. Of course, given the way our benefit function (B) is formulated, we are also unable to consider cases with Kw >1, but as mentioned above this should not be considered a major limitation given our interest in situations that at least permit sustainable outcomes via flushing.

Sensitivity analysis The above empirical formulation of the model was also used to investigate the sensitivity of the results to the variation of key parameters such as: price of water; construction cost, interest rate, and salvage value. Some of the results are shown below in Figures 4, 5 and 6. The base value for P is $0Ð03/m3 . An increase in P causes the boundary separating the sustainable and nonsustainable regions to move down, while the opposite happens when there is a decrease in P (see Figure 4). For example, when Kw D 0Ð01, the critical value of KT is 4 for PD0Ð03 as opposed to critical KT D3 when PD0Ð05 and critical KT D6 when P D 0Ð02 (for the same value of Kw ). Also note that any change in P influences ST through Equation (7) and T through equation (16). As a consequence, it is easy to see that, for nonsustainable dams, ST will go down and T will increase if P increases. This is quite intuitive because with a higher P, a lower value of ST will allow the annual fixed costs to be covered, thereby allowing T to increase.

100

Kt

Basson's boundary

10 P = 0.02 P = 0.03 P = 0.05

1 0.001

0.01

0.1 Kw

Figure 4.

Impact of variation in water price (P) on sustainable frontier.

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Kt

Basson's boundary

C2/S0 = 0.030

10

C2/S0 = 0.015

1 0.001

0.01

0.1

1

Kw

Figure 5.

Impact of variation in construction cost C2 /S0 ) on sustainable frontier. 100 Basson's boundary

Kt

160

10

r = 0.10 r = 0.05 r = 0.01

1 0.001

0.01

0.1

1

Kw

Figure 6.

Impact of variation in interest rate (T) on sustainable frontier.

The base value for unit cost of construction is $0Ð15/m3 . An increase in it causes the boundary to move up (see Figure 5). For example, with Kw D0Ð2 and the other parameters held at their base values, the critical value of KT rises from 9 to 19 when the unit cost of construction is raised from $0Ð015/m3 to $0Ð030/m3 . Note that any change in unit cost of construction also changes C1 because of our assumption that C1 D 0Ð02C2 . Thus, Equation (16) would imply that, for nonsustainable dams, ST should go up while T should decrease, and this does indeed happen. The base value for r is 0Ð05. The impact of varying r between 0Ð01 and 0Ð1 was studied. As one would expect, the above boundary tends to move up if r is raised—in other words, the sustainable region becomes

smaller. Such a shift is illustrated in Figure 6. Of course, if a dam is non-sustainable before and after the interest rate change, then its optimal life remains the same since the salvage value, V, is assumed to be zero, and we have the same sized reservoir. In this case, we are dealing with the corner solution Xt D0 for all t. Therefore, Equation (16) may be used to calculate the optimal life of the dam. Note that if V >0 in Equation (16), then an increase in r leads to a decrease in T, whereas the opposite happens if V <0. This is easy to interpret in economic terms. For example, a larger r will imply a larger interest cost of continuing with the dam if V >0, and this should put a downward pressure on T and an upward pressure on ST . The base value for V is 0. A negative V may be expected to cause the critical value

Economics of reservoir sedimentation and sustainable dam management

of KT to move down, while a positive V should move it up. The intuition is that a negative salvage value should encourage a move towards sustainability (since that is a way of avoiding this cost), and conversely, a positive salvage value would create an incentive towards a non-sustainable outcome. Our simulations support this intuition. For example, with V D 0Ð5C2 , Kw D 0Ð9, and the other parameters held at their base values, the critical value of KT is 65, while this critical value is 56 if V D 0 and 49 if V D 0Ð5C2 . The observation made above regarding corner solutions and impact on optimal life applies again: if a dam is in the non-sustainable region before and after any change in V, Equation (16) may be used to determine the impact of the change on T. For instance, if V increases, Equation (16) predicts a decrease in T —the economic explanation, once again, is that the increase in V increases the interest cost of continuing with the dam. The base value for MAR is 1000. The boundary is not sensitive to changes in MAR, which may be explained by the fact that, for given Kw and KT , an increase in MAR simultaneously increases S0 and M. The value of aggregate net benefits for any particular Kw and KT does, of course, change because of the change in S0 and M.

Conclusion and policy implications The evidence collected in dams around the world indicates a declining trend of storage capacity due to siltation in excess to the production of new water storage. With continuing siltation, not only is the capacity to store water reduced, but also the social cost of potential hazardous pollution associated with non-usable dams is increased. We believe that policy-makers throughout the world will need to address the worsening water situation and growing environmental hazard. This paper provides a tool to analyze and compare various sedimentation management options at the dam level. Other policy instruments are also needed to reduce the negative impact of non-sustainable management. This is especially critical as the stock of dams and reservoirs is aging and more countries face the inevitability of decommissioning more dams.

It is apparent that there are several variables that impact the direction and the value of the optimal solution. For example: r (interest rate), P (price of water), C1 (construction cost), MAR (mean annual runoff), and V (salvage value) have a significant impact on the optimal program, which is the life time and the concluding status of the dam. In this policy discussion we do not advocate sustainable versus nonsustainable policies because under suitable conditions either may be found preferable. The variable V, the salvage value of the dam, is a pivotal variable in the problem. Although not directly a policy variable, it significantly affects the solution. A substantial negative salvage value should encourage a move towards sustainability since that is a way of avoiding this cost. Conversely, a substantial positive salvage value would create an incentive towards a non-sustainable outcome. In cases where non-sustainable solutions is a stable preference what so ever value of V, if V increases, the life time T will decrease because the increase in V increases the interest cost of continuing with the dam. Two basic options may be taken into consideration to incorporate the salvage value of the dam in the project implementation plan: (1) retirement fund policy15 : If intergenerational equity is a concern and the salvage value (V) is negative, it may be desirable to set aside some of the net benefits (NB) at each instant as a retirement fund for use at time T. One simple plan would be to invest a constant amount, k, in each period so that the accumulated proceeds from the investment at time T are equal to V. Since this policy uses a market instrument (calculated by using the market rate of interest) to address intergenerational equity, it is likely to avoid the controversies associated with approaches that artificially lower the interest rate in benefit cost analysis to favor future generations. It must be pointed out that the magnitude of V may not be known with certainty at time 0. It might therefore be desirable to adjust the amount k up or down at some future date if more is learned about the true value of V. (2) Insurance policy: Another option for handling the intergenerational equity issue in the 15

Retirement here, includes a wide array of options: alteration of a dam to fit different purposes, raising, partial or total removal, etc.

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presence of uncertainty is to seek insurance against unexpectedly large negative values of V. This policy may be supplemented by a fund designed to cover the ‘normal’ cost of alteration retirement (and/or a possible co-payment to the insurance company). The value of water in storage is also an important policy variable. Higher values of P increase the range of conditions under which a sustainable solution is preferred. In case of a non-sustainable solution, higher values of P allow the annual fixed costs to be covered, thereby allowing T to increase. Several modifications of the model can be considered for various policy purposes. (1) Although the model is described in terms of planning a new dam, it is equally applicable to the modification both in structure and in management of an existing dam. One would simply replace the set-up cost with the appropriate initial cost of modifying the dam. (2) Although we assume that the excess of MAR over S0 has a zero price and its only possible use is in flushing, this assumption could be relaxed in a simple way, by introducing a price Ps .Ps

Acknowledgements The authors acknowledge comments from George Annandale, Gerrit Basson, Julian Dumanski, and Rodney White, and the editorial work by Lili Monk. Valuable and constructive comments were provided by three journal referees.

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