MedPower 2008 6th Mediterranean Conference on Exhibition Distribution, Thessaloniki, Greece, 2-5 November 2008

and

Power

Generation,

Transmission

and

Energy and Reserve Interaction in Greece’s Electricity Market Panagiotis Andrianesis*, George Liberopoulos, George Kozanidis Department of Mechanical & Industrial Engineering, University of Thessaly Pedion Areos,Volos, Greece * Tel. No.: +30 24210 74043, Fax: +30 24210 74059, E-mail: [email protected]

ABSTRACT: We model the Day-Ahead Scheduling (DAS) problem of Greece’s Electricity Market as a mixed integer linear programming (MILP) problem that co-optimizes energy and reserve. The shadow prices of the linear program, which results after fixing the integer variables of the original MILP to their optimal values, serve to analyze the resulting clearing prices and identify the interaction between energy and reserve. An 8-unit example that models Greece’s system is stated and solved, in order to demonstrate this interaction. Results are used to make practical remarks and suggest directions for further research. Keywords: Electricity market deregulation, reserves market, Day-ahead scheduling problem.

the installation of new generation units in the South, that is, near consumption. In this paper, we make several assumptions regarding the market rules and the numerous technical constraints of the generation units and the transmission system, in order to state a basic DAS model. The problem is formulated as a mixed integer linear programming model (MILP), which simultaneously optimizes energy, reserve, start-up and shut-down costs. A simple but rather indicative example is presented and solved using the mathematical programming language AMPL [3] and the optimization software package ILOG CPLEX 9.0. Results reveal the interaction between energy and reserve markets and provide the basis for more thorough research.

I. INTRODUCTION

Electricity market deregulation has triggered a number of significant changes in Greece’s energy sector, mostly by allowing private-owned companies to become producers and suppliers of electricity. The “Grid Control and Power Exchange Code for Electricity” [1] introduced the DayAhead Scheduling (DAS) problem, which forms the basis of the wholesale electricity market operation. The DAS aims at minimizing the total cost of serving energy load for the next day, under conditions of good and safe system operation, while ensuring adequate reserves. Greece’s main characteristic concerning the location of generation and consumption is that while most of its power plants, which also happen to be the cheapest ones (lignite units), are located in the North, the majority of energy consumption takes place in the South. To deal with this particularity of excess capacity in the North, Greece is divided in two operational zones (North – South) and producers are paid at different prices (Marginal Generating Prices) when, in case of high load, a transmission constraint is activated, prohibiting the transfer of the desired amount of energy from the North to the South. Suppliers, however, face a uniform price (System Marginal Price) regardless their location. The justification for this two-zone model was presented in [2], where it was shown that the two-zone model provides the right incentives for

II. THE DAY-AHEAD SCHEDULING PROBLEM The DAS problem is solved every day, simultaneously for all 24 hours of the next day. The objective is to minimize the cost of matching the energy to be absorbed with the energy to be injected in the system, under the transmission constraints, the generation units’ technical constraints and the reserve requirements. The DAS problem defines how each unit should operate in each hour, so that the social welfare of the electricity market is maximized. It also determines the clearing prices of the energy and reserve markets. In this paper, we focus only on thermal electricity– generation units. Hydro plants and renewable sources are subject to different rules and scheduling and have been excluded from our analysis. Imports and exports are also part of a different mechanism and are not included in our model. The demand is reduced to take into account the absence of energy injection from hydro plants, renewable sources and net imports. By the term “reserve”, we refer to the frequency-related ancillary services. In our model, we include only the tertiary spinning and non-spinning reserve. Primary reserve is not considered as the amount of it required is quite small. Secondary reserve is not included either, as it is usually provided by hydro units. The producers submit energy offers for each hour of the following day, as a stepwise function of price-quantity pairs, with successive prices being strictly non-decreasing. They also submit reserve bids as a price-quantity pair, and start-up and shut-down costs. The technical characteristics of the generation units that constitute the constraints of the DAS problem include the technical minimum and maximum output, the maximum reserve availability, the minimum up and down times, and the ramp up and down limits. Demand for energy and reserve requirement are exogenously determined by the Hellenic Transmission

MedPower 2008 6th Mediterranean Conference on Exhibition Distribution, Thessaloniki, Greece, 2-5 November 2008

and

System Operator (HTSO). A minimum zonal reserve requirement is also determined and an N – 1 criterion in case of a unit loss in the South is stated. There is also a transmission constraint that refers to the maximum allowable energy flow from North to South.

Shut-down for unit u , hour h . Dependent binary variable. 1: Shut-down Integer variable. Counter of hours unit u has been ON since last start-up Integer variable. Counter of hours unit u has been OFF since last shut-down Total generation (output) for unit u , hour h . Dependent variable Energy flow from the North to the South. Dependent variable

Sets – Subsets: U Generation units Un Generation units in the North (subset of U ) Us Generation units in the South (subset of U ) Parameters: u Generation unit h Time period (hour: 1..24) b Block steps of bid function (energy offer) Dhn Demand (load) in the North for hour h Dhs Dh

Vu , h X u ,h

Wu , h Gu , h Fh

Minimum reserve requirement in the North

min s

Pur, h

Minimum reserve requirement in the South Reserve requirement in case of a unit loss in the South Reserve requirement for hour h Quantity of energy offer for unit u , block b , hour h Price of energy offer for unit u , block b , hour h Price of reserve offer for unit u , hour h

u∈U

Qumax

Technical maximum for unit u

Fh + RnN −1 −

RsN −1 Fmax gbid u ,b , h

Q

Pug,b , h

Qumin

Technical minimum for unit u

R

Maximum reserve availability for unit u

SUCu

Start-up cost for unit u Shut-down cost for unit u Minimum up time for unit u Minimum down time for unit u Ramp up rate (in MW/hour) for unit u Ramp down rate (in MW/hour) for unit u

bid u

SDCu MU u MDu RU u RDu 0 u

and

u ,h

+ ∑ Yu , h ⋅ SUCu + ∑ Vu , h ⋅ SDCu }

min n

R

Transmission

{ ∑ Pug,b , h ⋅ Qug,b , h + ∑ Pur, h ⋅ Ru , h +

min

Qug,b ,h , Ru ,h , STu ,h ,Yu ,h , u , b , h Vu ,h , X u ,h ,Wu ,h , Gu ,h , Fh

req h

R

Generation,

The DAS problem is formulated as a mixed integer programming (MIP) problem, as follows:

Demand (load) in the South for hour h Demand (load) for hour h (equals the sum of demand in the North and in the South) Reserve requirement for hour h

R

Power

u ,h

subject to: ∑ Gu ,h = Dh

(1)

u,h

∀h

(2)

u∈U

∑R

u,h

≥ Rhreq

∀h

(3)

∑G

− Fh = Dhn

∀h

(4)

∑G

+ Fh = Dhs

∀h

(5)

u ,h

u∈U n

u,h

u∈U s

Fh ≤ Fmax

∀h

(6)

∑R

min n

≥R

∀h

(7)

∑R

≥ Rsmin

∀h

(8)

u,h

u∈U n

u,h

u∈U s

∑R

u∈U s

u ,h

Gu , h ≥ STu , h ⋅ Q

Gu , h + Ru , h ≤ STu , h ⋅ Q

Ru , h ≤ STu , h ⋅ R

(10)

∀u , h max u

bid u

(9)

∀u, b, h

Qug,b , h ≤ STu , h ⋅ Qugbid ,b, h min u

∀h

≤ Fmax

(11) ∀u, h

(12)

∀u , h

(13)

( X u , h −1 − MU u )( STu , h −1 − STu , h ) ≥ 0

∀u, h

(14)

(Wu , h −1 − MDu )( STu , h − STu , h −1 ) ≥ 0

∀u , h

(15)

Gu , h − Gu , h −1 + Ru , h ≤ RU u + Yu , h ⋅ Qumin

∀u, h

(16)

ST

Initial status of unit u (at hour 0)

X u0

Gu , h −1 − Gu , h ≤ RDu + Vu , h ⋅ Q

Hours unit u has been “ON” at hour 0

Yu , h = STu , h (1 − STu , h −1 )

∀u, h

(18)

Wu0

Hours unit u has been “OFF” at hour 0

Vu , h = STu , h −1 (1 − STu , h )

∀u , h

(19)

Gu0

Initial generation of unit u at hour 0

X u , h = ( X u , h −1 + 1) STu , h

∀u, h

(20)

Decision variables: Quantity of energy included in DAS for unit u , Qug,b , h block b , hour h Ru , h Reserve included in DAS for unit u , hour h

STu , h Yu , h

Status (condition) for unit u , hour h . Binary variable. 1: ΟΝ(LINE), 0:OFF(LINE) Start-up for unit u , hour h . Dependent binary variable. 1: Start-up

min u

Wu , h = (Wu , h −1 + 1)(1 − STu , h )

∑Q

= Gu , h ∀u, h

g u ,b , h

∀u , h

∀u, h

(17)

(21) (22)

b

STu ,0 = STu0

∀u

(23)

0 u

∀u

(24)

0 u

Wu ,0 = W

∀u

(25)

Gu ,0 = Gu0

∀u

(26)

X u ,0 = X

MedPower 2008 6th Mediterranean Conference on Exhibition Distribution, Thessaloniki, Greece, 2-5 November 2008

As it was already mentioned, the DAS problem aims at minimizing cost function (1), which includes expenses for producing energy and providing reserve, and costs for starting up and shutting down the power plants. Equation (2) states the energy balance, and constraint (3) ensures adequate reserve for the system. Constraints (4) and (5) describe the energy balance for each zone (North and South), while (6) defines the transmission constraint (flow limit) from North to South. Note that one of the constraints (2), (4) and (5) is redundant as it can be extracted by adding or subtracting the other two. However, they are all included in the formulation for clarity. Constraints (7) and (8) ensure a minimum reserve requirement for each zone, while (9) meets the N - 1 criterion in case of a unit loss in the South. Constraints (10)-(17) refer to the technical characteristics of generation units. More specifically, (10) ensures that the energy of each block that is included in the DAS does not exceed the block step that is declared in the energy offer. (11) and (12) define the technical minimum and maximum output constraints, while (13) refers to the maximum reserve availability. Constraints (14)–(15) refer to the minimum up and down times of the units, and (16)-(17) define the ramp up and down limits. Equations (18)-(22) define the dependent variables and (23)-(26) declare the initial values. All decision variables are nonnegative. Note that constraints (14)-(15) and (18)-(21) are non linear. To make them linear, we can replace them with equivalent inequalities, introducing auxiliary variables wherever necessary. For example, to eliminate the non linear term in equality (18), we can replace that equality, with the following two inequalities: Yu , h ≥ STu , h − STu , h −1 ∀u , h (27) STu , h − STu , h −1 + 1.1(1 − Yu , h ) ≥ 0.1 ∀u , h (28) After replacing all the non linear constraints with linear inequalities, the resulting model is an MILP problem. As the main purpose of this paper is to reveal the interaction between energy and reserve, we shall simplify the formulation of the DAS problem by assuming that transmission constraints, zonal reserve requirements and ramp constraints are not activated. Furthermore, we will assume that the energy offers have only one block that ranges from the technical minimum to the technical maximum. Under these assumptions, we can formulate the following linear programming (LP) problem that refers to a single zone.

min

Gu ,h , Ru ,h , STu ,h ,Yu ,h ,Vu ,h

u,h

u,h

+ ∑ Yu , h ⋅ SUCu + ∑ Vu , h ⋅ SDCu } u ,h

subject to: ∑ Gu ,h = Dh

(shadow price) ( phG ) (30)

∀h

∑R

u∈U

u,h

∀h

≥ Rhreq

Gu , h − STu , h ⋅ Q

min u

≥0

∀u , h

−Gu , h − Ru , h + STu , h ⋅ Q

≥0

− Ru , h + STu , h ⋅ Rubid ≥ 0

∀u , h

max u

STu , h

= STu′, h

∀u, h

∀u, h

Generation,

Transmission

and

∀u , h

( ρu , h )

(36)

Vu , h = Vu′, h

∀u, h

(σ u , h )

(37)

Parameters STu′, h , Yu′, h , and Vu′, h represent the optimal values of binary variables STu , h , Yu , h , Vu , h that are obtained if we solve the MILP problem. Next to each constraint, we have written its respective shadow price. The dual problem is stated as follows: max

phG , phR , λu ,h ,θu ,h ,ε u ,h ,π u ,h , ρu ,h ,σ u ,h

{∑ phG ⋅ Dh + ∑ phR ⋅ Rhreq + h

h

+ ∑ π u , h ⋅ STu′, h + ∑ ρu , h ⋅ Yu′, h + ∑ σ u , h ⋅ Vu′, h } (38) u ,h

u,h

u,h

subject to: phG + λu , h − θ u , h ≤ Pug, h

∀u, h

p − θu ,h − ε u , h ≤ P

∀u , h

R h

r u,h

−λu , h ⋅ Q

min u

+ θu , h ⋅ Q

max u

ρu , h ≤ SUCu σ u , h ≤ SDCu

(39) (40)

+ ε u,h ⋅ R

bid u

+ π u ,h ≤ 0

∀u , h (41)

∀u , h

(42)

∀u , h

(43)

p , p , λu , h , θ u , h , ε u , h , ≥ 0 and π u , h , ρu , h , σ u , h ∈ R G h

R h

∀u, h

We finally list the Karush-Kuhn-Tucker conditions for optimality: 0 ≤ Gu*, h ⊥ ( phG * + λu*, h − θ u*, h − Pug, h ) ≤ 0 0≤R

* u,h

⊥ (p

R* h

−θ

0 ≤ ST

⊥ (θ Q

0≤Y

0 ≤V

* u,h

* u,h * u ,h

0≤ p

G* h

* u,h

* u,h

−ε

* u,h

−P )≤0

−λ Q

max u

* u,h

min u

r u,h

+ε R * u,h

bid u

∀u , h

(44)

∀u, h

(45)

+ π ) ≤ 0 ∀u , h (46) * u,h

⊥ (ρ

* u,h

− SUCu ) ≤ 0

∀u, h

(47)

⊥ (σ

* u,h

− SDCu ) ≤ 0

∀u, h

(48)

∀h

(49)

⊥ (∑ G

* u,h

− Dh ) ≥ 0

u

0 ≤ phR* ⊥ (∑ Ru*, h − Rhreq ) ≥ 0

∀h

(50)

u

0 ≤ λu*, h ⊥ (Gu*, h − STu*, h ⋅ Qumin ) ≥ 0

∀u, h

0 ≤ θ u*, h ⊥ (−Gu*, h − Ru*, h + STu*, h ⋅ Qumax ) ≥ 0

0≤ε

π

* u,h

⊥ ( ST

* u,h

* u ,h

⋅ ( ST

* u ,h

⋅R

bid u

−R )≥0

− STu′, h ) = 0

* u,h

(51) ∀u, h

∀u , h

∀u, h

(52) (53) (54)

− Yu′, h ) = 0

∀u , h

(55)

σ u*, h ⋅ (Vu*, h − Vu′, h ) = 0

∀u, h

(56)

* u,h

⋅ (Y

* u,h

(29)

u,h

u∈U

Power

Yu , h = Yu′, h

ρ

{∑ Pug, h ⋅ Gu , h + ∑ Pur, h ⋅ Ru , h +

and

( phR )

(31)

(λu , h )

(32)

(θ u , h )

(33)

(ε u , h )

(34)

(π u , h )

(35)

Optimal values are marked with an asterisk (*) and 0 ≤ x ⊥ y ≥ 0 is shorthand for the conditions: 0 ≤ x; y ≥ 0; x ⋅ y = 0 .

III. NUMERICAL RESULTS In this section, we present some results from solving the DAS problem on an instance representing Greece’s Electricity Market The data needed as input to the DAS problem are listed in Tables 1-3. Quantities are given in MW and prices for energy and reserve bids in €/MWh. The

MedPower 2008 6th Mediterranean Conference on Exhibition Distribution, Thessaloniki, Greece, 2-5 November 2008

bids are considered to be the same for all the 24 hours. Minimum up/down times are given in hours and start-up/ shut-down costs in €. Over thirty thermal units are installed in the Greece’s system. The lignite units constitute the majority of installed thermal power plants and, because of their low cost, serve as base units. Actual competition is mainly limited to the gas units. With this in mind, unit u1 is an aggregate representation of all lignite units available for producing. This unit corresponds to about 80% of the installed lignite units, assuming that the rest 20% is not available due to scheduled maintenance or outages. Unit u2 is an aggregate representation of available oil units. Units u3 – u7 represent existing gas units, and unit u8 is a “peaker”, that can provide all its capacity for reserve.

and

Power

Qumax 4000 450 476 300 550 389 389 141

u1 u2 u3 u4 u5 u6 u7 u8

Qumin 2500 250 144 150 155 240 240 0

Pug, h

Rubid

35 60 52 70 55 50 65 150

300 50 150 80 150 149 149 141

Pur, h 10 5 4 4.5 6 3.5 3 2

Table 2. Units’ data and initial condition

Unit u1 u2 u3 u4 u5 u6 u7 u8

MU u =

SUCu =

= MDu

= SDCu

24 8 8 16 5 3 3 0

1.500.000 40.000 16.000 30.000 24.000 14.000 14.000 5.000

STu0

X u0

Wu0

1 0 1 0 1 1 0 0

24 0 24 0 24 24 0 0

0 24 0 24 0 0 24 24

1 7

2

3900

8

3

3800

9

4

3700

10

Unit

ON

u1

1-24

u2

10-24

u3

1-24

u4 u5

1-24

u6

1-24

u7 u8

11-14 1-24

Gu , h

Ru , h

1:3661, 2:3361, 3:3261, 4-5:3161, 6:3061, 7:3461, 8:3761, 9:3990, 10-23:4000, 24:3711 10-14:250, 15:294, 16-24:250 1-8:144, 9-10:326,11-14:366, 15:326, 16:315, 17:265, 18:215, 19:315, 20-21:326, 22:315, 23-24:144 1-8:155, 9:244, 10:344, 11:305, 12:255, 13-14:205, 15:400, 16-19:155, 20:344, 21:244, 22-24:155 1-9:240, 10:280, 11-14:389, 15-22:280, 23:241, 24:240 11-14:240 -

1-7:10, 8:239, 9:10, 24:289

10-24:50 1-10:150, 11-14:110, 15-23:150, 24:120 1-7:150, 8:70, 9-22:150, 23:121

1-7:149, 9:149, 10:109, 15-22:109, 23:138 11-14:149 1-24:141

We observe that the aggregate lignite unit (u1) serves as the system’s base. The aggregate oil unit (u2) produces during hours 10-24. Unit u4 is the most expensive gas unit (“peaker” not included) and stays offline. Units u3, u5 and u6 produce during all 24 hours, while unit u7 is included in the DAS only during hours 11-14 when the load reaches its peak. The “peaker” unit u8 is included only for reserve. In Tables 5 and 6, we list the shadow prices of constraints (30) and (31) that represent the marginal prices for energy and reserve respectively. Table 5. Energy marginal price for case 1

5

3700

11

6

3600

12

1

35

4

5

6

35

35

35

4800

5200

5550

5500

35

35

55

9

10

11

12

13

14

15

16

17

18

13

14

15

16

17

18

19

20

21

22

23

24

5450

5300

5000

4950

4900

19

20

21

22

23

24

5200

5100

5000

4800

4500

The total reserve requirement is set to 600 ΜW for all hours. We examine two cases, one with zero-priced and one with non-zero priced reserve offers. In both cases, the examples are solved using the mathematical programming language AMPL and the optimization software package ILOG CPLEX 9.0. A. Case 1

55 52

55

55

60

55

55

52

52

55

52

50

55

52

35

Table 6. Reserve marginal price for case 1

1

2

3

4

5

6

7

8 0

9

20

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

0 0 3 2

We first consider the DAS problem with zero-priced reserve offers, that is, Pur, h = 0 , ∀u , h . Under this assumption, there are multiple optimal solutions regarding

8

3

35

4300

5450

7

2

35

4000

5000

and

Table 4. Dispatching schedule for case 1

Table 3. Hourly demand data

4200

Transmission

the reserve allocation, when the available reserve is more than 600 MW, and normally we would have to establish a rule to allocate the reserve among the generating units. However, due to space considerations, we shall not address this issue in this paper. In Table 4, we list the optimal dispatching schedule for case 1. We considered the situation where constraint (31) is binding, i.e., the total reserves are exactly equal to the reserve requirement. The resulting allocation of reserve is one of the multiple solutions.

Table 1. Units’ energy and reserve offers

Unit

Generation,

0

3

5

0

10 5

0

5

2

2

0

3

2

0

0

3

2

0

We observe that the energy price generally “follows” the load, and provides a good economic sign. Perhaps, the most

MedPower 2008 6th Mediterranean Conference on Exhibition Distribution, Thessaloniki, Greece, 2-5 November 2008

interesting finding is the reserve price. Even though the units do not make a priced reserve offer, the shadow price associated to the reserve requirement constraint is positive in some hours. This indicates a reserve shortage. Namely, if we needed one extra MW for reserve we would have to exchange a cheap MW of energy with an expensive one, in order to “release” a MW for reserve. On the other hand, a reserve surplus forces reserve price to zero, implying that the need for an extra MW for reserve will not cause any additional cost. Note that the reserve price can be rather high; e.g. at hour 9 the reserve price is 20.

and

Power

Generation,

Transmission

and

We observe that, compared to case 1, energy prices are slightly higher at some hours, while reserve prices are considerably higher at all hours. Energy and reserve interaction becomes quite evident if we employ the analysis of the DAS problem which was provided in the previous section. For example, let us examine hours 15 and 16. We are interested in understanding the resulting energy and reserve marginal prices, by calculating the marginal values for these two commodities. The units’ dispatching and the shadow prices needed for the analysis are summarized in Tables 10 and 11.

B. Case 2 Table 10. Dispatching and selected shadow prices at hour 15

We now consider the case that producers submit both energy and reserve bids. The resulting optimal dispatching schedule is provided in Table 7. Table 7. Dispatching schedule for case 2 Unit

ON

u1

1-24

u2

10-24

Gu , h

u3

1-24

u4 u5

1-24

u6

1-24

u7 u8

11-14 1-24

Ru , h

1:3661, 2:3361, 3:3261, 4-5:3161, 6:3061, 7:3461, 8:3761, 9:3990, 10-23:4000, 24:3711 10-14:250, 15:294, 16-24:250 1-8:144, 9-10:326,11-14:366, 15-16:326,17:305, 18:255, 19-22:326,23:155, 24:144 1-8:155, 9:244, 10:344, 11:305, 12:255, 13-14:205, 15:400, 16-19:155, 20:344, 21:244, 22-24:155 1-9:240, 10:280, 11-14:389, 15:280, 16:269, 17-18:240, 19:269, 20-21:280, 22:269, 23-24:240 11-14:240 -

7

2

35

10-24:50 1-10:150, 11-14:110, 15-24:150 1-15:150, 16:139, 17-18:110, 19:139, 20-21:150, 22:139, 23-24:110 8-9:149, 10:109, 15:109, 16:120, 17-18:149, 19:120,20-21:109, 22:120, 23-24:149 11-14:149 1-24:141

4

5

6

35

35

35

35

35

35

55

9

10

11

12

13

14

15 60

16

52.5

17

18

19

20

21

22

23

24

55

52.5

8

3

55

55

55

55

52.5

55

52

52

55

52

35

Table 9. Reserve marginal price for case 2

1

10

7

2

10

4

5

6

10

10

10

10

10

10

30

9

10

11

12

13

14 7

15

13.5

16

17

18

19

20

21

22

23

24

7 6

8

3

8.5

8.5

8.5 6

6

Ru ,15

λu ,15

θ u ,15

ε u ,15

4000 294 326 400 280 0

0 50 150 150 109 141

0 0 0 0 0 0

25 0 8 5 10 11.5

0 8.5 1.5 2.5 0 0

Table 11. Dispatching and selected shadow prices at hour 16

Table 8. Energy marginal price for case 2

1

Gu ,15

u1 u2 u3 u5 u6 u8

1-9:10

We note that, compared to case 1, there are only some minor changes in the energy and reserve quantities. The units’ statuses are the same as in case 1. The energy and reserve marginal prices are listed in Tables 8 and 9.

35

Unit

7

6

6

7

6

6

Unit

Gu ,16

Ru ,16

λu ,16

θ u ,16

ε u ,16

u1 u2 u3 u5 u6 u8

4000 250 326 155 269 0

0 50 150 139 120 141

0 7.5 0 2.5 0 0

17.5 0 0.5 0 2.5 0

0 1 1.5 0 0 4

At hour 15, the energy price is set to 60. A quick look at the units’ data in Table 1 and the shadow prices is enough to see that the unit that sets the energy price is u2. However, u2 does not set the price for reserve. Reserve price is set to 13.5. The only units that can provide one extra MW for reserve are u1 and u6. All the other units, namely u2, u3, u5 and u8, are already scheduled to provide reserve at their maximum availability. However, units u1 and u6 have reached their technical maximum if we consider both energy and reserve as they appear in constraint (12). Consequently, to provide one extra MW for reserve, units u1 and u6 must lower their production by one MW, which has to be replaced by another unit. The cheapest way to achieve this is by lowering the output of u6 by one MW and by increasing the output of u2 by one MW, that results in a cost increase of 10 (60-50=10). If we add this to 3.5, which is the price for providing reserve by u6, we obtain the marginal price for reserve at hour 15, namely, 13.5. Note that this price is higher than the highest reserve bid, which is made by unit u1 and is rejected at hour 15. At hour 16, the energy price is set to 52.5 and is influenced by the reserve bids. We observe that this price does not match any of the bids for energy listed in Table 1. To compute the marginal cost for energy, suppose that we need one extra MW for energy. This MW cannot be produced by u1 as this unit is already scheduled at its maximum output. The cheapest way to produce this extra MW is by lowering the reserve of u6 by one MW, in order to produce one MW with cost 50, and increasing the reserve of u5 to replace the MW for reserve at an extra cost of 2.5 (6-3.5=2.5). Note that units u1, u5 and u6 are the only units that are not scheduled to provide reserve at their maximum availability. However, units u1 and u6 have

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reached their technical maximum if we consider both energy and reserve. As for the reserve price, it is set to 6 by u5, which is the unit that can provide an extra MW for reserve at the relatively lower cost. The shadow prices listed above are rather indicative of the units’ condition and combined with constraints (39) and (40) and conditions (44) and (45) help the reader to understand how each generating unit is involved in forming marginal prices for energy and reserve for all the 24 hours of the DAS problem.

and

Power

Generation,

Transmission

and

ACKNOWLEDGEMENTS This work was partly supported by a projected entitled “Investigation of the Interaction between the Energy and Reserves Market” which was funded by Greece’s Regulatory Authority for Energy (RAE). The authors wish to specially thank the Chairman of RAE, Professor M.C. Caramanis, for his guidance and encouragement. REFERENCES

IV. CONCLUDING REMARKS We formulated a model that represents Greece’s Electricity Market, and we solved an example that applies to Greece’s energy sector. Hydro plants, renewable sources and net imports were not included in our analysis as they are subject to special rules and conditions. Lignite and oil units were aggregated, in order to focus on the gas units that constitute the main field for actual competition. The mathematical formulation of the DAS problem aims at providing the basis to understand how the two commodities of energy and reserve interact, and how the marginal pricing theory [4] can be applied in a co-optimized energy and reserve market. Two cases, one with non-priced and another with priced reserve bids were examined. Our numerical results showed that energy and reserve markets are coupled and have strong interdependencies that influence marginal prices. In the case with priced reserve bids, reserve prices were higher, compared to the case of non-priced reserve bids, but this may not always be the case as the units’ commitment statuses may be different. Volatility of reserve prices was another finding that turned out to be common for the two cases. Our analysis gives rise to numerous questions about the establishment of the energy and reserves markets. In the absence of a balancing market, the introduction of reserves in the DAS problem is of great importance. The form of reserve bids (e.g. hourly or daily), if any, their weight on the objective function, payment and compensation schemes for both energy and reserve have to be thoroughly examined with the object to stimulate competition in the market. Whether the energy price should incorporate the impact of the energy-reserve interaction or not is a critical decision for the market design. Furthermore, the activation of transmission constraints and zonal reserve requirements, as well as the possibility for zonal reserve prices need further analysis. Providing incentives for the installation of new “peakers”, is also a matter of high significance, as their high installation and operational costs discourage new investments. However, the provision of non spinning reserve by these units can result in a more efficient dispatching schedule. This service would else be replaced by spinning reserve provided by expensive units which would have to remain scheduled and produce at their technical minimum in order to be able to provide reserve. Lastly, we should mention that the way that the producers’ bids can manipulate prices and the role each participant can play in the deregulated electricity market can be approached through a gaming analysis of the model that has been described in this paper. The examination of the issues stated above will be the object of our future research.

[1]

Regulatory Authority for Energy, “Grid Control and Power Exchange Code for Electricity”, Athens, Greece, 2005.

[2]

P.Andrianesis, G. Liberopoulos, G. Kozanidis, “Modeling the Greek Electricity Market”, 19th Hellenic Operations Research Society Conference, Arta, Greece, June 2007.

[3]

R. Fourer, D.M. Gay, B.W. Kernighan, “AMPL: A Modeling Language for Mathematical Programming”, Boyd & Fraser, Danvers, MA, 1993.

[4]

F.C. Schweppe, M.C. Caramanis, R.D. Tabors, R.E. Bohn, “Spot Pricing of Electricity”, Kluwer Academic Publishers, Boston, MA, 1988.

BIOGRAPHIES Panagiotis E. Andrianesis graduated from the Hellenic Military Academy in 2001, and received his B.Sc. degree in Economics from the National and Kapodistrian University of Athens, in 2004. He is currently pursuing a Ph.D. degree in the Department of Mechanical and Industrial Engineering at the University of Thessaly, Volos, Greece. His research interests include power system economics, electricity markets and optimization. George Liberopoulos received his B.S. and M.Eng. degrees in Mechanical Engineering from Cornell University, in 1985 and 1986, respectively, and his Ph.D. degree in Manufacturing Engineering from Boston University, in 1993. He is currently a Professor of Stochastic Methods in Production Management, Head of the Production Management Laboratory, Chairman of Department of Mechanical and Industrial Engineering, and Vice President of the Research Committee at the University of Thessaly, Volos, Greece. He is Associate Editor of the Focused Issue "Design and Manufacturing" of IIE Transactions and Co-editor of OR Spectrum. He has co-edited several collected volumes of books/journals with topics in the area of quantitative analysis of manufacturing systems. He is a member of INFORMS, the Hellenic Operations Research Society, and the Technical Chamber of Greece. His research interests include applied probability, operations research, and automatic control models and methodologies applied to operations management. George Kozanidis received his Diploma in Mechanical and Industrial Engineering from the University of Thessaly in 1997. In 1998, he was awarded the M.Sc. in Manufacturing Engineering from Boston University. He also holds a M.Sc. in Operations Research and a Ph.D. in Industrial Engineering, from Northeastern University, (2002). Since April 2007, he has been a Lecturer of Optimization Methods of Production/Service Systems in the Department of Mechanical and Industrial Engineering at the University of Thessaly, Volos, Greece. He is a member of the Hellenic Operations Research Society, INFORMS and the Technical Chamber of Greece. His research interests lie in the area of Operations Research (Integer Programming, Design and Analysis of Optimization Algorithms, Stochastic Analysis of Production/Transportation Systems, Multicriteria Optimization).