The Influence of Environmental Constraints on the Water Value †

Ignacio Guisández *, Juan I. Pérez-Díaz and José R. Wilhelmi Department of Hydraulic, Energy and Environmental Engineering, Escuela de Ingenieros de Caminos, Canales y Puertos, Technical University of Madrid (UPM), c/ Profesor Aranguren s/n, Madrid 28040, Spain; ji.perez@upm.es (J.I.P.-D.); joseroman.wilhelmi@upm.es (J.R.W.) * Correspondence: i.guisandez@upm.es; Tel.: +34-913-366705 † This is an expanded version of the conference paper entitled “Influence of the maximum flow ramping rates on the water value” presented in the 5th International Workshop on Hydro Scheduling in Competitive Electricity Markets held in Trondheim, Norway, 17–18 September 2015.


Introduction
This research article is an expanded study based on the conference paper entitled "Influence of the maximum flow ramping rates on the water value" presented in the 5th International Workshop on Hydro Scheduling in Competitive Electricity Markets celebrated in Trondheim, Norway, during the 17th and the 18th of September of 2015 whose proceedings were published in [1].
Water has an economic value in all its competing uses [2].In the framework of a competitive electricity market, the hydroelectric use of this good, so special [3], may define that value as the marginal change in the hydropower producer's expected revenue for a marginal change in its available hydro resources [4].Therefore, in this context, the water value (WV) can be understood as an opportunity benefit which plays an important role between the long-and short-term hydro scheduling models by bringing the consequences of the expected future into the present [5].WV depends on the time of the year and the water content stored in the reservoir of each hydropower plant [6].It is usually described as monotonically decreasing when increasing both the stored water volume and the amount of water inflow [7], because of the subsequent growing risk of spill [8].
It has been more than two decades since the United Nations agreed to encourage the sustainable use of renewable natural resources [9], and during this time hydroelectricity has kept its position Energies 2016, 9, 446 2 of 21 as the first source of primary energy among renewable energies [10].Unfortunately, a plethora of studies [11][12][13][14][15][16][17][18][19] have found that hydropower plants can yield undesirable effects on the ecosystems where they are located, such as changing evapotranspiration, flow regime, reoxygenation of surface waters, sediment transport, temperature, total gas pressure or water depth, and impeding faunal migration.Hence, many countries in the world have imposed [20] or are considering imposing [21] new constraints on hydropower operation in order to achieve that goal.These environmental constraints are usually: minimum environmental flows (ϕ), seasonal minimum values of water release, and maximum ramping rates (ρ), maximum interhourly increments and decrements of water release.On the one hand, ϕ reduce the water volume and head available to produce electricity, and, on the other hand, ρ limit the hydropower plant's ability to change power output levels [22]; that is, these ramps introduce some kind of inertia in the plant operation.
Despite the increasingly important contribution of hydroelectricity to the proper operation of power systems as a result of the rise of the other intermittent renewable technologies [23] and the abovementioned environmental policies, [1] is the only work published so far on the influence of ρ on WV to the best of our knowledge.Hence, this study has two objectives: first, to start filling the knowledge gap of how ϕ affects WV, and second, to compare this influence with that caused by ρ described in [1].This is the reason why a very similar optimisation model and the same case study presented in [1] are used in this study in order to make a fair comparison between the results shown here and those of [1].
In order to compute the WV of the hydropower plant under study, an optimisation model similar to the one proposed in [24] is used.The aim of this model is to solve the annual power generation scheduling of the hydropower plant.The selected case study is a real hydropower plant that participates in the Spanish day-ahead electricity market and is located at the head of its river basin, allowing easy identification of its available water inflows.
This paper on the influence of environmental constraints on WV is organised in the following manner.In Sections 2 and 3, the optimisation model and the case study are described.In Section 4, the analysis (both numerical and analytical) of the new findings (related to ϕ) and the comparison with the results (related to ρ) obtained in [1] are shown.The conclusions of the study are drawn in Section 5, which can be summarised by the following statement: ϕ and ρ have opposite effects on WV, the former increases it and the latter decreases it.Finally, the Appendix 5 contains the applied nomenclature.

Optimisation Model
The optimisation model used to understand how ϕ affects WV is practically identical to the one proposed in [1]; the only difference is the inclusion of ϕ.Nevertheless, for the sake of completeness, it is fully described in this section.The optimisation model is a master-slave algorithm whose basic idea is as follows (see Figure 1): 1.
The master module, based on stochastic dynamic programming (SDP), decomposes the annual scheduling problem, defined by a specific pair of ϕ and ρ, into weekly subproblems.

2.
The slave module, based on deterministic mixed integer linear programming (DMILP), solves each weekly subproblem in the considered week.The weekly subproblems are aimed at maximizing the weekly revenue plus the future expected revenue (given by WV) from a linearized generation characteristic.The process starts from the last week.

3.
The master module recalculates, with the real generation characteristic of the plant, the power outputs according to the decisions provided by the slave module and obtains the resulting revenue.

4.
The master module calculates WV at the beginning of the week.

5.
Second, third and fourth steps are repeated backwards until the beginning of the year for the rest of the weeks using the last-obtained WV.

6.
The master module checks if the relative interannual variations of the expected revenue at the beginning of each week converge (see Equation ( 4)).If the answer is yes, the process stops, otherwise the process is repeated from the second step but now considering the last-obtained WV at the end of the last week.In what follows, the master and slave modules are described.

Master Module
The purpose of the master module is to calculate the steady-state policy [25] (p.107) of a hydropower plant that sells energy in a day-ahead electricity market.Based on SDP, it has a yearly planning period with weekly time steps and its state variables are: stored water volume in the reservoir at the beginning of each week, water inflow of the current week, and average energy price of the current week.According to [26], the first variable is discretised in nine equidistant values from the dead volume to the maximum one.The two latter variables are modelled by means of Markov chains following the approaches of [27,28], respectively.
The order of the Markov chains is selected through the Akaike information criterion [29].The number of classes of water inflows (five) and energy prices (three) are determined following the recommendation in [30] (p.51), and according to the length of the available historical series of these variables.As suggested in [31,32], the weekly extreme values of both variables are represented by their respective extreme classes in order to improve the robustness of the plant operation.
The master module uses the volume released from the reservoir throughout each week as decision variable [30] (p.49) and is based on the recursive relationship: Equation ( 1) is the stochastic Bellman equation in which ̃ , , represents the optimum cumulative revenue at when the weekly water inflow is a and the weekly average energy price is b from the week k to the end of the planning period, , , , is the revenue corresponding to the decision to go from to when those inflow and price occur during the considered week, , In what follows, the master and slave modules are described.

Master Module
The purpose of the master module is to calculate the steady-state policy [25] (p.107) of a hydropower plant that sells energy in a day-ahead electricity market.Based on SDP, it has a yearly planning period with weekly time steps and its state variables are: stored water volume in the reservoir at the beginning of each week, water inflow of the current week, and average energy price of the current week.According to [26], the first variable is discretised in nine equidistant values from the dead volume to the maximum one.The two latter variables are modelled by means of Markov chains following the approaches of [27,28], respectively.
The order of the Markov chains is selected through the Akaike information criterion [29].The number of classes of water inflows (five) and energy prices (three) are determined following the recommendation in [30] (p.51), and according to the length of the available historical series of these variables.As suggested in [31,32], the weekly extreme values of both variables are represented by their respective extreme classes in order to improve the robustness of the plant operation.
The master module uses the volume released from the reservoir throughout each week as decision variable [30] (p.49) and is based on the recursive relationship: Equation ( 1) is the stochastic Bellman equation in which r z i,a,b k .represents the optimum cumulative revenue at V i k when the weekly water inflow is a and the weekly average energy price is b from the week k to the end of the planning period, z i,a,b,l k . is the revenue corresponding to the decision to go from V i k to V l k`1 when those inflow and price occur during the considered week, ε a,x k signifies the probability that the inflow at the week k + 1 is x given that the inflow at week k was a, and ε b,y k means the probability that the price at week k + 1 is y given that the price at week k was b.
The state-transition equation of the master module is the mass balance equation: pW t ´et ´qt ´qbo t ´qsp t q; @ti, lu where CF1 is a conversion factor for converting m 3 /s into Mm 3 /h (0.0036), W t represents the water inflow to the reservoir during the hour t, e t refers to the evaporation losses during the hour t, and q t , qbo t and qsp t are, respectively, the flows released through the hydro units, the bottom outlets and the spillways during the hour t.
The convergence criterion of the master module is determined by: where r z i,a,b k q n represents the optimum cumulative revenue at V i k when the weekly water inflow is a and the weekly average energy price is b from the week k to the end of the planning period of the iteration n of the SDP.
Once the iterative process of the SDP converges, the master module calculates the WV through the year given by: where WV j k represents WV at the end of the week k along the j-th portion of the water volume stored in the reservoir (j-th reservoir segment of the storage-WV curve), r z i k`1 and r z i`1 k`1 are the optimum cumulative revenue at the extremes of that reservoir segment from the next week to the end of the planning period, V i k`1 and V i`1 k`1 are the stored volumes corresponding to r z i k`1 and r z i`1 k`1 , Ω k contains the set of feasible states evaluated during the considered week, J signifies the total number of used segments in the storage-WV curve, and K comprises the set of weeks of the year.

Slave Module
The purpose of the slave module is to determine the weekly schedule of a hydropower plant that sells energy in a day-ahead electricity market.Based on DMILP, it has a weekly planning period with hourly time steps.The weekly rate of hourly evaporated water volume per flooded area as well as the hourly water inflows and energy prices are input variables to this module.The disaggregations of these two latter variables from the weekly values considered in the Markov chains of the master module to hourly ones are performed by weighting the historical average weekly profile of each variable in every week by a ratio between the considered value in its respective chain and the mean value of the The first term in Equation ( 5) represents the revenue obtained in the day-ahead market (hourly energy price, P t , times hourly generated power, pow t ).The first term in square brackets is the cost for interhourly power variations (wear and tear cost of hydro units, α, times hourly decrements, pow dec t , and increments, pow inc t , in generated power), the second one is the units start-up and shut-down cost (start-up and shut-down cost of hydro units, β, times operating state changes of these units, on u t and o f f u t ) and the third one penalises the interhourly variations in the flow released through the spillways and the bottom outlets (penalisation parameter of bottom outlets and spillways, γ, times hourly flow decrements, qbosp dec t , and increments, qbosp inc t , through said elements).The final term refers to the future expected revenue at the end of the week (WV of the j-th reservoir segment at the end of the week k given the inflow a and the average energy price b, WV j,a,b k , times the stored volume in the j-th segment at the end of the considered week, vwv j ).
Energies 2016, 9, 446 5 of 21 , and increments, , in generated power), the second one is the units start-up and shutdown cost (start-up and shut-down cost of hydro units, β, times operating state changes of these units, and ) and the third one penalises the interhourly variations in the flow released through the spillways and the bottom outlets (penalisation parameter of bottom outlets and spillways, γ, times hourly flow decrements, , and increments, , through said elements).The final term refers to the future expected revenue at the end of the week (WV of the j-th reservoir segment at the end of the week k given the inflow a and the average energy price b, , , , times the stored volume in the j-th segment at the end of the considered week, ).
(a) (c) (b) The terms α and β can be estimated either by means of specific experimental studies in the considered plant or from the information contained in [33,34], respectively, as it was done in this study.γ is a small (0.01) and artificial cost considered in the slave module with the aim of obtaining a more realistic use of the bottom outlets and spillways (this parameter is neglected by the master module during the recalculation of the power outputs with the real generation characteristic of the The terms α and β can be estimated either by means of specific experimental studies in the considered plant or from the information contained in [33,34], respectively, as it was done in this study. γ is a small (0.01) and artificial cost considered in the slave module with the aim of obtaining a more realistic use of the bottom outlets and spillways (this parameter is neglected by the master module during the recalculation of the power outputs with the real generation characteristic of the plant).The objective function of the slave module is subject to the following constraints:

‚
Water mass balance (Equation ( 6)): v t " v t´1 `CF1 ¨pW t ´et ´qt ´qbo t ´qsp t q ; @t P T This equation is analogous to Equation ( 2) where v t means the stored volume at the end of the hour t and T contains the set of hours of the week.

‚
Hourly evaporation losses (Equation ( 11)): e t " CF2 ¨Ek ¨pK1 e ¨vt `K2 e q ; @t P T ^@k P K where CF2 is a conversion factor for converting m 3 /s into Mm 3 /week (1/0.6048),E k signifies the rate of hourly evaporated water volume per flooded area during the week k, and K1 e and K2 e are the coefficients of the linear approximation of the storage-flooded area curve of the reservoir.

‚
Use of the bottom outlets and spillways (Equations ( 12)-( 18)): qbo t ď K1 bo ¨vt `K2 bo ; @t P T (13) Equation ( 12) computes the hourly flow increments, qbosp inc t , and decrements, qbosp dec t , through the bottom outlets and spillways.Equation ( 13) limits the maximum water release through the bottom outlets according to the hourly stored volume where K1 bo and K2 bo are the coefficients of the linear approximation of the storage-maximum bottom outlet flow curve.Equation ( 14) is analogous to Equation ( 13) for the case of the spillways where K sp represents the coefficient of the linear approximation of the storage-maximum spillway flow curve and vasp t means the stored volume at the end of the hour t above the spillways minimum level.Equations ( 15)-( 18) are used to calculate vasp t where vbsp t is the stored volume at the end of the hour t below the Energies 2016, 9, 446 7 of 21 spillways minimum level, V sp is the stored volume above which the spillways can operate, and sp t indicates whether (1) or not (0) the stored volume is above V sp during the hour t.

‚
Maximum plant flow according to the stored volume and minimum technical stored volume for power generation (Equations ( 19)-( 21)): v t ě ÿ cPC pVQ c ¨vq c t q; @t P T (20) Equation ( 19) is analogous to Equations ( 13) and ( 14) for the case of the hydro units where Q VQ c is the length (measured in terms of flow) of the c-th segment of the storage-maximum plant flow curve and vq c t indicates whether (1) or not (0) the stored volume is above the minimum storage of the c-th segment during the hour t.It is important to clarify that Equation ( 19) is added since the maximum plant flow may vary substantially within each week at low reservoir levels.Equations ( 20) and ( 21) are used to calculate vq c t , where VQ c is the length (measured in terms of volume) of the c-th segment of the storage-maximum plant flow curve and o 1 t indicates whether (1) or not (0) at least one hydro unit is on-line during the hour t.

‚
Power-discharge piecewise non-concave linear curve as in [35], in terms of the initial and estimated final stored volumes (Equations ( 22)-( 25)): q s t ě Q s ¨ou t ; @s P S ˇˇs u´1 ď s ă s u ^u P U ˇˇu ą 1; @t P T (24 pow t " POW ¨o1 t `ÿ sPS pR s ¨qs t q; @t P T In Equation (22), the first term means the minimum flow of a single hydro unit (Q u ) and the second one the sum of the flow in all segments (q s t ) in which the power-discharge curve is divided.Equations ( 23) and ( 24) define the maximum and minimum values of each q s t .In Equation ( 25), the first term signifies the minimum power output of one hydro unit (POW) and the second one refers to the sum of the power in all segments of the power-discharge curve, each calculated as the product of q s t and the slope of each segment (R s ).

‚
Start-ups and shut-downs of the hydro units as in [36] (Equations ( 27)-( 31)): Energies 2016, 9, 446 8 of 21 ; @u P U ˇˇu ă U ^@t P T (29) ´ou t ; @u P U ^@t P T ˇˇt ă 168 (30) Equations ( 27)-( 29) are used to calculate the operating state of the hydro units (o u t ) according to q t where Q u means the plant flow above which the u-th hydro unit starts up and U represents the total number of hydro units of the plant.Equations ( 30) and ( 31) compute the changes in the operating state of the hydro units.

‚
Future expected revenue at the end of the week (Equations ( 32)-( 35)): vwv j ď VWV j ¨wv j ; @j P J (33) vwv j ě VWV j ¨wv j`1 ; @j P J ˇˇj ă J (34) wv j ě wv j`1 ; @j P J ˇˇj ă J where VWV j is the length (measured in terms of volume) of the j-th segment of the storage-WV curve, vwv j is the stored volume in the j-th segment of the said curve at the end of the week, and wv j indicates if the stored volume at the end of the week is above the minimum storage of the j-th segment.

‚
Up and down ρ as in [37] (Equations ( 36) and ( 37)): q t`1 `qbo t`1 `qsp t`1 ´qt ´qbo t ´qsp t ď ρ up ; @t P T|t ă 168 (36) q t`1 `qbo t`1 `qsp t`1 ´qt ´qbo t ´qsp t ě ´ρdown ; @t P T ˇˇt ă 168 (37) where ρ up and ρ down refer to the maximum interhourly increment and decrement of water release, respectively.The fulfillment of ρ between consecutive weeks is not considered and is one of our ongoing works.The interested reader is referred to [38] where the fulfillment of the interweekly ρ was considered in a deterministic context.

‚
Seasonal ϕ (Equation ( 38)): Equation (38) forces the total water release (through the hydro units, bottom outlets and spillways) to be larger than or equal to the minimum of ϕ and the current water inflow into the reservoir, W t [39].

Case Study
A real hydropower plant, located in the Northwest area of Spain (see Figure 3) was selected to analyse the influence of the two abovementioned environmental constraints on WV.The plant is a dam-based scheme; the power house is located at the toe of the dam and the water is conveyed to the turbines via three different penstocks that go through the body of the dam.The technical data of the hydropower plant were provided by the company that owns and operates the plant; its main design parameters are included in Table 1.The historical series of daily water inflow and hourly energy price were taken, respectively, from  The magnitudes of the environmental constraints ϕ, considered in this study, and ρ, evaluated in [1], were selected to cover a wide range of values.As in [44], ϕ is the average of its seasonal values ( -, -, -and -) expressed as the percentage of the maximum flow of the hydropower plant, and ρ is expressed as the number of hours necessary for the hydropower plant to "go" from 0 to maximum flow (or vice versa) at a rate equal to the average of its up and down values ( and ).The scenarios analysed in this study as well as in [1] are summarised in Table 2.The technical data of the hydropower plant were provided by the company that owns and operates the plant; its main design parameters are included in Table 1.The historical series of daily water inflow and hourly energy price were taken, respectively, from

Results and Discussion
The averages across the water content and the seasons of WV in relative terms with respect to WV in the scenario without environmental constraints (i.e., [WV(ϕ, ρ) − WV(0%, 0 h)]/WV(0%, 0 h)) are included in Figure 5 for every considered ϕ and ρ (see Table 2).This figure seems to indicate that WV increased (approximately linearly) with ϕ and decreased (approximately quadratically) with ρ.The magnitudes of the environmental constraints ϕ, considered in this study, and ρ, evaluated in [1], were selected to cover a wide range of values.As in [44], ϕ is the average of its seasonal values (ϕ 1-13 , ϕ 14-26 , ϕ 27-39 and ϕ 40-52 ) expressed as the percentage of the maximum flow of the hydropower plant, and ρ is expressed as the number of hours necessary for the hydropower plant to "go" from 0 to maximum flow (or vice versa) at a rate equal to the average of its up and down values (ρ up and ρ down ).The scenarios analysed in this study as well as in [1] are summarised in Table 2.

Results and Discussion
The averages across the water content and the seasons of WV in relative terms with respect to WV in the scenario without environmental constraints (i.e., [WV(ϕ, ρ) ´WV(0%, 0 h)]/WV(0%, 0 h)) are included in Figure 5 for every considered ϕ and ρ (see Table 2).This figure seems

Results and Discussion
The averages across the water content and the seasons of WV in relative terms with respect to WV in the scenario without environmental constraints (i.e., [WV(ϕ, ρ) − WV(0%, 0 h)]/WV(0%, 0 h)) are included in Figure 5 for every considered ϕ and ρ (see Table 2).This figure seems to indicate that WV increased (approximately linearly) with ϕ and decreased (approximately quadratically) with ρ.In order to explain the abovementioned opposing effects, the weekly scheduling problem considered in Section 2.2 is reformulated in a simpler manner as follows: In order to explain the abovementioned opposing effects, the weekly scheduling problem considered in Section 2.2 is reformulated in a simpler manner as follows: min « ´ÿ tPT pP t ¨pow t pv t , q t qq ´r z v 168 ff ; (39) subject to: v t ´vt´1 `CF1 ¨pq t ´Wt q " 0; @t P T|t ă 168 ^v0 " V 0 (40) v t ´V ď 0; @t P T ˇˇt ă 168 (41) V dead ´vt ď 0; @t P T ˇˇt ă 168 (42) q t ´Q ď 0; @t P T ˇˇt ă 168 (43) ϕ k ´qt ď 0; @t P T|t ă 168 (44) q t ´qt´1 ´ρup ď 0; @t P T|t ă 168 (45) ´qt `qt´1 ´ρdown ď 0; @t P T ˇˇt ă 168 (46) Equation ( 39) is the objective function in which r z v 168 represents the future expected revenue starting from the volume stored at the final hour of the week (hour 168).The rest are constraints: the water mass balance (Equation ( 40)), the limits of the stored volume (Equations ( 41) and ( 42)), the upper limit of the released flow (Equation ( 43)), and the environmental constraints (Equations ( 44)-( 46)).
The resulting Lagrangian function of the above-described problem is: In Equation ( 47), λ's and µ's refer to the Lagrange multipliers of the equality and inequality constraints, respectively.As it is well-known, these multipliers mean the incremental prices of the constraint requirements expressed in units of the objective function [45].Thus, from the Karush-Kuhn-Tucker conditions, it can be obtained:  The abovementioned opposing effects can be also understood, in a more intuitive manner, from the storage-head relationship.Suppose two separate problems aimed at maximising the future revenue of a hydropower plant starting from two different stored volumes (V 1 and V 2 |V 2 > V 1 ).Regardless of the future values of the inputs (evaporations, inflows and prices), the trajectory (in a time-volume diagram) followed by the reservoir from V 2 would have a higher average head than the one corresponding to V 1 , and therefore the impact of ϕ would be higher for V 1 than for V 2 (i.e., WV would increase) since the detraction of water (−ΔV) represents a greater reduction of head for V 1 for V 2 (see the example in Figure 7).The greater ϕ (i.e., −ΔV), the greater its impact on WV.As regards ρ, its impact would be larger for V 2 than for V 1 (and therefore WV would decrease) because the average head is bigger at V 2 than V 1 .The greater ρ, the greater its impact on WV.The increasing trend of WV with ϕ can be to some extent inferred also from the few numerical evidences of WV reported in the literature [6][7][8]46,47], where WV was shown for different reservoir levels and periods within the hydrological year (usually weeks).In said references, it can be seen that WV increases as the reservoir level or water inflow decrease.ϕ can be understood as a reduction both in the reservoir level or water inflow, and consequently, the increasing trend of WV with ϕ might be anticipated.However, to the authors' opinion, the quantification of the impact of ϕ on WV is nowadays a relevant contribution for the hydropower sector.
It is interesting to highlight that the trends of the annual WV observed in Figure 5 are in agreement with the trends of the average revenue losses obtained in [44].Nonetheless, the impact of both ϕ and ρ on WV is smaller than that on the average revenue losses obtained in [44].Without the aim of dismissing the results presented in [44], the results presented in this paper can be considered more realistic since they were obtained as a result of a stochastic approach, whereas those presented in [43] were obtained as a result of a deterministic one.The abovementioned opposing effects can be also understood, in a more intuitive manner, from the storage-head relationship.Suppose two separate problems aimed at maximising the future revenue of a hydropower plant starting from two different stored volumes (V 1 and V 2 |V 2 > V 1 ).Regardless of the future values of the inputs (evaporations, inflows and prices), the trajectory (in a time-volume diagram) followed by the reservoir from V 2 would have a higher average head than the one corresponding to V 1 , and therefore the impact of ϕ would be higher for V 1 than for V 2 (i.e., WV would increase) since the detraction of water (−ΔV) represents a greater reduction of head for V 1 than for V 2 (see the example in Figure 7).The greater ϕ (i.e., −ΔV), the greater its impact on WV.As regards ρ, its impact would be larger for V 2 than for V 1 (and therefore WV would decrease) because the average head is bigger at V 2 than V 1 .The greater ρ, the greater its impact on WV.The increasing trend of WV with ϕ can be to some extent inferred also from the few numerical evidences of WV reported in the literature [6][7][8]46,47], where WV was shown for different reservoir levels and periods within the hydrological year (usually weeks).In said references, it can be seen that WV increases as the reservoir level or water inflow decrease.ϕ can be understood as a reduction both in the reservoir level or water inflow, and consequently, the increasing trend of WV with ϕ might be anticipated.However, to the authors' opinion, the quantification of the impact of ϕ on WV is nowadays a relevant contribution for the hydropower sector.
It is interesting to highlight that the trends of the annual WV observed in Figure 5 are in agreement with the trends of the average revenue losses obtained in [44].Nonetheless, the impact of both ϕ and ρ on WV is smaller than that on the average revenue losses obtained in [44].Without the aim of dismissing the results presented in [44], the results presented in this paper can be considered more realistic since they were obtained as a result of a stochastic approach, whereas those presented in [43] were obtained as a result of a deterministic one.The increasing trend of WV with ϕ can be to some extent inferred also from the few numerical evidences of WV reported in the literature [6][7][8]46,47], where WV was shown for different reservoir levels and periods within the hydrological year (usually weeks).In said references, it can be seen that WV increases as the reservoir level or water inflow decrease.ϕ can be understood as a reduction both in the reservoir level or water inflow, and consequently, the increasing trend of WV with ϕ might be anticipated.However, to the authors' opinion, the quantification of the impact of ϕ on WV is nowadays a relevant contribution for the hydropower sector.
It is interesting to highlight that the trends of the annual WV observed in Figure 5 are in agreement with the trends of the average revenue losses obtained in [44].Nonetheless, the impact of both ϕ and ρ on WV is smaller than that on the average revenue losses obtained in [44].Without the aim of dismissing the results presented in [44], the results presented in this paper can be considered more realistic since they were obtained as a result of a stochastic approach, whereas those presented in [43] were obtained as a result of a deterministic one.
Returning to Figure 5, it can also be observed that the effect of ϕ was more significant in spring and winter, whereas ρ had more influence in spring and summer.In order for these results not to make the reader draw misleading conclusions, it is important to note that Figure 5 shows relative values but that as shown in Figure 8, WV in absolute terms is higher in spring and summer than in autumn and winter.On the other hand, the reader should bear in mind that ϕ has its largest value during spring and winter (see Table 2).Even though ϕ usually varies seasonally, Figure 9a,b show for the sake of clarity the same results as Figures 5a, 6a, 7a and 8a but considering ϕ constant along the year (all seasonal values are identical).In addition, Figure 9c shows the impact of ϕ constant along the year on WV (i.e., WV("constant ϕ", 0 h)-WV(0%, 0 h)) in k€/Mm 3 .As regards ρ, its higher impact on WV in Energies 2016, 9, 446 14 of 21 spring and summer is to a certain extent expected, since the lower the water inflow, the higher the economic value of the the operational flexibility.make the reader draw misleading conclusions, it is important to note that Figure 5 shows relative values but that as shown in Figure 8, WV in absolute terms is higher in spring and summer than in autumn and winter.On the other hand, the reader should bear in mind that ϕ has its largest value during spring and winter (see Table 2).Even though ϕ usually varies seasonally, Figure 9a,b show for the sake of clarity the same results as Figures 5a-8a but considering ϕ constant along the year (all seasonal values are identical).In addition, Figure 9c shows the impact of ϕ constant along the year on WV (i.e., WV("constant ϕ", 0 h)-WV(0%, 0 h)) in k€/Mm 3 .As regards ρ, its higher impact on WV in spring and summer is to a certain extent expected, since the lower the water inflow, the higher the economic value of the the operational flexibility.The results in Figures 5 and 7 would indicate that the use in the short-term scheduling of WV determined considering ϕ or ρ would result in an operation with a higher or lower average reservoir level.These effects are coherent as regards the risk of spillage since it decreases as the severity of ϕ increases and increases as the severity of ρ increases.and winter, whereas ρ had more influence in spring and summer.In order for these results not to make the reader draw misleading conclusions, it is important to note that Figure 5 shows relative values but that as shown in Figure 8, WV in absolute terms is higher in spring and summer than in autumn and winter.On the other hand, the reader should bear in mind that ϕ has its largest value during spring and winter (see Table 2).Even though ϕ usually varies seasonally, Figure 9a,b show for the sake of clarity the same results as Figures 5a-8a but considering ϕ constant along the year (all seasonal values are identical).In addition, Figure 9c shows the impact of ϕ constant along the year on WV (i.e., WV("constant ϕ", 0 h)-WV(0%, 0 h)) in k€/Mm 3 .As regards ρ, its higher impact on WV in spring and summer is to a certain extent expected, since the lower the water inflow, the higher the economic value of the the operational flexibility.The results in Figures 5 and 7 would indicate that the use in the short-term scheduling of WV determined considering ϕ or ρ would result in an operation with a higher or lower average reservoir level.These effects are coherent as regards the risk of spillage since it decreases as the severity of ϕ increases and increases as the severity of ρ increases.The results in Figures 5 and 7 would indicate that the use in the short-term scheduling of WV determined considering ϕ or ρ would result in an operation with a higher or lower average reservoir level.These effects are coherent as regards the risk of spillage since it decreases as the severity of ϕ increases and increases as the severity of ρ increases.
In Figure 10, the level curves of WV according to the water content across the year are represented for the scenario without environmental constraints, with ϕ = 5%, with ρ = 60 h, and with ϕ = 5% and ρ = 60 h (values proposed by the river basin authority).As it can be seen in the figure, on the one hand, the lowest average absolute values appeared in winter and the highest ones in summer, and, on the other hand, the lowest relative values of each week were given at the maximum stored volumes in winter whereas these ones during summer were located at the minimum stored volumes.It also shows how the effects of ϕ and ρ on WV approximately counteracted each other.
In Figure 10, the level curves of WV according to the water content across the year are represented for the scenario without environmental constraints, with ϕ = 5%, with ρ = 60 h, and with ϕ = 5% and ρ = 60 h (values proposed by the river basin authority).As it can be seen in the figure, on the one hand, the lowest average absolute values appeared in winter and the highest ones in summer, and, on the other hand, the lowest relative values of each week were given at the maximum stored volumes in winter whereas these ones during summer were located at the minimum stored volumes.It also shows how the effects of ϕ and ρ on WV approximately counteracted each other.These results are only partially in agreement with the few numerical evidences of WV reported in the literature.As discussed above, according to said evidences WV increases as the reservoir level or water inflow decrease.However, as it can be seen in Figure 10, during summer WV increases as the reservoir level increases.This interesting phenomenon which, to the authors' knowledge, had not been reported in the literature yet until [1], occurs not only because of the lack of risk of spill during summer, but also because of the subsequent prominence of the plant generation characteristic.In order to better understand this effect, we invite the reader to think about another theoretical problem aimed at maximising the future revenue of a hydropower plant starting from three different stored volumes (V 1 , V 2 = V 1 + ∆V, V 3 = V 2 + ∆V), assuming that future expected water inflows are null.In this extremely dry scenario, it is obvious that the difference in revenue between V 1 and V 2 , and between V 2 and V 3 , is due to both the plant generation characteristic and the difference in the initial available water volume (∆V), whereas the difference in WV depends only on the former.
With the aim of supporting this discussion, the WV was calculated in five additional cases, each with a constant throughout the year weekly profile of both hourly water inflow and price, and with a different water inflow (very dry, dry, normal, wet and very wet).The five weekly profiles of hourly water inflow were selected by clustering the historical series.The weekly profile of hourly energy price was formed with the average hourly prices of the historical series.It is easily seen in Figure 11 that the trend reported in the literature (increasing WV with decreasing reservoir level) disappears and even inverts as the water inflow decreases.These results are only partially in agreement with the few numerical evidences of WV reported in the literature.As discussed above, according to said evidences WV increases as the reservoir level or water inflow decrease.However, as it can be seen in Figure 10, during summer WV increases as the reservoir level increases.This interesting phenomenon which, to the authors' knowledge, had not been reported in the literature yet until [1], occurs not only because of the lack of risk of spill during summer, but also because of the subsequent prominence of the plant generation characteristic.In order to better understand this effect, we invite the reader to think about another theoretical problem aimed at maximising the future revenue of a hydropower plant starting from three different stored volumes (V 1 , V 2 = V 1 + ΔV, V 3 = V 2 + ΔV), assuming that future expected water inflows are null.In this extremely dry scenario, it is obvious that the difference in revenue between V 1 and V 2 , and between V 2 and V 3 , is due to both the plant generation characteristic and the difference in the initial available water volume (ΔV), whereas the difference in WV depends only on the former.
With the aim of supporting this discussion, the WV was calculated in five additional cases, each with a constant throughout the year weekly profile of both hourly water inflow and price, and with a different water inflow (very dry, dry, normal, wet and very wet).The five weekly profiles of hourly water inflow were selected by clustering the historical series.The weekly profile of hourly energy price was formed with the average hourly prices of the historical series.It is easily seen in Figure 11 that the trend reported in the literature (increasing WV with decreasing reservoir level) disappears and even inverts as the water inflow decreases.

Conclusions
This paper has offered an analysis of the influence of the most common environmental constraints (seasonal minimum flows and maximum ramping rates) on the water value of a real hydropower plant that sells energy in the Spanish day-ahead electricity market.For this purpose, an annual scheduling model, based on stochastic dynamic programming and deterministic mixed integer linear programming, has been developed.
The empirical results of the case study point out that water value is very sensitive to the presence of the considered environmental constraints, as well as their magnitudes.According to said results, water value would follow an upward (approximately) linear trend and a downward (approximately) quadratic one as the severity of the seasonal minimum flows and the maximum ramping rates increase, respectively.These opposing trends have been analytically deduced from the Karush-Kuhn-Tucker conditions, and inferred in a more intuitive manner, from two simple problems and the storage-head curve of the reservoir.Finally, it has been also found that the impacts of these

Conclusions
This paper has offered an analysis of the influence of the most common environmental constraints (seasonal minimum flows and maximum ramping rates) on the water value of a real hydropower plant that sells energy in the Spanish day-ahead electricity market.For this purpose, an annual scheduling model, based on stochastic dynamic programming and deterministic mixed integer linear programming, has been developed.
The empirical results of the case study point out that water value is very sensitive to the presence of the considered environmental constraints, as well as their magnitudes.According to said results, water value would follow an upward (approximately) linear trend and a downward (approximately) quadratic one as the severity of the seasonal minimum flows and the maximum ramping rates increase, respectively.These opposing trends have been analytically deduced from the Karush-Kuhn-Tucker conditions, and inferred in a more intuitive manner, from two simple problems and the storage-head curve of the reservoir.Finally, it has been also found that the impacts of these environmental constraints on water value counteract each other, and that significantly vary throughout the year.
To conclude, the following line of work is proposed in order to continue the research presented in this paper: to study the usefulness of considering the seasonal minimum flows and maximum ramping rates to estimate water value, by calculating the weekly generation schedule and revenue of a hydropower plant throughout a long enough set of consecutive weeks, using the water values determined with and without considering said environmental constraints.

J
Total number of segments in the storage-WV curve.
K1 bo , K2 bo Coefficients of the linear approximation of the storage-maximum bottom outlet flow curve ((m 3 /s)/Mm 3 ; m 3 /s).

K1 e , K2 e
Coefficients of the linear approximation of the storage-flooded area curve (km 2 /Mm 3 ; Mm 3 ).

K sp
Coefficient of the linear approximation of the storage-maximum spillway flow curve ((m 3 /s)/Mm 3 ).

P t
Energy price during the hour t (€/MW).

POW
Minimum power output of the power-discharge curve (MW).

Q
Maximum plant flow (m 3 /s).Q s Maximum flow of the s-th segment of the power-discharge curve (m 3 /s).

Q u
Plant flow above which the u-th hydro unit starts up (m 3 /s).

Q VQ c
Plant flow corresponding to the c-th interval of the storage-maximum plant flow curve (m 3 /s).R s Slope of the s-th segment of the power-discharge curve (MW/(m 3 /s)).

U
Total number of hydro units of the plant.

V
Maximum physical stored volume (Mm 3 ).

V dead
Dead reservoir volume (stored volume below the bottom outlets) (Mm 3 ).V i k Stored volume i at the beginning of the week k (Mm 3 ).

V legal k
Maximum legal stored volume at the end of the week k (Mm 3 ).

V sp
Stored volume above which the spillways can operate (Mm 3 ).

VQ c
Minimum stored volume of the c-th interval of the storage-maximum plant flow curve (Mm 3 ).

VWV j
Stored volume of the j-th segment of the storage-WV curve (Mm 3 ).

W t
Water inflow to the reservoir during the hour t (m 3 /s).

WV j k
WV of the j-th reservoir segment of the storage-WV curve at the end of the week k (€/Mm 3 ).Plant flow during the hour t (m 3 /s).
Plant flow corresponding to the s-th segment of the power-discharge curve during the hour t (m 3 /s).

Figure 1 .
Figure 1.Flowchart of the optimisation model.

Figure 1 .
Figure 1.Flowchart of the optimisation model.

Figure 2 .
Figure 2. (a) Historical weekly profiles (grey lines) and their average (black line) of the hourly energy prices of the first week of the water year; (b) Calculation of the weighting factors for disaggregation of the classes of the Markov chain of the energy price of the first week of the water year; (c) Hourly disaggregations of the three classes of the Markov chain of the energy price of the first week of the water year.

Figure 2 .
Figure 2. (a) Historical weekly profiles (grey lines) and their average (black line) of the hourly energy prices of the first week of the water year; (b) Calculation of the weighting factors for disaggregation of the classes of the Markov chain of the energy price of the first week of the water year; (c) Hourly disaggregations of the three classes of the Markov chain of the energy price of the first week of the water year.

Figure 3 .
Figure 3. (a) Location of the hydropower plant (image taken from Google Earth™); (b) Approximate ground plant of the reservoir water basin.
[40]  (years: 1963-1965, 1966-2005)  and[41]   (years: 1998-2005).The weekly rates of hourly evaporated water volume per flooded area were estimated in two steps.First, from the application of the first empirical temperature-based formula proposed in[42] to the average monthly temperatures taken from [43] (years: 1931-1960), the monthly surface evaporations (mm/month) and second the conversion of these values into the said rates.The average values of these inputs are depicted in Figure4.

Figure 3 .
Figure 3. (a) Location of the hydropower plant (image taken from Google Earth™); (b) Approximate ground plant of the reservoir water basin.
[40]  (years: 1963-1965, 1966-2005)   and [41] (years: 1998-2005).The weekly rates of hourly evaporated water volume per flooded area were estimated in two steps.First, from the application of the first empirical temperature-based formula proposed in[42] to the average monthly temperatures taken from [43] (years: 1931-1960), the monthly surface evaporations (mm/month) and second the conversion of these values into the said rates.The average values of these inputs are depicted in Figure4.Energies 2016, 9, 446 10 of 21

-1
Values proposed by the river basin authority; 2 Scenarios calculated in[1].

Figure 5 .
(a) Average relative WV in terms of ϕ; (b) Average relative WV in terms of ρ.

Figure 5 .
Figure 5. (a) Average relative WV in terms of ϕ; (b) Average relative WV in terms of ρ.

Figure 8 .
Figure 8.(a) Average WV in terms of ϕ; (b) Average WV in terms of ρ.

Figure 9 .
Figure 9. (a) Average relative WV in terms of ϕ constant along the year; (b) Average WV in terms of ϕ constant along the year; (c) Difference in average WV in terms of ϕ constant along the year.

Figure 11 .
Figure 11.Average profiles of WV for five different, and constant through the year, weekly water inflows (Very dry = lowest cluster; Dry = second lowest cluster; Normal = medium cluster; Wet = second highest cluster; Very wet = highest cluster).

Figure 11 .
Figure 11.Average profiles of WV for five different, and constant through the year, weekly water inflows (Very dry = lowest cluster; Dry = second lowest cluster; Normal = medium cluster; Wet = second highest cluster; Very wet = highest cluster).
WV of the j-th reservoir segment of the storage-WV curve at the end of the week k given the inflow a and the average energy price b (€/Mm3

Table 1 .
Main design parameters of the hydropower plant.

Table 2 .
Scenarios of environmental constraints.

Table 1 .
Main design parameters of the hydropower plant.

Table 2 .
Scenarios of environmental constraints.
to indicate that WV increased (approximately linearly) with ϕ and decreased (approximately quadratically) with ρ.

Table 2 .
Scenarios of environmental constraints.
µ ρ down t ¨pow t pv t , q t qq `λv t ¨rv t ´vt´1 `CF1 ¨pq t ´Wt qs `µV t ¨`v t ´Vμ µ ρ down Bv

t
). Feasible states within the state diagram of the master module during the week k.=1 if the u-th hydro unit is shut-down during the hour t. on u =1 if the u-th hydro unit is started-up during the hour t. o u

t
=1 if the u-th hydro unit is on-line during the hour t. sp t =1 if the stored volume is above V sp during the hour t.=1 if the stored volume is within the c-th interval of the storage-maximum plant flow curve during the hour t.wv j =1 if the stored volume is above the minimum storage of the j-th segment of the storage-WV curve.Flow of evaporation during the hour t (m 3 /s).Lagrange multipliers of the equality constraints during the hour t (€).Lagrange multipliers of the inequality constraints during the hour t (€).pow t Generated power during the hour t (MW).Decrease in generated power between the hours t and t + 1 (MW).Increase in generated power between the hours t and t + 1 (MW).q t