Primal and Dual Stochastic Dynamic Programming in Long Term Hydrothermal Scheduling Luciana Martinez
Secundino Soares
Departament of Electrical Engineering Escola Polit´ecnica Federal University of Bahia Aristides Novis 02, 40210-630 Salvador, BA, Brazil Email:
[email protected]
Departament of Engineering System School of Electrical and Computer Engineering State University of Campinas P.O. Box 6101, 13083-970 Campinas, SP, Brazil Email:
[email protected]
Abstract— This work presents a comparative study between primal and dual stochastic dynamic programming in the hydrothermal scheduling problem. The comparison is made by simulation using historical inflows records of Brazilian hydroelectric system, to the specific case of systems comprising a single hydro plant. The stochastic variable of the system is modeled by a lag-one parametric autoregressive model. In the case of the dual approach, a parametric autoregressive model of superior order is also considered. Results have demonstrated that the performance of the primal and dual stochastic dynamic programming is not similar.
I. Introduction Dynamic Programming (DP) [1] has been extensively used in the optimization of hydrothermal scheduling problems in particular, and water resource systems in general. The popularity and success of this technique can be attributed to the fact that the nonlinear and stochastic features of such problems can be adequately handled by a DP formulation [2]. However, the usefulness of the DP is limited by the socalled “curse of dimensionality”, since the computational burden increases exponentially with the number of state variables. Various approaches have been suggested to overcome the problem of dimensionality, including the aggregation of the hydroelectric system through a composite representation [3], [4], [5], [6], and the use of Dual Stochastic Dynamic Programming (DSDP), based on Bender’s decomposition. [7], [8], [9]. In contrast of the Stochastic dynamic Programming (SDP), DSDP does not require the discretization of the state space in the solution of the recursive equation of the DP. This approach is based on the approximation of the expected-cost-to-go function of SDP at each stage by piecewise linear functions. These approximate functions are obtained from the dual solution of the problem at each stage, based on the Benders Decomposition. In this way, in systems composed by a single hydro plant or a single equivalent system, it is expected that solution provided by the DSDP be compatible with the solution provided by the SDP.
In the hydrothermal scheduling problem, what make the DSDP an attractive approach in relation to SDP, is the possible benefit associated with the use of models more efficient than the lag-one parametric autoregressive model to represent the actual stochastic process of the inflows. The SDP cannot cope with these kind of representations on account of the “course of dimensionality” associated with this its methodology. The goal of the present paper is to compare the DSDP and SDP policies in long term hydrothermal scheduling, using the Brazilian system as a case study. The comparison involves systems composed of a single hydro plant and, therefore, the modeling manipulations usually necessary to implement a DP for multiple hydro plant systems is not required. The paper is structured as follows: Section II presents the formulation of the long term hydrothermal scheduling problem. Section III presents the SDP approach. Section IV presents the DSDP approach. Section V presents the comparison between SDP and DSDP approaches and Section VI presents the conclusions of the study. II. Problem Formulation The deterministic version of the long term hydrothermal scheduling of a single hydroelectric plant can be formulated as the following nonlinear programming problem: min
T
ψt (Gt ) + αT (xT )
(1)
t=1
subject to Pt + Gt = Dt
(2)
Pt = k · (φ(xt ) − θ(ut ) − pc) · qt xt = xt−1 + (yt − ut ) ∆t ut = qt + st
(3) (4) (5)
xt ≤ xt ≤ xt
(6)
ut ≤ ut ≤ ut
(7)
qt ≤ qt ≤ qt
(8)
st ≥ 0
(9)
x0 given
(10)
In the objective function (1), the operational cost ψt represents the minimum cost from complementary nonhydraulic sources such as thermoelectric generation, imports from neighboring systems, and load shortage. As a consequence of this minimization, ψt is a convex decreasing function of the hydro generation ht and depends on the system load demand dt . The function αT (xT ) represents future operational cost as a function of the final reservoir storage. This term is essential for the equilibrium between the use of water during the planning period and its use afterward. Hydro generation in stage t is a nonlinear function represented by equation (3), where xt is the water storage in the reservoir, qt is the water discharge through the turbines and st the water spillage from the reservoir. The constant k is the product of water density, gravity acceleration and average turbine/generator efficiency, φ(.) is the forebay elevation as a function of water storage, θ(.) is the tailrace elevation as a function of total water release, and pc is the average penstock head loss. The equality constraints in (4) represent the water balance in the reservoir at each stage t, where yt is the incremental water inflow. Other terms such as evaporation and infiltration have not been considered for the sake of simplicity. Lower and upper bounds on variables, expressed by constraints (6)-(9), are imposed by the physical operational constraints of the hydro plant, as well as other constraints associated with the multiple uses of water, such as irrigation, navigation and flood control. III. Stochastic Dynamic Programming The goal of the SDP is to determine a rule for decisionmaking at each stage of the planning period which provides the optimal decision for each possible state of the system. Mathematically, the SDP finds a sequence of decision functions mapping the states into decisions so as to minimize the expected costs. In some applications, the system state is constituted only by the storage variable, which is the case when the stochastic variable is considered independent in time. In other situations, however, when the stochastic variable is modeled by autoregressive models, the system state must be increased to include the water inflows from previous stages in order to represent the time dependence of the inflows, a procedure which makes the “course of dimensionality” even more crucial to this approach. It is assumed, in this paper, that the stochastic variable representing the inflow in stage t depends only on the inflow from the previous stage t − 1. This means that the inflows are represented by a lag-one parametric autoregressive model, PAR(1), describing the stochastic process of the hydrologic variable yt as a Markov chain [10]. For reservoir operation, the state variables are the water stored in the reservoir at the beginning of each stage, xt ,
and the water inflow during the previous stage, yt−1 , which represents the hydrological trend. The control variables are the amount of water discharged, qt , and spilled, st from the reservoir during the time stage t. The long term hydrothermal scheduling problem, in its stochastic version, can be formulated as min Eyt /yt−1
T −1
ψt (ht , dt ) + αT (xT )
(11)
t=1
subject to the constraints in (2)-(10), where Eyt /yt−1 {.} is the expected value with respect to the inflow during stage t conditioned by the inflow during stage t − 1. At each stage, decisions are ranked based on the minimization of the sum of the present cost plus that of the expected future cost, assuming optimal decision-making for all subsequent stages. This cost function is additive in the sense that the cost incurred at time t accumulates over time. According to Bellman’s Optimality Principle [1], the optimal decision is obtained by solving the following recursive equation: min [ψt+1 (Gt+1 )+ Ft (xt , yt ) = {ut+1 }∈Ωt+1 ∞ + Ft+1 (xt+1 , yt+1 ).f (yt+1 |yt )dyt+1 ]
(12)
−∞
s. to
FT (xT , yT ) = αT (xT ) ∀t, t = T − 1, T − 2 , . . . , 0
(13)
where, Ωt+1 = (uk+1 , qk+1 ) {subject to (2)-(10)}, Ft (xt , yt−1 ) represents the minimum expected operational cost from stage t till the end of the planning period T , assuming that the system is at the state (xt , yt−1 ) and ft (yt |yt−1 ) is the probability density function of the inflow in stage t conditioned by the inflow in the previous stage t − 1. The resolution of (12)-(13) requires the discretization of the state and control variables and the conditioned probability density function of the inflows, which leads to the “course of dimensionality” in DP. IV. Dual Stochastic Dynamic Programming In order to present the Dual Dynamic Programming (DDP), a simplified notation of the problem (1)-(10) is considered. Let a two-stages deterministic hydrothermal optimization be represented as: mincx + dy Ax ≥ b subject to : Ex + F y ≥ g
(14)
where, the variables x represents the decision about hydro and thermal generation in the first stage, cx represents the associated cost decision x and Ax ≥ b represents the constraints on system operation (hydraulic constraints, upper and lower bounds on outflows, etc...).
Given a feasible solution x∗ for the first-stage problem (Ax∗ ≥ b), the operation problem in the second stage can be represented as: min dy subject to : F y ≥ g − Ex∗
(15)
where, the variables y represents the decision about hydro and thermal generation in the second stage, dy represents associated cost y and F y ≥ g − Ex represents the constraints on system operation. It is assumed that second-stage problem is always feasible for any x∗ . The objective is to minimize the sum of operation costs cx + dy. The decomposition methodology is based on the following observations [7]: 1. The second-stage cost dy ∗ , where y ∗ is the optimal solution of problem (15), can be seen as a function α(x) of the first-stage decision x, that is, α(x) = min dy subject to : F y ≥ g − Ex
(16)
2. The two-stage problem (14) can be rewritten as min cx + α(x) subject to : Ax ≥ b
(17)
where, α(x) is the value of the optimal solution of the (16) for each x. DDP is based on the approximation of the expectedcost-to-go functions of SDP at each stage by piecewise linear functions. The function α(x) is characterized without state space discretization, therefore avoiding the usual dimensionality problems associated with DP. The Benders decomposition [11] is a technique for building the function α(x), which is characterized as a convex polyhedron [12], [7], [13]. Let the dual problem (16) be written as: w = max π(g − Ex) subject to : πF ≤ d
(18)
where, π is a row dual variables (Simplex multipliers). The feasible regionπF ≤ d does not depend on the firststage decision. Let be the set of p vertives that characterize the region πF ≤ d, that is, Π = {π 1 , π 2 , . . . , π p }. Because the optimal solution of a linear programming problem is always in a vertex of the feasible region, (18) can, in principle, be solved by enumeration max π i (g − Ex) subject to : π i ∈ Π
Because (18) is the dual of (17), one can conclude that the constraint in (20) define the function α(x) in the twostage optimization problem. In this way, the two-stage optimization problem can then be written as
(19)
min cx + α ⎧ ⎪ ⎪ Ax = 1b ⎪ ⎪ ⎪ ⎨ α ≥ π 2 (g − Ex) α ≥ π (g − Ex) subjecto to : ⎪ .. ⎪ ⎪ . ⎪ ⎪ ⎩ α ≥ π p (g − Ex)
(21)
In the usual dynamic programming recursion, the future cost function αt (xt ) is calculated for discretized values of x, while the objective of the DDP is to construct this function directly from the dual information of the linear problem. Associated to the solution of the second-stage problem there is a set Simplex multipliers which measure the change in system operating cost caused by marginal changes in system operating cost caused by marginal changes in the trial first-stage decisions. Theses multipliers are used to form a Benders cut, that corresponds to a new approximation to α ˆ and is returned to the first-stage problem: w∗ = π ∗ (g − Ex∗ )
(22)
α ˆ ≥ π ∗ (g − Ex)
(23)
and,
In the first iteration of the solution algorithm, all constraints concerning future costs (Benders cuts) are relaxed. Linear functions to a new approximation to α ˆ are determined by iterative way, in the similar scheme of the SDP, t = T, T − 1, . . . , 1 (backward recursion). The optimal objective function value of problem for the first stage, cx + α ˆ (x) is a lower bound to the solution of scheduling problem, because α ˆ (x) is a lower bound to the future cost function α(x). A simulation of system operation is then realized, t = 1, 2, . . . , T (forward simulation). In this simulation, the state variables are actualized and the feasible solution is obtained. The total solution, cx + dy is an upper bound to the solution of scheduling problem, because the solution obtained is feasible, but not necessarily optimal solution of problem. The process is terminated when a difference between upper and lower bounds achieve tolerance specified.
Problem (19) can be rewritten as min α ⎧ α ≥ π 1 (g − Ex) ⎪ ⎪ ⎪ ⎨ α ≥ π 2 (g − Ex) subject to : .. ⎪ . ⎪ ⎪ ⎩ α ≥ π p (g − Ex)
A. Independent Random Vectors
(20)
The Benders algorithm is able to handle stochastic problem. Suppose that the vector g in (14) can assume values g1 , g2 , . . . , gm , with associated probabilities p1 , p2 , . . ., pm respectively (p1 +p2 +. . .+pm = 1). In this case, the optimization problem is to find the strategy that minimize
the expected value of the operation cost, that is min cx + p1 dy1 + p2 dy2 + . . . pm dym ⎧ Ax ≥ b ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ Ex + F y1 ≥ g1 Ex + F y2 ≥ g2 subject to : ⎪ .. ⎪ ⎪ . ⎪ ⎪ ⎩ Ex + F ym ≥ gm
B. Linearly Dependente Random Vectors
(24)
gt = ϕ1 gt−1 + ϕ2 gt−2 + . . . ϕp gt−p + ζt
Problem (24) corresponds to the following two-stage decision process [7]: 1) Determine a feasible decision x∗ such that Ax∗ ≥ b ∗ 2) Look for the solutions y1∗ , y2∗ , . . . , ym that optimize:
min p1 dy1 + p2 dy2 + . . . pm dym ⎧ F y1 ≥ g1 − Ex∗ ⎪ ⎪ ⎪ ⎨ F y2 ≥ g2 − Ex∗ subject to : .. ⎪ . ⎪ ⎪ ⎩ F ym ≥ gm − Ex∗
(25)
Problem (25) can be decomposed in m independent optimizations: w1 = min dy1 subject to : F y1 ≥ g1 − Ex∗
Let the lag-pt parametric autoregressive model PAR(pt ), where pt represents order or numbers of autoregressive terms of the model in the month t. In this case, the stochastic process of the inflows at the stage t, gt , can be represented by: (31)
where, ϕi represents the i-th autocorrelation coefficient of the normalized series and e ζt is a sequence of uncorrelated −1 −1 2 ), τm = σm . random variables with distribution N (0, τm In case of linearly dependente random vectors, the state variables are the water stored in the reservoir at the beginning of each stage, xt , and pt water inflows during the previous stage, {yt−1 , . . . , yt−pt }, which represents the hydrological trend. In the algorithm DSDP, the uncorrelated random variables {ζt , t = 1, . . . , T } are sampled and the {gt , t = 1, . . . , T } values are obtained according to (31) and substituted in the problem. The Benders cut at the stage t, associated with {xt−1,i , gt−1,i , . . . , gt−pt ,i } is represented as [14] ∗ wt,i + (π x )t,i (x∗t,i − xt,i ) + +(π gt−1 )t,i (gt−1,i − gt−1,i )+ ∗ + . . . + (π gt−pt )t,i (gt−p − gt−pt ,i ) ≤ α t ,i
(26)
(32)
where, for t = T , w2 = min dy2 subject to : F y2 ≥ g2 − Ex∗
(27)
(π gt−s )t,i = 0, s = pt , . . . , q
.. . wm = min dym subject to : F ym ≥ gm − Ex∗
(π yt−s )t,i = ϕs,t (π x )t,i , s = 1, . . . , pt otherwise,
(28) +
where the solutions of (26)-(28) are weighted by the probabilities p1 , p2 , . . ., pm , respectively. As in the deterministic case, the solution of each scenario in the second stage is a function of the first-stage decision x, and (24) can thus be rewritten as min cx + α(x) subject to : Ax ≥ b
(29)
n
(π gt−1 )t,i = ϕ1,t (π x )t,i + λt,j (ϕ1,t (π gt−1 )t+1,j + (π gt−2 )t+1,j ) .. .
n j=1
(π gt−pt +1 )t,i = ϕpt −1,t (π x )t,i + λt,j (ϕpt −1,t (π gt−1 )t+1,j + (π gt−pt )t+1,j )
(π gt−pt )t,i = ϕpt ,t (π x )t,i +
(30)
∗ where, w = p1 w1∗ + p2 w2∗ + . . . + pm wm is the expected value of the solution of the second-stage problems and π = ∗ is the expected value of the p1 π1∗ + p2 π2∗ + . . . + pm πm multipliers Simplex associated.
(34)
j=1
+
where, α(x) is the expected value of the solution of the (26)-(28) for each specified gm . Let π1 , π2 , . . ., πm the multipliers Simplex associated with constraints of the problems (26)-(28) end w1∗ , w2∗ , ∗ the corresponding optimal solution. The Benders . . ., wm cut associated with the (29) can be represented as w + πE(x∗ − x) ≤ α (x)
(33)
n
(35)
λt,j ϕpt ,t (π gt−1 )t+1,j
j=1
(36) (π gt−s )t,i = 0
∀s = pt+1 , . . . , q and q = max{p, ∀t}
(37)
V. Test Results This section provides a comparative analysis of SPD and DSDP policies in long term hydrothermal scheduling through simulation. The two approaches have been applied to the specific case of systems comprising a single hydro plant. Two plants located in different Brazilian river basins selected for the study were: Furnas, located on the Grande
TABLE I
Hydro plant characteristics.
Furnas Emborca¸ c˜ ao
Installed Capacity (MW) 1312 1192
Storage Capacity (hm3 ) 17217 12521
Discharge max/min (hm3 /mˆ es) 515/4446 202/2754
river and Emborca¸c˜ao on the Paranaiba. The main operational characteristics of these hydro plants are given in Table I. As is standard for planning studies in the Brazilian power system, the forebay elevation φ(.) and the tailrace elevation θ(.) are fitted by fourth degree polynomial functions of the water storage and discharge in the reservoir, where ai and bi are the coefficients of the terms with the exponent i of each polynomial function, respectively. Table II gives the polynomial coefficients, the value of
MW , and the average the constant k, given in (hm3 /month)m penstock head loss, in meters, respectively, for each hydro plant considered. TABLE II Hydro generation characteristics.
a0 a1 a2 a3 a4 b0 b1 b2 b3 b4 k hp
Furnas 7.3525×102 3.4966×103 -1.9744×10−7 6.9170×10−12 -9.7736×10−17 6.7163×102 3.8713×106 -2.6059×10−12 1.3847×10−18 0.0 0.008633 0.00981
Emborca¸c˜ ao 5.6809×102 1.4506×10−2 -1.2028×10−6 5.8303×10−11 -1.1245×10−15 5.1977×102 1.5208×10−5 -1.5908×10−11 1.2913×10−17 -3.6995×10−24 0.008731 0.0165481
the determination of the inflow future scenarios used during the optimization process and (2) as inflow forecasting model in the simulation process of this approach. In order to prove the benefit associated with the DSDP approach in coping with models more efficient than PAR(1) model to represent the actual stochastic process of the inflows, the PAR(pt ) model is also associated with this approach. In this paper, the DSDP optimization algorithm considers six possible future scenarios of inflows in each stage of optimization. The Benders cuts are determined based on four system states in each stage, selected by Monte Carlo simulation. The simulation process is based on information of state of reservoir storage in the begging of the month, hydrologic series simulated until the previous month and Benders cuts determined by the optimization algorithm. In order to compare the SDP and DSDP approaches, initially is analyzed if the Benders cuts determined by SDPD are a good approximation for the expected future cost determined by SDP. Let the optimization of the Furnas operation by DSDP. The cost at the end of the horizon is fixed as being the same expected future cost determined by SDP for the operation of this hydro plant. May is the initial month of operation. The figure 1 shows the expected future cost curve determined by SDP and the approximations of this curves determined by the DSDP, for the may month, considering optimization horizons of two and four months. The results show that the quality of the adjustment of the Benders cuts to the function of expected future cost of SDP tends to get worse when the stage of optimization is moved away of the final of the horizon optimization. In general, the results show that the Benders cuts determined by the DSDP do not reproduce suitability the expected future cost function determined by SDP. 7
7
x 10
2.25
2.2
2.2
2.15
2.15
Expected Future Cost ($)
The operational cost ψt is, in general, obtained by the optimal dispatch of the non-hydraulic sources available. Optimization ranks these sources according to their marginal costs, which results in a convex increasing operational cost function. For the non-hydraulic aspects of the Brazilian system, an estimate of the operational cost is given by the quadratic function (ψt (Gt ) = 0.5(Gt )2 ). The load demand Dt was considered both constant during the planning period and equal to the installed capacity of the hydro plant. This assures a balanced hydrothermal system since, for these three hydro plants, the firm energy is approximately 50% of the installed capacity. In the SDP, the PAR(1) model is used for provide the conditional probability density functions of the inflows. The optimal decision rule is the result of decision tables which provide the optimal hydro generation decision and the future expected operational cost for each possible state of the system. PAR(1) model is also associated with the DSDP: (1) in
Expected Future Cost ($)
2.25
2.1
2.05
2
1.95
x 10
2.1
2.05
2
1.95
1.9
1.9 SDP DSDP
1.85 0
20
40 60 Storage (%)
SDP DSDP 80
100
1.85 0
20
40 60 Storage (%)
80
100
Fig. 1. Expected Future Cost (SDP) and Aproximate Function Determined (DSDP).
The comparison between SDP and DSDP policies has been made using a simulation model which reproduces the behavior of the hydrothermal system. This simulation model provides the response of the system for specific inflow sequences, according to the policy adopted, thus
allowing the comparison of the two techniques for the same computational environment. Assuming that the inflows are known exactly for the planning period, the deterministic optimization of the scheduling of the system is also conducted (Optimal Solution (OS)). The statistics of interest for the simulations and OS are the values of the mean and standard deviation of hydroelectric generation and operational cost. The results obtained in the simulations using historical inflow records, available since 1931, for the hydro plants of Furnas and Emborca¸c˜ao are presented in Tables III e IV, respectively.
On the other hand, the use of a lag-pt parametric autoregressive model did not favor the performance of the dual dynamic programming the point to make the performance of this approach surpass the primal approach, which is restricted to models of lag-one. Although the dual programming does not require the discretization of the state space in the solution of the problem, the number of scenarios considered in the process of optimization of this approach increases exponentially with the number of stage of planning horizon. In this case, the stochastic representation of the system is limited the use of a reduced number of scenarios.
TABLE III
Acknowledgment
Hydroelectric Generation and Operational Cost of Furnas hydro plant.
The authors would like to acknowledge the support provided by the S˜ ao Paulo State Foundation for the Support of Research (FAPESP), and the Brazilian National Research Council (CNPq).
Policy OS SDP DSDP(PAR1) DSDP(PARpt )
Hyd.Gen. (M W ) Mean St.Dev. 743.0 174.0 719.6 198.2 703.0 334.2 708.3 296.8
Cost (105 $) Mean St.Dev. 1.7695 0.8984 1.9509 1.0113 2.4125 2.0538 2.2619 1.3270
TABLE IV
Hydroelectric Generation and Operational Cost of ˜o hydro plant. Emborca¸ ca Policy OS SDP DSDP(PAR1) DSDP(PARpt )
Hyd.Gen. (M W ) Mean St.Dev. 562.7 134.0 548.3 180.2 547.5 323.2 540.3 217.7
Cost (105 $) Mean St.Dev. 2.0698 0.8122 2.2339 0.9235 2.5981 0.1678 2.3601 0.9609
The results revealed lower average hydroelectric generation and higher operational costs with the use of the DSDP for all simulations and two hydroelectric plants considered. Based on the PAR(1) model, the DSDP presented mean operational cost about 46% higher in relation to SDP, in the case of Furnas, and about 15% higher in relation to SDP in the case of Emborca¸c˜ao. This differences are lower when the DSDP is associated whit the PAR(pt ) model. In this case, the difference of mean operational cost in relation to SDP is about 16% in the case of Furnas end 5,6% in the case of Emborca¸c˜ao. VI. Conclusion This paper has compared the primal and dual stochastic dynamic programming approaches in long term hydrothermal scheduling for hydrothermal systems composed of a single hydro plant. The comparison was made through simulation using historical inflow records. The dual stochastic dynamic programming did not reveal to equivalent the primal approach, as it expected in the case of the operation of a single hydro plant, assuming it same form of representation the actual stochastic process of the inflows, lag-one parametric autoregressive model.
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