1. Choosing among infinite alternatives NRMLec17 Andrea Castelletti Politecnico di Milano

2. 2 The pure planning problem This is the more general formulation of the problem, accounting for both the transient period and the steady state conditions of the system. Often the DM is only interested in the steady state conditions

3. 3 Transient vs. steady-state A project has been proposed by Israel for exploiting the depression (-400 m a.s.l.) for hydropower production. Let’s call u p the capacity of the canal connecting the two seas. Let’ assume that the flow through the canal is constant Dead Sea Mediterranean Sea Jordan El Araba Jerusalem 40 km Mediterranean Sea Dead Sea Israel Jordan Example: the Dead Sea Proj.

4. 4 Transient vs. steady-state Hp: deterministic and periodic inflow. There are not outflow Storage oscillations are peridodic with T Example: the Dead Sea Proj. What happens when the canal is constructed? The level of the lake rise until a new equilibrium (cycle) has been reached Dead Sea Mediterranean Sea Jordan El Araba Jerusalem 40 km

5. 5 Transient vs. steady-state Transient or steady-state? The choice is up to the DM, but she will presumably be interested in the staedy state conditions. - With the TDC formulation more importance is given to the transient. - With the AEV formulation only the steady state conditions are accounted for, while the transient ignored otherwise re-formulation of the planning problem. Example: the Dead Sea Proj.

6. 6 Pure planning problem

7. 7 The Long-term pure planning problem With the AEV indicator this formulation produces the same optimal alternative as the previous one. However, for some particular forms of the objectives and the constraints the alternative can be singled out more easily.

8. 8 The disturbance scenario Very often, when dealing with the long-term planning problem the average or the worst year is considered as the trajectory of the deterministic disturbances However this relationship holds only for the linear case. average year worst year based on the idea that the average perfomance can be computed by evaluating the system perfomance in the average conditions.

9. 9 Solution algorithm When U p includes few elements: the objective is evaluted for each alternative and that corresponding to the best value selected. Finite horizon:

10. 10 When U p includes few elements: the objective is evaluted for each alternative and that corresponding to the best value selected. Solution algorithm Infinite horizon: a simulation based approach is not suitable as the simulation would last forever…... future costs are discounted increasingly, the further away they are in time, so that the variations of the value of the indicator, sampled every T steps, become progressively smaller with the time t.... if the system holds a ciclostationary condition in the long term, the variations of the value of the indicator become progressively smaller with the time t.

11. 11 When U p includes few elements: the objective is evaluted for each alternative and that corresponding to the best value selected. Solution algorithm Infinite horizon: a simulation based approach is not suitable as the simulation would last forever… Operationally, the simulation can be stopped when the variations in the value of the indicator – sampled every T steps – are small enough. Termination condition The bigger the value of  the shorter the simulation, but the more approximate the evaluation of the indicator will be.

12. 12 Solution algorithm When U p includes a very high, or infinite, number of elements the exhaustive, simulation-based procedure is not feasible: one should settle for a sub-optimal solution. Constrained evolutionary methods are used for this purpose: Via simulation the value of the indicator is computed for a limited number of elements in U p ; The algorithm evolves in U p searching for a solution as closes as possible to the optimal one; When the number of components of u p is high, this approach quickly becomes computationally unfeasible.

13. 13 The Sinai Plan

14. 14 The Sinai Plan GOAL: improving the quality of life by increasing the cultivated surface. Actions: To reclaim desertic lands in the 7 zones Z j (j=1-7) by exploiting the water from the 4 sources S s (s=1-4) and by choosing for each one the more appropriate crop rotation R i (i=1-5) and irrigation techniques I h (h=1- 3). Evaluation criterion: Net benefit in steady state conditions.

15. 15 The Sinai Plan jReclaimable zone Z j A j [feddan] 1Costa Nord-Est del Sinai515 000 2El Tina Plains135 000 3Territory West of the Suez Canal235 000 4Territory North of the Salhia Canal102 000 5Territory between the Ismailia and Salhia Canals 120 000 6Territory South of the Ismailia Canal45 000 7Territory South of Lake Manzala29 000 max reclaimable area in zone Z j 4 212 m 2 sCanal S S w s [10 9 m 3 ] 1Ismailia4.4 2Bahr Hadous2.4 3Bahr El Baqar1.4 4Damietta-Salam2.3 average annual volume derivable from S S

16. 16 The Sinai Plan jReclaimable zone Z j A j [feddan] 1Costa Nord-Est del Sinai515 000 2El Tina Plains135 000 3Territory West of the Suez Canal235 000 4Territory North of the Salhia Canal102 000 5Territory between the Ismailia and Salhia Canals 120 000 6Territory South of the Ismailia Canal45 000 7Territory South of Lake Manzala29 000 max reclaimable area in zone Z j 4 212 m 2 sCanal S S w s [10 9 m 3 ] 1Ismailia4.4 2Bahr Hadous2.4 3Bahr El Baqar1.4 4Damietta-Salam2.3 avearge annual volume derivable from S S iCrop Rotation R i b i [ EGP /feddan] 1Clover-Rice298 2Clover-Cotton374 3Clover-Maize152 4Clover-Soya158 5Wheat-Maize184 hIrrigation Techniques I h L h [ EGP /feddan] 1Flood650 2Sprinkler1500 3Drip2500 unitary agricoltural benefit produced by rotation R i unitary installation cost for irrigation technique I h

17. 17 ICT Tools yes Final (political) decision reasonable alternatives 2. Conceptualisation 3. Designing Alternatives 4. Estimating effects Stakeholders 1. Reconnaissance 5. Evaluation no Mitigation, and compensation, Agreement? 6. Comparison or negotiation

18. 18 The Sinai Plan: Phase 1 - Reconnaissance Conditions imposed by the Ministry: the Plan must be prepared in a non-participatory way; the DM is the Ministry; the evaluation must be carried out using the Cost Benefit Analysis; the Plan must be evaluated on the long term (steady state). Conditions identified by the Analyst: the Stakeholders are the farmers and the current water users (industries and hydropower) ; the quality of the water withdrawn plays a crucial role: salinity Q s [ppm] might strongly affect productivity.

19. 19 ICT Tools yes Final (political) decision reasonable alternatives 2. Conceptualisation 3. Designing Alternatives 4. Estimating effects Stakeholders 1. Reconnaissance 5. Evaluation no Mitigation, and compensation, Agreement? 6. Comparison or negotiation Defining Actions (measures) Identifying the Model Defining Criteria and Indicators

20. 20 The Sinai Plan: Phase 2a – Defining actions The actions proposed by the Ministry concern: area to be reclaimed in each zone crop rotation irrigation technique area [feddan] to be devoted to rotation R i in zone Z j with the irrigation technique I h annual water volume [m 3 ] to be conveyed into the canal S s in zone Z j supply volume source for the supply They can be formalized through these variables However not all the combinations are feasible (e.g. drip irrigation is not allowed for rice). Let’s call F the set of the feasible combinations of i,j,h,s The vector u p of the planning decisions is the following

21. 21 ICT Tools yes Final (political) decision reasonable alternatives 2. Conceptualisation 3. Designing Alternatives 4. Estimating effects Stakeholders 1. Reconnaissance 5. Evaluation no Mitigation, and compensation, Agreement? 6. Comparison or negotiation Defining Actions (measures) Identifying the Model Defining Criteria and Indicators

22. 22 The Sinai Plan: Phase 2b – Def. criteria and indicators unitary agricoltural benefit productivity coefficient [%]: the productivity decreases as the salinity Q j increases i crop j zone h irrigation technique The project indicator (objective) has to be formulated with the CBA max (Social benefits- Social costs)

23. The Sinai Plan: Phase 2b – Def. criteria and indicators The project indicator (objective) has to be formulated with the CBA max (Social benefits- Social costs) installation cost of the irrigation techniques Unitary installation costs of technique I h transportation costs of the water to the reclaimed zone opportunity costs the specific cost [EGP /m 3 km] constant over all the territory distance [km] from the canal S s and the zone Z j opportunity costs [EGP /feddan] of the water in canal S s In conclusion the project indicator is the following vector of the volumes w s that can be withdrawn from the canals

24. 24 ICT Tools yes Final (political) decision reasonable alternatives 2. Conceptualisation 3. Designing Alternatives 4. Estimating effects Stakeholders 1. Reconnaissance 5. Evaluation no Mitigation, and compensation, Agreement? 6. Comparison or negotiation Defining Actions (measures) Identifying the Model Defining Criteria and Indicators

25. 25 The Sinai Plan: Phase 2c – Identifying the model The interest is in the long-term situation (according to the Ministry’s request) and the annual volumes w s are known: Design problem = long-term pure planning prolem; Time step = year; The model is the state is the salinity at the equilibrium distance [km] between S s and the zone Z j unitary loss [m 3 /km] Set of the feasible actions. Let’s see how it is defined.

26. 26 The Sinai Plan: the feasibility set It is defined by 3 constraints 1. The total volume withdrawn from S s must not be greater than the volume available w s 2. The total reclaimed area in zone Z j must not be greater than the maximum reclaimable area A j 3. The components of vector u p cannot assume negative meaningless values (N.B. The alternative zero is the null vector.)

27. 27 The Sinai Plan: Phase 2c – Identifying the model The interest is in the long-term situation (according to the Ministry’s request) and the annual volumes w s are known: Design problem = long-term pure planning prolem; Time step = year; The model is: volume w s that can be withdrawn from the canal All the constraints not included above, precisely

28. 28 The Sinai Plan: Phase 2c – Identifying the model The unitary benefit b i has been estimated by assuming that the crop water demand is totally covered by the water supply. Therefore in order for the agricoltural benefit to be estimated in the correct way, the total water volume supplied to a zone must be greater than the water demand of the crop rotation considered unitary water demand [m 3 /feddan] of rotation R i irrigated with technique I h

29. 29 ICT Tools yes Final (political) decision reasonable alternatives 2. Conceptualisation 3. Designing Alternatives 4. Estimating effects Stakeholders 1. Reconnaissance 5. Evaluation no Mitigation, and compensation, Agreement? 6. Comparison or negotiation

30. 30 The Sinai Plan: Phase 3 – Designing the alternatives The design problem has the following form subject to Non-linear mathematical programming 55 decision variables 80 constraints

31. 31 The Sinai Plan: Phase 3 – Designing the alternatives The design problem has the following form subject to REMARKS: if the value of Q j (j=1,...,7) were given the problem would be linear REMARKS: if the value of Q j (j=1,...,7) were given the problem would be linear idea for the solution Evolutionary algorithm in the space Q j (j=1,...,7) minimizing Subject to the same constraints of the original problem simplex algorithm

32. 32 The Sinai Plan: the optimal alternative The solution of the problem is the optimal Alternative (we assumed o s = 10 [EGP/1000 m 3 ]) Total reclaimed land = 448 000 feddan Annual net benefit (objective) = 80 000 000 EGP ZoneReclaimed area [feddan] Canals used Crop Rotation Irrigation techniques 10*** 2135 0001 e 352 3235 0002 e 321 40*** 50*** 645 000352 729 0002 e 421

33. 33 ICT Tools yes Final (political) decision reasonable alternatives 2. Conceptualisation 3. Designing Alternatives Stakeholders 1. Reconnaissance 4. Estimating effects 5. Evaluation no Mitigation, and compensation, Agreement? 6. Comparison or negotiation OPTIMAL ALTERNATIVE

34. 34 Reading IPWRM.Theory Ch. 8, 10