1
Self-Scheduling of a Joint Hydro and Pumped-Storage Plants in Energy, Spinning Reserve and Regulation Markets S. Jalal Kazempour, Student Member, IEEE, Majid Hosseinpour, Student Member, IEEE, and Mohsen Parsa Moghaddam, Member, IEEE Abstract— This paper addresses the self-scheduling problem for a price-taker hydro generating company. This company is comprised of several cascaded hydro plants along a river basin as well as a pumped-storage plant. Due to existence of a suitable zone as a natural reservoir, it is assumed that the hydro generating company has constructed a pumped-storage plant using the mentioned natural zone as upper reservoir and one of its hydro dams as lower reservoir. The goal is maximizing the profit of company through participating in the day-ahead energy and ancillary service markets. In order to reach this goal, it is essential to have an appropriate approach to self-schedule of company. The spinning reserve and regulation markets are considered as ancillary services in which the company can participate. The self-scheduling problem of hydro generating company is therefore formulated and solved as a mixed integer non-linear programming (MINLP) problem. Numerical results for a case study are discussed. Index Terms— Hydro generating company, pumpedstorage plant, self-scheduling, energy market, spinning reserve market, regulation market
NOMENCLATURE Functions: φh,i ( xi , ui , t )
Power generation function of i-th hydro plant φ p,s ( x p , u p , t ) Power generation function of pumped-storage plant in its selling (discharging) mode φ p, p ( x p , wi , t ) Power consumption function of pumpedstorage plant in its purchasing (charging) mode
Variables: λe(t) λs(t) λr(t)
Market-clearing price of energy market on hour t ($/MWh) Market-clearing price of spinning reserve market on hour t ($/MWh) Market-clearing price of regulation market on hour t ($/MWh)
λspot(t)
Phe,i(t) Phs,i(t) Phr,i(t) Ppe,s(t) Ppe,p(t) Pps,s(t)
Pps,p(t)
Ppr(t) Pt,i(t) Phexp,i(t) Ppexp.(t) SUi(t) xi(t)
The authors are with the Electrical Engineering Department, Tarbiat Modares University (TMU), P.O. Box: 14115-143, Tehran, Iran. Emails:
[email protected] &
[email protected] (S.J. Kazempour),
[email protected] (M. Hosseinpour), and
[email protected] (M. Parsa Moghaddam).
978-1-4244-4241-6/09/$25.00 ©2009 IEEE
ui(t) xp(t)
Market-clearing price of spot market on hour t ($/MWh) Amount of power participated by i-th hydro plant to sell in the energy market at hour t (MW) Amount of power participated by i-th hydro plant in the spinning reserve market at hour t (MW) Amount of power participated by i-th hydro plant in the regulation market at hour t (MW) Amount of power participated by pumpedstorage plant to sell in the energy market at hour t (MW) Amount of power participated by pumpedstorage plant to purchase in the energy market at hour t (MW) Amount of power participated by pumpedstorage plant in the spinning reserve market at hour t, when pumped-storage operates in its selling mode (MW) Amount of power participated by pumpedstorage plant in the spinning reserve market at hour t, when pumped-storage operates in its purchasing mode (MW) Amount of power participated by pumpedstorage plant in the regulation market at hour t (MW) Amount of power consumed by pumpedstorage plant which is supplied by i-th hydro plant at hour t (MW) Amount of expected power to be generated by i-th hydro plant in energy and ancillary service markets on hour t (MW) Amount of expected power to be generated by pumped-storage plant in energy and ancillary service markets on hour t (MW) Start-up cost of i-th hydro plant on hour t ($) Reservoir storage of i-th hydro plant on hour t (Hm3) Water discharge of i-th hydro plant on hour t (Hm3) Reservoir storage of pumped-storage plant on
2 hour t (Hm3) Water discharge of pumped-storage plant on hour t (Hm3) Water spillage of i-th regulating hydro plant on hour t (Hm3) Independent inflow into reservoir of i-th hydro plant on hour t (Hm3) Dependent inflow into reservoir of i-th hydro plant on hour t due to discharge and spillage of upstream plants (Hm3) Water charge of pumped-storage plant supplied by i-th hydro plant on hour t (Hm3)
up(t) vi(t) yi(t) zi(t) wi(t)
Binary variable: Indicates whether the i-th hydro plant is on-line di(t) or not at hour t ds(t) Indicates whether the pumped-storage plant is in its selling mode at hour t dp(t) Indicates whether the pumped-storage plant is in its purchasing mode at hour t Constants: Pdel. Pr,up Pr,down SUi β
X i, X i
U i ,U i X p, X
p
U p ,U p
V i ,V i X i0
X iend X
0 p
X end p
τ m,i
Mu,i
The probability of calling plants to generate in the spinning reserve market The probability of being in the regulation-up state in the regulation market The probability of being in the regulation-down state in the regulation market Start-up cost of i-th hydro plant ($) Adjusting constant of stored water for subsequent time interval Bounds on reservoir storage of i-th hydro plant (Hm3) Bounds on water discharge of i-th hydro plant (Hm3) Bounds on reservoir storage of pumped-storage plant (Hm3) Bounds on water discharge of pumped-storage plant (Hm3) Bounds on water spillage of i-th hydro plant (Hm3) Amount of reservoir storage of i-th hydro plant in the beginning of concerned time interval (Hm3) Amount of reservoir storage of i-th hydro plant in the end of concerned time interval (Hm3) Amount of reservoir storage of pumped-storage plant in the beginning of concerned time interval (Hm3) Amount of reservoir storage of pumped-storage plant in the end of concerned time interval (Hm3) Water transport delay time from m-th reservoir to i-th reservoir Number of upstream hydro plants directly above hydro plant i
T I K1 K2
Concerned time interval (24 hrs) Number of hydro plants The fixed term of O&M costs of pumpedstorage plant The coefficient of variable term in O&M costs of pumped-storage plant I.
INTRODUCTION
A. Aim OWER producers participate in the power pool trading aim to maximize their profit in day-ahead (DA) energy and ancillary service markets. In a pool-based environment, power producers face challenging problems with the ultimate goal of maximizing their profits. One of those problems is determining, in the short term, the optimal self-schedule of the units belonging to the generating company considering production bids which are submitted based on the forecasted market clearing-prices (MCPs) in the energy and the ancillary service markets. In fact, the profit due to participate in the energy and the ancillary service markets is maximized ignoring the power balance in the system [1-2]. This problem is referred to as self-scheduling (SS), and it is the problem addressed in this paper.
P
B. Literature Review References [3-7] are among the early works in which SS problem has been addressed based on the day-ahead forecasted prices. In [3-5], the optimal operation scheduling problem for a price-taker power plant has been investigated. In [6], the SS problem for a hydro generating company in a pool-based energy market has been addressed. No pumpedstorage plant was considered in this paper. In [7], the optimal operation scheduling of individual pumped-storage plant in the energy and the spinning reserve markets was investigated without considering stored energy constraint in the upper reservoir. In [8], a simple heuristic method for optimal operation scheduling problem of hydro-electric and also pumped-storage plants has been proposed. In this method, the optimal operation scheduling problem is performed to maximize the potential of revenues in each hour. Although this method is very simple, it can not achieve to a global optima. C. Contribution This paper is focused on a comprehensive self-scheduling problem development for a joint hydro and pumped-storage plants in a pool-based electricity market to bid in day-ahead energy and ancillary service markets. In fact, we consider a hydro generating company which is comprised of several cascaded hydro plants along a river basin as well as a pumped-storage plant. Due to existence of a suitable zone as a natural reservoir, it is assumed that the hydro generating company has constructed a pumped-storage plant using the mentioned natural zone as upper reservoir and one of its
3
P he,i(t)
P hs,i(t)
Ppe,p (t) Ppe,s (t) Pps ,s (t) P ps,p(t) Ppr(t)
Phr,i(t) P t,i(t)
Hydro Plants
w i(t)
Pumped-Storage Plant
up(t) zi(t)
y i(t)
ui(t)
Power flow Water inflow
Fig.1.The concept of joint hydro plants and pumped-storage plant
hydro dams as lower reservoir. The concept of joint hydro plants and pumped-storage plant is depicted in Fig. 1. The power flow and water inflow in joint hydro and pumped-storage plants are separately shown in Fig. 1. As it can be seen in this figure, the hydro plants can transfer energy to the pumped-storage plant as well as participating in the energy, spinning reserve and regulation markets. Consequently, not only the pumped-storage plant can be charged through energy market, but also it can be charged through hydro plants. The pumped-storage plant can participate in the energy, spinning reserve and regulation markets when it operates in its selling (discharging) mode. In purchasing (charging) mode, the pumped-storage plant purchases electricity but it can be committed for spinning reserve, because it can readily reduce its purchasing power and consequently reduce the overall system load. The pumped-storage plant can only be participated in the spinning reserve market when it is in its off-line mode. The analysis will be performed considering the energy, the spinning reserve and the regulation markets, simultaneously. The power producer is considered as price-taker, i.e., a plant with no capability of altering the MCPs. The profit maximization problem faced by the hydro generating company is therefore formulated and solved as a MixedInteger Non-Linear Programming (MINLP) problem. D. Paper Organization The rest of this paper is organized as follows. Section II describes the different states of spinning reserve and regulation markets. Section III is devoted to self-scheduling problem formulation of hydro generating company. Section IV illustrates the numerical results of a case study. Section V is dedicated to conclusions of the paper. II.
ANCILLARY SERVICE MARKETS
When the hydro generating company participates in the ancillary service markets for a specific hour, it receives hourly ancillary service price and also hourly spot price if company is called to generate. Generally, if the hydro generating company commits in spinning reserve market, following states may occur: • Company is called to generate: in this state, the company receives both of hourly spinning reserve and spot prices.
The amount of latter income depends on the amount of extra energy that delivers as spinning reserve power. The probability of being in this state is presented by Pdel.. • Company is not called to generate: in this state, the plant receives according to hourly spinning reserve price. It is obvious that the probability of being in this state is equal to (1-Pdel.). In addition, if the company participates in a DA regulation market, following states may occur: • Regulation-up: In this state, the amount of generated power must be increased. The company receives all of hourly energy, regulation and spot prices. The latest income depends on the amount of extra energy that is requested. Pr,up shows the probability of being in the regulation-up state. • Regulation-down: In this state, the amount of generated energy must be decreased. In regulation-down state, the company receives according to hourly energy and regulation prices, but must repay for power not generated according to hourly spot price. The probability of being in regulation-down state in a specific hour is presented by Pr,down. • No-regulation: In this state, the amount of generated power is not changed and the plant receives hourly energy and regulation prices. The probability of being in this state is calculated by (1- Pr,up - Pr,down). III.
FORMULATION OF SELF-SCHEDULING PROBLEM
The self-scheduling problem is executed to optimally determine the hourly production bids in the energy and the ancillary service markets to maximize company’s profit while all the operational constraints are satisfied. The analysis will be performed considering the energy, the spinning reserve and the regulation markets, simultaneously. In the case of hydro plants, the production costs are negligible. The start-up costs have real impact on the short-term scheduling of hydro plants. Start-up costs are mainly caused by the increased maintenance of windings and mechanical equipment and by malfunctions of the control equipment [9-10]. However, in the case of pumped-storage plant, the O&M costs are considered using fixed and variable terms [11]. By assuming the incomes, payments, start-up and O&M costs of hydro generating company, the objective function and respected constraints of self-scheduling problem over a concerned time interval can be represented by (1)-(30). The first six terms of (1) represent the revenues of hydro generating company earned by participating in energy, spinning reserve and regulation markets, respectively. As discussed in section II, the company expects to receive extra income when it is called to generate in the ancillary service markets. These expected revenues are shown by seventh to tenth terms. Finally, two last terms show the start-up and O&M costs of company.
4 Maximize:
∑∑ ∑ T
I
Phe,i (t ).λe (t ) +
t =1 i =1 T
∑ ∑∑ T
t =1
Ppr (t ).λr (t ) + Pdel..
+
( Ppe,s (t ) − Ppe, p (t )).λe (t ) +
t =1
∑ T
+ ( Pr ,up − Pr , down).
T
I
∑∑ ∑ ∑{ T
Ppr (t ).λspot (t ) −
t =1
∑∑ T
SUi (t ) −
t =1 i =1
∑ T
t =1
∑∑ ∑∑
(Pps, s (t ) + Pps, p (t )).λs (t ) +
( Pps,s (t ) + Pps, p (t )).λspot (t ) + ( Pr ,up − Pr ,down).
t =1 T
I
Phs,i (t ).λs (t ) +
t =1 i =1 T
Phs,i (t ).λspot (t ) +Pdel..
t =1 i =1
I
T
T
I
Phr ,i (t ).λr (t )
t =1 i =1 I
Phr,i (t ).λspot (t )
(1)
t =1 i =1
[
]}
K1 + K 2. (Ppe, s (t ) + Ppe, p (t ) + ( Pr ,up − Pr , down).Ppr (t ) + Pdel..(Pps,s (t ) − Pps, p (t ))
t =1
s.t. X i ≤ xi (t ) ≤ X i
X
p
≤ x p (t ) ≤ X
(2) (3)
p
U i .d i ( t ) ≤ u i ( t ) ≤ U U
p
.d s ( t ) ≤ u
p
(t ) ≤ U
i
.d i ( t )
(4)
p
.d s ( t )
(5)
V i .di (t ) ≤ vi (t ) ≤ V i .di (t )
(6)
Ph exp,i (t ) = Phe,i (t ) + Pt ,i (t ) + Pdel. .Phs ,i (t ) + ( Pr ,up − Pr ,down ).Phr ,i (t )
(7)
Pp exp (t ) = Ppe,s (t ) + Pdel. .Pps ,s (t ) + ( Pr ,up − Pr ,down ).Ppr (t )
(8)
x i (t + 1) = x i (t ) + y i (t ) + z i (t ) − u i (t ) − v i (t )
xi (t + 1) = xi (t ) + y i (t ) + z i (t ) + u p (t ) − u i (t ) − vi (t ) − wi (t )
i ∉ lower reservoir of pumped − storage plant i ∈ lower reservoir of pumped − storage plant
x p (t + 1) = x p (t ) + wi (t ) − u p (t ) M u ,i
z i (t ) =
∑ (u
m (t
− τ m,i ) + vm (t − τ m ,i ))
(9) (10) (11) (12)
m=1
wi (t ) ≤ Min[ xi (t ), X p − x p (t ), wmax ]
(13)
Ph exp,i (t ) = φ h,i ( xi , ui , t ) = d i (t ).(c1,i .xi2 (t ) + c 2,i .ui2 (t ) + c3,i .xi (t ).ui (t ) + c4,i .xi (t ) + c5,i .ui (t ) + c6,i )
(14)
Pp exp (t ) = φ p,s ( x p , u p , t ) = d s (t ).(c1, ps .x 2p (t ) + c 2, ps .u 2p (t ) + c3, ps .x p (t ).u p (t ) + c 4, ps .x p (t ) + c5, ps .u p (t ) + c6, ps )
(15)
φ p , p ( x p , wi , t ) = d p (t ).(c1, pp .x 2p (t ) + c2, pp .wi2 (t ) + c3, pp .x p (t ).wi (t ) + c4, pp .x p (t ) + c5, pp .wi (t ) + c6, pp )
(16)
φ p, p ( x p , wi , t ) = Ppe, p (t ) +
∑ I
pt ,i (t )
(17)
i =1
Phr ,i (t ) ≤ Phe,i (t )
(18)
Ppr (t ) ≤ Ppe,s (t )
(19)
0 ≤ Phr ,i (t ) ≤ φ h,i ( xi ,U i , t ) 2
(20)
0 ≤ Ppr (t ) ≤ φ p,s ( x p ,U p , t ) 2
(21)
Ppe,s (t ) + Pps,s (t ) + Ppr (t ) ≤ φ p ,s ( x p ,U p , t )
(22)
Phe,i (t ) + Phs ,i (t ) + Phr ,i (t ) ≤ φ h,i ( xi ,U i , t )
(23)
SU i (t ) = SU i .d i (t ).(1 − d i (t − 1))
(24)
0 ≤ Pps , p (t ) ≤ d p (t ).Ppe, p (t )
(25)
d s (t − 1) + d p (t ) ≤ 1
(26)
d p (t − 1) + d s (t ) ≤ 1
(27)
d s (t ) + d p (t ) ≤ 1
(28)
X iend X end p
= β . X i0 = β . X 0p
(29) (30)
5 Hydro and pumped-storage plants must adjust their water storage, discharge and spillage in the acceptable ranges. These cases are applied by (2)-(6). The amount of hourly expected power which must be generated to response in the energy, spinning reserve and regulation markets by hydro and pumped-storage plants are represented by (7) and (8), respectively. The hourly water amount of each hydro plant in its reservoir can be calculated by (9). This equation is modified by (10) for hydro reservoir which is considered as lower reservoir of pumped-storage plant. The hourly water amount of upper reservoir of pumped-storage plant can be calculated by (11). In equations (9)-(10), the input water of each hydro plant is categorized to independent and dependent inflows. The natural inflow of each hydro plant is considered as independent inflow, while the dependent inflow is caused due to discharge and spillage of upstream hydro plants. The amount of dependent inflow of each hydro plant considering water transport delay time is determined by (12). The water is transported between lower and upper reservoirs of pumped-storage plant through a special pipeline named “penstock”. As it is presented in (13), the hourly amount of water charge of pumped-storage plant is limited due to the available water in lower reservoir, the vacant capacity in upper reservoir and penstock’s capacity (wmax). Equations (14)-(15) characterize the hydro-electric generation function of each hydro plant and pumped-storage plant, respectively [12]. Also, the amount of required power of pumped-storage in its charging mode is determined by (16) which must be provided by internal generated power of company or by purchasing from energy market referred in (17). In order to ensure regulation-down service, constraints (18)-(19) are applied. In (20)-(21), the lower and upper limits of regulation power which can be produced by hydro and pumped-storage plants are presented where the upper limit is considered as half of maximum capability of power generation, in order to response to both regulations up and down requests. The lower limit of water stored in each hydro plant and in upper reservoir of pumped-storage plant must be adjusted to allow the company to response to the worst condition from the viewpoint of energy stored level. The worst condition may occur when company is called to generate in the spinning reserve and the regulation-up markets, simultaneously. Eqs. (22)-(23) are applied to consider this condition. The start-up times of each hydro plant are identified in (24). Also, the upper limit of spinning reserve power in purchasing mode of pumped-storage plant is represented by (25). In (26)-(27), changeover times of the pumped-storage plant are modeled. The changeover time of a pumped-storage plant is typically between 15 to 30 minutes. For a DA market operated on an hourly basis, this constraint translates to a plant having a buffer of at least one hour at zero generation between selling and purchasing modes [7]. Eq. (28) is considered to eliminate conflict between different modes in a specific hour. Eqs. (29)-(30) are contemplated in order to reserve enough energy stored in each hydro plant and upper reservoir of pumped-storage plant for the subsequent time interval. The parameter β adjusts the amount of water that should be stored for the subsequent time
interval. If lower prices for the next time interval are forecasted, the company will choose a low value for β. This parameter can be varied while the stored water constraints are satisfied. The optimization problem in (1)-(30) is a MINLP problem that can be solved by any commercial software. In this paper, it will be solved using SBB [13] under GAMS [14]. IV.
NUMERICAL RESULTS
A. Case Study The numerical results which are presented in this section consist of self-scheduling for a price-taker hydro generating company in day-ahead energy, spinning reserve and regulation markets. The time horizon comprises 24 hrs. The test hydro company is comprised of four cascaded hydro plants where one of them is regulating plant. Also, the third hydro plant is considered as lower reservoir of pumpedstorage plant as shown in Fig. 2. The coefficients of generation function for each hydro plant and pumped-storage plant as well as consumption function of pumped-storage plant are represented in Table I. Bounds of reservoir’s storage, water discharge, water spillage and initial storage for each hydro plant and upper reservoir of pumped-storage plant are depicted in Table II. The assumed forecasted prices for the energy, the spinning reserve and the regulation markets are shown in Table III. Price data are adopted from electric energy market of Mainland, Spain [15] with a few adjustments. Also, Table III shows the forecasted hourly independent water inflow for each hydro plant. The time delays for water transportation are τ1,3 = 2, τ2,3 = 3 and τ3,4 = 4. The probability of calling plants to generate energy in the spinning reserve market (Pdel.) is assumed to be 3% [16]. Also, the probabilities of regulation-up and regulation-down states are considered 40% and 35% respectively, which seem rational assumptions [17]. TABLE I COEFFIECIETS OF POWER GENERATION/CONSUMPTION FUNCTION OF PLANTS
C1
C2
C3
C4
C5
C6
Plant 1
-0.001
-0. 1
0.01
0.40
4.0
-30.0
Plant 3
-0.001
-0. 1
0.01
0.30
3.0
-30.0
Plant 4
-0.001
-0. 1
0.01
0.45
4.5
-30.0
-0.001
-0. 1
0.01
0.42
4.2
-30.0
-0.001
-0. 1
0.01
0.70
7.0
-30.0
Pumped-Storage (discharging) Pumped-Storage (charging)
TABLE II PLANTS DATA
X
X
U
U
X0
Plant 1
80
150
5
15
100
0
0
Plant 2
60
120
0
0
80
10
30
Plant 3
50
300
10
30
170
0
0
Plant 4
70
160
13
25
100
0
0
40
250
10
30
100
0
0
PumpedStorage
V
V
6
Fig.2. Configuration of hydro and pumped-storage plants
TABLE III FORECASTED PRICES AND INDEPENDENT INFLOWS
Hour
λe(t)
λs(t)
λr(t)
y1(t)
y2(t)
y3(t)
y4(t)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
55.3 50.0 45.7 44.0 42.9 42.6 46.9 56.8 70.6 75.9 70.9 77.0 73.4 75.9 67.8 66.0 64.8 65.2 68.6 72.5 77.7 75.9 68.4 70.3
43.2 30.0 27.5 29.2 29.2 29.2 44.1 39.9 28.5 20.2 22.3 19.1 13.2 23.4 30.0 30. 9 30.0 30.0 30.0 30.0 22.7 28.2 28.6 27.5
34.0 10.0 10.0 10.0 10.0 10.0 10.0 51.1 60.0 57.3 65.0 62.5 73.0 70.0 65.0 62.5 51.8 49.8 57.3 69.4 66.9 66.9 61.4 60.0
10 9 8 7 7 7 8 9 10 10 10 10 0 0 0 0 0 0 0 0 0 0 0 0
28 28 29 25 22 20 16 14 12 10 8 8 8 8 8 8 8 8 8 16 20 28 28 28
20 20 10 10 5 4 5 5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 20 20
20 20 20 18 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 18 20 20 20
The start-up costs of plant1, plant2 and plant 3 are considered to be $550, $1500 and $1000, respectively. The adjusting constant (β) is assumed to be 1. Also, the maximum
penstock transfer capability (wmax) is assumed to be 30MW. The random-based method shown by (31) is used to forecast the hourly spot price [18-20]. The hours between 9 and 18 are contemplated as the peak period. In order to present the spike price in the peak hours, a number of spikes are randomly generated using Frechet distribution [21]. t ∈ [9,18] ⎧ (1 + γ )λe (t ) 0 ≤ γ ≤ 0.25 ( 1 + μ ) λ ( ) − 0 . 1 ≤ μ ≤ 0 . 1 t otherwise e ⎩
λ spot (t ) = ⎨
(31)
B. Results of SS problem Here, the SS problem results are presented for a mentioned case study. Table IV presents the results of self-scheduling problem for the considered hydro generating company. In this table, the hourly production bids of hydro and pumpedstorage plants in the energy, spinning reserve and regulation markets are shown. The Expected daily profit is equal to $319888.198. The fluctuations of water storage amount in hydro and pumped-storage plants are shown in Table V. As it can be seen, assuming β=1 forces the hydro and pumped-storage plants to equalize the initial and final amounts of their water storage. Also, this table shows the discharge amount of hydro and pumped-storage plants, spillage amount of hydro plant 2 (regulating plant) and charge amount of pumped-storage plant during concerned time interval. Finally, Table VI shows the company’s strategy to provide charging energy of pumped-storage plant in its purchasing mode.
7 TABLE IV SELF-SCHEDULING RESULTS
Hour 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Pe,1(t) 22.7 23.7 24.4 0.0 0.0 0.0 0.0 27.5 28.4 29.2 30.0 30.8 30.0 29.2 28.4 27.5 0.0 0.0 26.5 25.6 24.5 23.4 22.3 21.1
Hydro Plant1 Ps,1(t) 31.4 31.8 32.1 57.2 0.0 60.0 60.9 7.0 6.7 6.4 6.1 5.9 6.1 6.4 6.7 7.1 0.0 0.0 7.5 7.9 8.4 8.9 9.6 10.2
Pr,1(t) 0.0 0.0 0.0 0.0 0.0 0.0 0.0 27.5 28.4 29.2 30.0 30.8 30.0 29.2 28.4 27.5 0.0 0.0 26.5 25.6 24.5 23.4 22.3 21.1
Pe,3(t) 43.1 43.1 38.9 0.0 0.0 0.0 0.0 13.3 16.7 24.9 31.0 26.0 26.1 25.7 12.3 21.4 24.8 25.5 27.2 27.3 39.2 35.0 36.6 43.1
Hydro Plant3 Ps,3(t) 0.0 0.0 0.0 34.4 27.6 19.6 8.9 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Pr,3(t) 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 8.8 7.9 12.3 1.7 6.3 7.5 11.1 11.3 0.0 0.0 0.0 0.0
Pe,4(t) 61.5 80.4 79.5 0.0 0.0 0.0 0.0 94.1 96.2 98.4 100.4 102.5 104.4 98.9 46.8 95.9 89.2 82.1 78.5 70.6 72.2 74.9 77.4 80
Hydro Plant4 Ps,4(t) Pr,4(t) 21.9 46.9 1.5 0.0 75.9 0.0 87.2 0.0 89.6 0.0 91.8 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Ppe,s(t) 40.0 36.6 0.0 0.0 0.0 0.0 0.0 0.0 62.5 61.1 59.5 57.7 55.7 0.0 0.0 0.0 59.5 57.7 55.7 53.5 51.1 48.5 45.7 42.7
Pumped-Storage Plant Pps,s(t) Pps,p(t) Ppr(t) 22.6 0.0 0.0 20.6 0.0 0.0 0.0 0.0 0.0 0.0 56.8 0.0 0.0 98.9 0.0 0.0 99.2 0.0 0.0 125.7 0.0 0.0 0.0 0.0 0.0 0.0 44.2 0.0 0.0 42.1 0.0 0.0 40 0.0 0.0 37.9 0.0 0.0 35.8 0.0 0.0 0.0 0.0 173.3 0.0 0.0 0.0 0.0 0.0 0.0 40.0 0.0 0.0 9.0 0.0 0.0 35.8 0.0 0.0 33.7 0.0 0.0 31.5 0.0 0.0 29.4 0.0 0.0 27.3 0.0 0.0 25.2
TABLE V HOURLY WATER STORAGE, DISCHARGE, SPILLAGE AND CHARGE AMOUNTS OF PLANTS
Hour 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
x1(t) 100 105 109 112 114 121 123 126 130 135 140 145 150 145 140 135 130 130 130 125 120 115 110 105 100
x2(t) 80 78 76 75 70 62 60 60 60 72 60 68 60 68 60 68 60 68 60 68 84 86 84 82 80
x3(t) 170 170 170 155 140 120 99 74 84 92 112 129 115 141 137 112 108 130 136 153 154 156 142 147 170
x4(t) 100 107 104 99 92 115 120 125 130 135 140 145 150 155 141 129 134 119 104 97 82 85 90 95 100
xp(t) 100 90 80 80 110 140 170 200 200 190 180 170 160 150 150 180 180 170 160 150 140 130 120 110 100
u1(t) 0 5 5 5 5 0 5 5 5 5 5 5 5 5 5 5 5 0 0 5 5 5 5 5 5
u3(t) 0 30 30 30 30 30 30 30 30 30 11 12 30 10 10 17 10 10 10 10 10 30 30 30 30
u4(t) 0 13 23 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25
up(t) 0 10 10 0 0 0 0 0 0 10 10 10 10 10 0 0 0 10 10 10 10 10 10 10 10
v2(t) 0 30 30 30 30 30 22 16 14 0 20 0 16 0 16 0 16 0 16 0 0 18 30 30 30
w3(t) 0 0 0 0 30 30 30 30 0 0 0 0 0 0 0 30 0 0 0 0 0 0 0 0 0
TABLE VI PROVIDED MW TO CHARGE PUMPED-STORAGE PLANT IN ITS PURCHASING MODE
Charging hours 4 5 6 7 15
Consumed MW supplied from energy market Ppe,p (t) 56.7 98.9 99.2 125.7 173.3
Consumed MW supplied from internal generation of company Pt,1(t) Pt,3(t) Pt,4(t) 24.1 33.4 73.6 0.0 26.7 84.6 25.9 19.0 86.9 26.5 8.6 89.1 0.0 19.7 44.5
8
V.
CONCLUSION
A comprehensive approach is developed in this paper to optimally schedule a price-taker hydro generating company. The day-ahead energy, spinning reserve and regulation markets are considered. This company is comprised of several cascaded hydro plants along a river basin as well as a pumped-storage plant. Due to existence of a suitable zone as a natural reservoir, it is assumed that the company has constructed a pumped-storage plant using the mentioned natural zone as upper reservoir and one of its hydro reservoirs as lower reservoir. The developed approach in this paper guarantees the maximum expected profit achievement during the concerned time interval while all of the technical constraints are satisfied. In order to make self-scheduling problem more precise, it is necessary to consider the profit risk due to important uncertainties such as forecasted price and independent water inflow uncertainties which will be included in our future works.
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Seyyed Jalal Kazempour was born in Marand, Iran, on 1985. He received his B.Sc. degree in Electrical Engineering from Tabriz University, Tabriz, Iran in 2006. He is presently a M.Sc. student in the Department of Electrical Engineering, Tarbiat Modares University (TMU), Tehran, Iran. His research interests are mainly operation, distributed resources and power systems
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economics. Majid Hosseinpour was born in Ardabil, Iran, on 1983. He received his B.Sc. degree with honors in Electrical Engineering from Tabriz University, Tabriz, Iran in 2005, the M.Sc. degree from the Tarbiat Modares University (TMU), Tehran, Iran in 2008. He is presently a Ph.D. student in the Department of Electrical Engineering, TMU, Tehran, Iran. Mohsen Parsa Moghaddam was born in Iran, on 1956. He received the B.Sc. degree in electrical engineering from Sharif University of Technology, Tehran, Iran, in 1980, the M.Sc. degree from the Toyohashi University of Technology, Japan, in 1985, and the Ph.D. degree from Tohoku University, Japan, in 1988. Currently, he is an Associate Professor in the Department of the Electrical Engineering, Tarbiat Modares University (TMU), Tehran, Iran. His research interests include power system planning & control, optimization, and restructuring.
