Corrective economic dispatch in a microgrid

INTERNATIONAL JOURNAL OF NUMERICAL MODELLING: ELECTRONIC NETWORKS, DEVICES AND FIELDS

Int. J. Numer. Model. 2013; 26 26:140 :140–150 Published online 18 June 2012 in Wiley Online Library (wileyo (wileyonlinelibr nlinelibrary.com) ary.com).. DOI: 10.1002/jnm.1847 10.1002/jnm.1847

Corrective economic dispatch in a microgrid Yu-Chi Wu1,*,†, Meng-Jen Chen2, Jin-Yuan Lin 1, Wen-Shiush Chen 1 and 2 Wen-Liang Huang 1

Department of Electrical Engineering, National United University, Miao-Li, Taiwan Department of Electrical Engineering, National Kaohsiung University of Applied Science, Kaohsiung, Taiwan

2

ABSTRACT In this paper, a corrective economic dispatch problem formulation for distributed generators in a microgrid is presented, which considers the transition from grid-connected operation to islanded operation. Various constraints related relate d to the operation of a micro microgrid grid are modele modeled, d, such as spinning reserve requirement requirement for the variation of load demand and the output of intermittent sources, the flow limits between the control areas, and ramping limits of distributed generators. The resulting problem is then solved by an interior point method. Numerical tests based on a three-area ten-unit microgrid system show the effects of these constraints on the generation cost. Copyright © 2012 John Wiley & Sons, Ltd. Received 9 May 2011; Revised 15 February 2012; Accepted 26 April 2012 KEY WORDS:

economic econom ic dispatc dispatch; h; microg microgrid; rid; distributed generator; interior point method

1. INTRODUCTION A microgrid is a low-voltage distribution system with various distributed generators (DGs) and controllable loads together with storage devices ( flywheels, energy capacitors, and batteries). It can be operated either in a non-autonomous way (grid-connected) or in an autonomous way (islanded mode). Significant research projects have been conducted about the design, control, and operation of microgrids [1–4]. A generally accepted concept about a microgrid is that it has enough local power generation to supply a local load demand (or at least a signi ficant portion of it) and has the ability to work in grid-connected or islanded mode [5]. To obtain the maximum bene fit from the operation of a microgrid, many researchers propose the central central controller controller or energy manageme management nt system (EMS) for the the microgrid gri d [3, [3,5 5–8]. Th Thee pur purpo pose se of th thee EM EMS S in in a mi micro crogri grid d is is to ma make ke dec decisi isions ons for th thee best best use of th thee ge gener nerat ators ors to pro produ duce ce ele elect ctric ric po powe werr and hea heat, t, on th thee ba basis sis of var vario ious us inf inform ormati ation on such as th thee lo local cal elect electric rical al and hea heat t demands, weather, price of electric power, fuel cost, power quality requirements, demand side management requests, and congestion levels [7,9]. In this paper, a post-islanding corrective economic dispatch (ED) problem formulation for the EMS in a microgrid is presented to provide the optimal power reference of DGs in a microgrid. The advantage of this proposed problem formulation is that it takes into account the microgrid corrective capabilities after islanding has occurred. The corrective capability considered in this paper is the generation rescheduling [10], which is then represented in the presented corrective ED formulation by ramping limits for the pre-islanding and post-islanding DG generations. The DG generations obtained from the corrective ED are economical for grid-connected operation with the ramping limits as well as capable to be rescheduled to make the system stay in a secure mode when islanding occurs. Besides ramping limits, other constraints are also modeled in the presented corrected ED, such as the flow limits between the control areas for both grid-connected and islanded modes, spinning reserve *Correspondence to: Yu-Chi Wu, Departm *Correspondence Department ent of Electrica Electricall Engineer Engineering, ing, National United University, Miao-Li, Taiwan. † E-mail: [email protected]

Copyright © 2012 John Wiley & Sons, Ltd.

CORRECTIVE ECONOMIC DISPATCH IN A MICROGRID

141

requirement for the variation of load demand and the output of intermittent sources. When a gridconnected microgrid switches to the islanded mode, the DGs then increase or decrease power in order to compensate the power loss of the grid. The feasible increased/decreased power is warranted by ramping constraints. However, if the power flow in the inter-area line is near its maximum value under the grid-connected mode, then the power flow might not be able to be transmitted under islanded mode, and this may cause the system to become unstable [9]. To avoid this situation, flow limits between the control areas are imposed for both grid-connected and islanded modes. The difference between the ED of a conventional power system and the ED of a microgrid relies on the renewable energy sources, such as photovoltaic (PV) and wind generators. In a conventional power system, most utility generator units are dispatchable. However, in a microgrid, there exist PVs and wind generators that are intermittent energy sources, and therefore they are not dispatchable. In a microgrid, the total power production of a PV plant or a wind farm in a speci fied future period cannot be determined because of the variations of weather conditions, control strategies, and so on. For this reason, any grid-connected PV plant and wind farm have to be considered as an uncontrollable and non-dispatchable power generator in the utility network. However, output power forecasting for PV and wind generator has been studied extensively [11 –21]. References [11–14] adopted time series method incorporated with arti ficial intelligence method to forecast the average wind speed; the forecasted output wind power is calculated from the wind speed according to the power characteristic of the wind generator. References [15 –18] took advantage of artificial intelligence method to directly forecast the output wind power from the meteorological data such as the average wind speed and the wind direction. References [19–21] address the forecasting methods for PV output power. The presented corrective ED problem in this paper is solved by an interior point method. Numerical tests based on a three-area ten-unit microgrid system are conducted to show the effects of these constraints on the microgrid generation cost.

2. CORRECTIVE ECONOMIC DISPATCH PROBLEM IN A MICROGRID 2.1. Con figuration of the example microgrid

For illustration purpose, the test system used in [9] is adopted in this paper but with no limitation for the use of the presented formulation. The presented ED problem formulation is suitable for any microgrid topology. Figure 1 shows the con figuration of this system. It consists of three control areas. The weather-driven output of an intermittent energy source such as PV cell and wind turbine cannot be dealt as a dispatchable source; therefore, it is dealt as a negative load and assumed to be predicable within some range of uncertainty [2,9]. The variation of loads and the output of non-dispatchable DGs in each area are compensated by the dispatchable DGs in the same area [9]. In Figure 1, FL pcc is the power from the main grid. FL 1–2 and FL 2–3 are the line flows of inter-areas. 2.2. Corrective ED problem formulation

The objective of the corrective ED problem in a microgrid is to minimize the total energy cost for the whole microgrid, subject to a set of physical and security constraints. The total cost includes the fuel cost of DGs, and the cost of the energy purchased from the main grid or the pro fit by selling energy to the main grid. For the grid-connected mode, DGs should produce more power to minimize the total cost when the market prices are high. In the opposite condition, when the market price is low, DGs would reduce the power output. The presented corrective ED problem can be expressed as follows:

Figure 1. Configuration of test system. Copyright © 2012 John Wiley & Sons, Ltd.

Int. J. Numer. Model. 2013; 26:140–150 DOI: 10.1002/jnm

142

Y.-C. WU ET AL . G

À ÁþX À Á

min F ¼ f pcc FL pcc

(1)

f i Pgi

i¼1

s.t. G

J

X

FL pcc þ

Pgi À FL 1À2 ¼ L 1

i¼1 i 2 Area1

X À

j ¼ 1 j 2 Area1

G

FL 1À2

Pgi À FL 2À3 ¼ L 2

i¼1 i 2 Area2

Pgi ¼ L 3

i¼1 i 2 Areak

Pgi ≥r Â ð L k

i¼1 i 2 Areak

X À

P NDj Þ; 8k ¼ 1; 2; 3

Pgimin≥r Â

ð L k

i¼1 i¼1 i 2 Areak i 2 Areak

X À

P NDj Þ; 8k ¼ 1; 2; 3

(6)

j ¼ 1 j 2 Areak

G

J

X

Pisl gi

À

FL isl 1À2

X À

¼ L 1

i¼1 i 2 Area1

j ¼ 1 j 2 Area1 J

Pisl gi

À

FL isl 2À3

¼ L 2

i¼1 i 2 Area2

X þ

X À

P NDj

(8)

j ¼ 1 j 2 Area2

G

FL isl 2À3

(7)

P NDj

G

X þ

J

Pisl gi

¼ L 3

i¼1 i 2 Area3

X À

P NDj

Pgimin≤Pgi ≤Pgimax ;

(9)

j ¼ 1 j 2 Area3

Àrampi ≤Pisl gi À Pgi ≤rampi ;8i

(10)

max Pgimin≤Pisl gi ≤Pgi ;8i

max max ÀFL pcc ≤FL pcc ≤FL pcc


(5)

J

X À

FL isl 1À2

(4)

P NDj

j ¼ 1 j 2 Areak

G

Pgi

X À J

X À

G

X

(3)

j ¼ 1 j 2 Area3

G

Pgimax

P NDj

J

X þ

i¼1 i 2 Area3

G

X À

j ¼ 1 j 2 Area2

G

X

(2)

J

X þ

FL 2À3

P NDj

(11)

(12) Int. J. Numer. Model. 2013; 26:140–150 DOI: 10.1002/jnm


143

max ÀFL ðmax k À1ÞÀk ≤FL ðk À1ÞÀk ≤FL ðk À1ÞÀk 8k ¼ 2; 3

(13)

isl max ÀFL ðmax k À1ÞÀk ≤FL ðk À1ÞÀk ≤FL ðk À1ÞÀk 8k ¼ 2; 3

(14)

where Pgi and Pisl gi are generations of dispatchable DGs for grid-connected mode and islanded mode, respectively; L k is the total load of area k ; FL isl ðk À1ÞÀk is the inter-area line

flow

between area ( k À 1)

and area k under islanded mode; PND j is the output power of non-dispatchable DG j ; G and J are the total numbers of dispatchable DGs and non-dispatchable DGs, respectively; Pgimin and Pgimax are the generation lower and upper bounds of DG i, respectively; FL ðmax is the inter-area (area k À 1 to area k ) k À1ÞÀk line flow limit; rampi is the ramping limit of DG i; and f pcc and f i are cost functions of FL pcc and Pgi, respectively. Equations (2)–(4) are load balance equations under grid-connected mode for area 1, 2, and 3, respectively. Equations (7) –(9) are load balance equations under islanded mode for area 1, 2, and 3, respectively. Equations (5) and (6) are spinning reserve requirements for area k , where r accounts for the uncertainty of load forecasting and the variation of intermittent DGs. Equation (10) is the ramping constraints for DG i. Equations (11)–(14) are physical limits for Pgi, Pisl gi , FL pcc, FL (k À 1) À k , and FL isl ðk À1ÞÀk . The spinning reserve requirements (5) and (6) are modified from [9,22] where (15) and (16) are used instead with parameters v and u to account for the uncertainty of the load forecasting and variation of intermittent DGs, respectively. As both v and u are user-defined parameters according to the system characteristics, they are combined as one parameter r in this paper. G

G

X

Pgimax

i¼1 i 2 Areak

J

X À

Pgi ≥v Â L k À u

i¼1 i 2 Areak

G

X

Pgi À

i¼1 i 2 Areak

X

(15)

j ¼ 1 j 2 Areak

G

X

P NDj ; 8k ¼ 1; 2; 3

J

Pgimin≥v

X

Â L k À u

i¼1 i 2 Areak

P NDj ; 8k ¼ 1; 2; 3

(16)

j ¼ 1 j 2 Areak

Equations (1)–(14) can be rewritten as follows:  min cT x  ¼ beq s:t : Aeq x  ≤ bineq Aineq x ≤u l ≤ x

Â ¼ Â ¼

 where x

PG FL

Pisl G

isl T

FL

Ã

, PG = [Pg1 Pg2 . . . PgG ],Pisl G

h ¼

(17)

Pisl g1

isl Pisl g2 ⋯PgG

i

, FL = [FL pcc FL 1 À 2 FL 2 À 3],

isl  ¼ beq represents equalities of (2)–(4) and (7)–(9). Aineq x ≤bineq represents FL isl FL isl 1À2 FL 2À3 . Aeq x inequalities of (5), (6), and (10). 0 ≤ x ≤ u represents variable bound constraints corresponding to (11)– (14). By introducing slack and surplus variables, one can further reformulate inequalities (5), (6), and (10) as equalities. T isl Let x ¼ PG FL spu spl Pisl sr spr . Linear programming (LP) problem (17) can be G FL rewritten as follows:

Ã

Â

Ã

min cT x s:t : Ax ¼ b l ≤ x ≤ u

(18)

where spu = [spu1 spu2 spu3] are the surplus variables for inequality (5); spl = [spl 1 spl 2 spl 3] are the surplus variables for inequalities (6); sr = [sr 1 sr 2 ⋯ srG ] are the slack variables for ramping constraints (10); spr = [spr 1 spr 2 ⋯ sprG ] are the surplus variables for ramping constraints (10). N is the dimension of x , and M is the number of equalities. Copyright © 2012 John Wiley & Sons, Ltd.


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Y.-C. WU ET AL .

3. INTERIOR POINT METHOD In this paper, predictor –corrector primal-dual interior point method (PCPDIPM) [23,24] is used to solve the corrective ED problem. The theoretical foundation of PCPDIPM consists of three crucial methods: Newton ’s method for solving nonlinear equations and hence for the unconstrained optimization, Lagrange’s method for optimization with equality constraints, and Fiacco and McCormick ’s barrier method for optimization with inequality constraints. Consider the LP problem (18), and the resulting Lagrangian functions with barrier terms (19a) and (19b) for the corresponding primal and dual LP problems, respectively: N

T

T

T

N

X ½

T

L p ¼ c x þ y ðb À Ax Þ À v ðu À x À sÞw ð x À p À l Þ À m

X ð Þþ

ln x i

i

N

Á X

L d ¼ bT y þ l T w À uT v À x T AT y À v þ w À c þ m½

À

lnðsi Þ

(19a)

lnðvi Þ

(19b)

i

N

lnðwi Þ þ

i

X i

where s and p are slack and surplus variables for l ≤ x ≤ u respectively; v, w, and y are dual variables or Lagrangian multipliers (also known as shadow prices) for constraints u À x À s = 0, x À l À p = 0, and Ax = b, respectively. The Karush –Kuhn –Tucker first-order necessary conditions for L p and L d are as follows: (20a) ðprimal constraintsÞ Ax ¼ b x þ s ¼ u

ðprimal constraintsÞ

(20b)

x À p ¼ l

ðprimal constraintsÞ

(20c)

ðdual constraintsÞ

(20d)

AT y À v þ w ¼ c WPe ¼ me

ðapproximated complementary slacknessÞ

(20e)

VSe ¼ me

ðapproximated complementary slackness Þ

(20f)

where e is a column vector with all elements being 1; W , S , P, and V are diagonal matrices with elements wi, si, pi, and vi, respectively. In the spirit of the barrier method, m starts at some positive value and approaches zero (by Fiacco and McCormick ’s theorem [25]: x (m) approaches x *, the constrained minimizer, as it decreases towards zero). Conditions (20a) –(20f) are then solved by some iterative method. In this paper, the predictor –corrector method is implemented: Aðx þ Δ x Þ ¼ b

⇒

AΔ x ¼ b À Ax

(21a)

ð x þ Δ x Þ þ ðs þ ΔsÞ ¼ u ⇒ Δ x þ Δs ¼ u À x À s

(21b)

ð x þ Δ x Þ À ðp þ Δ pÞ ¼ l ⇒ Δ x À Δ p ¼ l À x þ p

(21c)

AT ð y þ Δ yÞ À ðv þ ΔvÞ þ ðw þ ΔwÞ ¼ c T T ⇒ A Δ y À Δv þ Δw ¼ c À A y þ v À w

(21d)

ðW þ ΔW ÞðP þ ΔPÞe ¼ me ⇒W Δ p þ PΔw ¼ me À WPe À ΔW ΔPe ~ ΔPe ~ ⇒W Δ p þ PΔw ffi me À WPe À ΔW ðV þ ΔV ÞðS þ ΔS Þe ¼ me ⇒V Δs þ S Δv ¼ me À VSe À ΔV ΔSe ~ Δ~ V Δs þ S Δv ¼ me À VSe À ΔV Se

(21e)

(21f)

⇒

where ΔP, ΔW , ΔS , ΔV are diagonal matrices with unknown elements Δ pi, Δwi , Δsi, Δvi, and ~ ; ΔW ~ ; Δ~ ~ are the approximated matrices for ΔP, ΔW , ΔS , ΔV . As ΔP , ΔW , ΔS , ΔV S ; ΔV ΔP are unknown, approximations for them in (21e) and (21f) are calculated by solving Equations (22a)– (22f). Copyright © 2012 John Wiley & Sons, Ltd.


145


~ ¼ b À Ax AΔ x

(22a)

~ þ Δ~ s Δ x

¼ u À x À s

(22b)

~ À Δ p ~ Δ x

¼ l À x þ p

(22c)

T ~ À Δ~ ~ ¼ c À A y þ v À w AT Δ y v þ Δw

(22d)

~ þ PΔw ~ ¼ ÀWPe W Δ p

(22e)

V Δ~ s þ S Δ~ v ¼ ÀVSe

(22f)

The procedures of PCPDIPM are stated as follows: Beginning of iteration loop Step 1: Check convergence If criteria (23) are satisfied, an optimal solution is found; otherwise, continue. gapÃ

≤e1

1 þ jdobj j kthe largest mismatch of KKT k <

(23) e2 ;

where dobj ¼ cT x þ yT ðb À Ax Þ À vT ðu À x Þ À wT ð x À l Þ

(24)

Step 2: Calculate the primal-dual af fine direction Solve equation (25) for the primal-dual af fine direction.

2 666 666 4

0 PÀ1 W À1 0 S V 0 I 0 I 0 0 0 0

I 0 0 0 À I 0

0 I 0 0 I 0

0 0 À I I 0 À A

0 0 0 0 À AT 0

32 777666 777666 54

Δ p ~

s Δ~ ~ Δw Δ~ v

~ Δ x ~ Δ y

3 2 À 777 666 À 777 ¼ 666 ÀÀ ÀÀ 5 4À þ À

w v x p l u x s c AT y v þ w Àb þ Ax

3 777 777 5

(25)

Step 3: Calculate the barrier parameter,

m

¼

2

   g^ ap

gapÃ

g^ ap 2n

(26)

gapÃ ¼ vT s þ wT p T

(27) T

~ Þ ð pþ^ a Δ~ a Δ~ a Δw a Δ~ g^ ap ¼ ðvþ^ vÞ ðsþ^ sÞþðwþ^ pÞ

(28)

where ^ a is the step size based on the predictor direction. Step 4: Compute the final search direction Solve Equation (29) for the primal-dual direction:

2 666 666 4

0 PÀ1 W À1 0 S V 0 I 0 I 0 0 0 0

I 0 0 0 À I 0

0 I 0 0 I 0


0 0 À I I 0 À A

0 0 0 0 À AT 0

32 777666 777666 54

Δ p Δs Δw Δv Δ x Δ y

3 2 777 666 777 ¼ 666 5 4

~ ΔPe ~ e À w À PÀ1 ΔW À1 À1 ~ ~ ΔV e mS e À v À S ΔS x À p À l u À x À s Àc þ AT y À v þ w Àb þ Ax

À1

mP

3 777 777 5

(29)


146

Y.-C. WU ET AL .

Step 5: Update the solution x  x þ aΔ x ; s  s þ aΔs; p  p þ aΔ p; w  w þ aΔw; v  v þ aΔv

y  y þ ad Δ y;

where a is the step size to keep primal and dual variables positive. End of iteration loop As the big matrix in (29) is symmetrical and the same as (25), the Chelosky factorization on it is only used once per iteration. The computational effort can be saved, and convergence performance can be enhanced.

4. NUMERICAL RESULTS The three-area ten-unit test system in [9] is adopted in this paper. Table I lists the cost function of 10 DGs in the system, which are adopted from [9,26]. The last column in Table I indicates the area that DG is located in, and the cost function is expressed as (30). As it is very dif ficult to obtain the cost data of the new energy sources, the cost functions of the conventional power units are used in this study. Although the ratings of the conventional units are higher than those of typical DGs, the algorithm can be verified without limiting its generality. Table II lists the area load data for the total load 1800 MW. The user-defined parameter r is set as 0.1 in this study, but without loss of generality, it can be adjusted by the user according to the characteristics of the system that he or she is dealing with. f i Pgi ¼ ai þ bi Â Pgi þ ci Â P2gi

À Á

(30)

Tables III and V show the ED solutions for three different prices of main grid power. For the one with highest price (Table III), DGs in the microgrid produce more power than those with lower prices (Tables IV and V). In Table III, the microgrid sells the surplus power as much as possible to the grid through the line flow FL pcc (=À50 MW) till the flow hits its limit (50 MW). The total DG generation cost is 3887.3, and the net cost is 3667.3 (3887.3 À 220 = 3667.3). In this case, two DGs (units 5 isl and 9) reach the ramping limits, Pg1 and Pisl g1 hit the generation lower limits, and FL 1À2 hits its upper limit (50 MW). The Lagrangian multipliers (shadow prices) of ramping limits for units 5 and 9 are 0.04234 and 0.04239, respectively. That is, the incremental cost of DG unit 5 is 0.04234 if it violates its limit by 1 MW. The shadow prices of Pg1 and Pisl g1 for their lower limits are 0.00346 and 0.04236, respectively. The shadow prices of FL pcc for its lower limit ( À50 MW) and FL isl 1À2 for its upper limit are 2.6891 and 0.04236, respectively. On the other hand, for the case with the lowest price (Table IV), the microgrid buys power (+50 MW) as much as possible from the grid. The net cost for this case is 3746.9 (3666.9 + 80 = 3746.9). FL 11 – 2 is at its upper bound (50 MW). DG unit 1 is still at its lower bound. The shadow prices of FL pcc for its upper limit (50 MW) and FL 1 À 2 for its upper limit are 1.1213 and 0.1717, respectively. The shadow price of Pg1 for its lower limit is 0.1840. For the case

Table I. Cost funcation data of 10 DGs. Cost data

Output limits

DG

a

b

c

1 2 3 4 5 6 7 8 9 10

15 25 40 32 29 72 49 82 105 100

2.2034 1.9161 1.8518 1.6966 1.8015 1.5354 1.2643 1.2163 1.1954 1.1285

0.00510 0.00396 0.00393 0.00382 0.00212 0.00261 0.00289 0.00148 0.00127 0.00135


max

min

P

P

60 80 100 120 150 280 320 445 520 550

10 20 30 25 50 75 120 125 250 250

Area

1 2 3 3 1 3 2 3 1 2


147


Table II. Area load data. Area load (MW) Total load (MW) 1800

1

2

3

450

720

630

Table III. Economic dispatch solution for the case with total load = 1800 MW, F PCC(FL PCC) = 5 Â FL PCC +30. i

Pgi

1 2 3 4 5 6 7 8 9 10 ΣGen FL pcc FL 1–2 FL 2–3

10.000 48.717 57.276 79.244 108.049 146.855 179.523 366.786 418.951 434.599 1850.000 À50.000 37.000 À20.161

Pisl gi

10.000 48.865 56.479 82.365 98.049 157.565 187.198 338.011 391.951 429.517 1800.000 0.000 50.000 À4.420

Pgi À Pisl gi

0.000

À0.148 0.797

À3.122 10.000

À10.709 À7.674 28.775 27.000 5.082 50.000 À50.000 À13.000 À15.741

min

max

cost

rampi

10.0 20.0 30.0 25.0 50.0 75.0 120.0 125.0 250.0 250.0 955.0 À50.0 À50.0 À50.0

60.0 80.0 100.0 120.0 150.0 280.0 320.0 445.0 520.0 550.0 2625.0 50.0 50.0 50.0

37.5 127.7 159.0 190.4 248.4 353.8 369.1 727.2 828.7 845.4 3887.3 À220.0

5.0 6.0 7.0 9.5 10.0 20.5 20.0 32.0 27.0 30.0

Table IV. Economic dispatch solution for the case with total load = 1800 MW, F PCC(FL PCC) = 1 Â FL PCC +30. i

1 2 3 4 5 6 7 8 9 10 ΣGen FL pcc FL 1–2 FL 2–3

Pgi

10.000 47.593 56.137 78.068 75.443 145.142 177.983 363.764 364.557 431.311 1750.000 50.000 50.000 À13.112

Pisl gi

Pgi À Pisl gi

min

max

Cost

rampi

14.067 49.171 58.195 83.202 84.460 158.155 193.933 338.184 384.733 435.901 1800.000 0.000 32.260 À7.736

À4.067 À1.577 À2.058 À5.133 À9.017 À13.013 À15.950

10.0 20.0 30.0 25.0 50.0 75.0 120.0 125.0 250.0 250.0 955.0 À50.0 À50.0 À50.0

60.0 80.0 100.0 120.0 150.0 280.0 320.0 445.0 520.0 550.0 2625.0 50.0 50.0 50.0

37.5 125.2 156.3 187.7 177.0 349.8 365.6 720.3 709.6 837.9 3666.9 80.0

5.0 6.0 7.0 9.5 10.0 20.5 20.0 32.0 27.0 30.0

25.581

À20.176 À4.590 À50.000 50.000 16.740 À5.377

with f pcc(FL pcc)=2.15 Â FL pcc + 30, the microgrid buys some power from the grid and the line flow FL pcc = 31.979 < 50 is within its limit, and the net cost is 3804.1 (3705.4 + 98.6 = 3804.1). No unit hits its ramping limit in this case. These three tables show that the price of grid power affects the net cost of the microgrid. For the case that the inter-area line flow limits are relaxed, the ED solution for f pcc(FL pcc) = 5 Â FL pcc + 30 is listed in Table VI, where the net cost is 3353.0 (4158 À 805 = 3353) and FL pcc = À167 (microgrid sells power to the grid). As the price is high, the microgrid sells power as much as possible to the main grid through line flow FL pcc. In this case, all units reach their ramping limits in this case, and the net cost of the microgrid is cheaper than that inTable III. Tables VI and III show the impact of line flow limits to the microgrid cost. When the price of the grid power is reduced to f pcc(FL pcc) = 1 Â FL pcc + 30 and no inter-area line flow limits are imposed in the ED, the net cost is 3598.5 and FL pcc = +167 (microgrid buys power from the grid), and all units hit their ramping limits but in the opposite way as the case with f pcc(FL pcc) = 5 Â FL pcc +30. Copyright © 2012 John Wiley & Sons, Ltd.


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Y.-C. WU ET AL .

Table V. Economic dispatch solution for the case with total load= 1800 MW, F PCC(FL PCC)=2.15 Â FL PCC +30. i

Pisl gi

Pgi À Pisl gi

min

max

Cost

rampi

14.960 51.721 55.527 82.810 91.679 157.472 183.407 342.524 382.754 437.146 1800.000 0.000 39.393 À8.333

À4.960 À4.126

10.0 20.0 30.0 25.0 50.0 75.0 120.0 125.0 250.0 250.0 955.0 À50.0 À50.0 À50.0

60.0 80.0 100.0 120.0 150.0 280.0 320.0 445.0 520.0 550.0 2625.0 50.0 50.0 50.0

37.5 125.2 156.3 187.7 191.4 349.8 365.6 720.3 733.6 837.9 3705.4 98.6

5.0 6.0 7.0 9.5 10.0 20.5 20.0 32.0 27.0 30.0

Pgi

1 2 3 4 5 6 7 8 9 10 ΣGen FL pcc FL 1–2 FL 2–3

10.000 47.595 56.137 78.069 82.195 145.142 177.982 363.764 375.826 431.312 1768.021 31.979 50.000 À13.111

0.611

À4.741 À9.485 À12.331 À5.424 21.240

À6.927 À5.835 À31.979 31.979 10.607 À4.778

Table VI. Economic dispatch solution for the case with total load = 1800 MW, F PCC(FL PCC) = 5 Â FL PCC +30. i

1 2 3 4 5 6 7 8 9 10 ΣGen FL pcc FL 1–2 FL 2–3

Pisl gi

Pgi À Pisl gi

min

max

Cost

rampi

10.000 47.878 55.478 75.104 117.733 134.127 166.646 348.648 424.679 419.707 1800.000 0.000 102.412 16.643

5.000 6.000 7.000 9.500 10.000 20.500 20.000 32.000 27.000 30.000 167.000 À167.000 125.000 À69.000

10.0 20.0 30.0 25.0 50.0 75.0 120.0 125.0 250.0 250.0 955.0 N/A N/A N/A

60.0 80.0 100.0 120.0 150.0 280.0 320.0 445.0 520.0 550.0 2625.0 N/A N/A N/A

49.2 139.7 171.1 202.9 293.7 371.8 385.7 759.4 904.0 880.5 4158.0 À805.0

5.0 6.0 7.0 9.5 10.0 20.5 20.0 32.0 27.0 30.0

Pgi

15.000 53.878 62.478 84.604 127.733 154.627 186.646 380.648 451.679 449.707 1967.000 À167.000 À22.588 À52.357

Table VII. Economic dispatch solution for the case with total load= 1800 MW, F PCC(FL PCC)=2.15 Â FL PCC +30. i

1 2 3 4 5 6 7 8 9 10 ΣGen FL pcc FL 1–2 FL 2–3

Pgi

10.000 38.928 47.407 69.084 82.186 131.993 166.112 340.579 375.828 405.897 1668.014 131.986 150.000 40.937

Pisl gi

14.938 38.114 50.089 69.758 91.949 152.454 186.110 370.202 399.001 427.385 1800.000 0.000 55.888 À12.503

Pgi À Pisl gi

À4.938 0.815 À2.682 À0.674 À9.763 À20.461 À19.998 À29.624 À23.173 À21.488 À131.986 131.986 94.112 53.441

min

max

Cost

rampi

10.0 20.0 30.0 25.0 50.0 75.0 120.0 125.0 250.0 250.0 955.0 À150.0 À150.0 À150.0

60.0 80.0 100.0 120.0 150.0 280.0 320.0 445.0 520.0 550.0 2625.0 150.0 150.0 150.0

37.5 105.6 136.6 167.4 191.4 320.1 338.8 667.9 733.6 780.5 3479.5 313.8

5.0 6.0 7.0 9.5 10.0 20.5 20.0 32.0 27.0 30.0

Table VII shows the ED solution for the case where the inter-area line flow limit is changed to 150 MW and f pcc(FL pcc)=2.15 Â FL pcc + 30. As the price 2.15 is cheaper than some DGs in the microgrid, the microgrid buys power from the main grid till the line flow FL 11 – 2 hits its limit, 150. In this case, no unit hits its ramping limit. The net cost is 3793.3 (3479.5 + 313.8 = 3793.3), which is lower than the one in Table V. Copyright © 2012 John Wiley & Sons, Ltd.



149

On the basis of the aforementioned numerical tests, it is found that the price of grid power and interarea line flow limits affect the economic solution of the microgrid. With the presented corrective ED problem formulation, one can manipulate different parameter settings ( r , rampi, and f pcc(FL pcc)) to observe the economic impact to the microgrid.

5. CONCLUSIONS In this paper, a corrective ED problem formulation for the EMS in a microgrid and an interior point method for solving it has been presented. Various constraints related to the microgrid operation were modeled, such as spinning reserve requirement for the variation of load demand and the output of intermittent sources, the flow limits between the control areas, and generation ramping limits for the transition from grid-connected operation to islanded operation. Several numerical tests based on a three-area 10-unit microgrid system were conducted to show the effects of these constraints on the microgrid generation cost. With the help of the presented formulation, one can manipulate different parameter settings to observe the economic impact to the microgrid. ACKNOWLEDGEMENT

The financial support (project no. NSC98-3114-E-007-004) from the National Science Council in Taiwan is gratefully acknowledged. REFERENCES

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Y.-C. WU ET AL .

AUTHOR ’S BIOGRAPHIES

Yu-Chi Wu received his electrical engineering diploma in 1984 from the National Kaohsiung Institute of Technology, Taiwan, and his MS and PhD degrees both in 1993 from the Georgia Institute of Technology (Georgia Tech), USA. He has been involved in activities in both academic and industrial areas since 1986. His industrial work experience with Pacific Gas and Electric Company, EDS/Energy Management Associates, and EDS/ China Management Systems includes development and study of EMS applications and power system planning. At present, he is a professor at the Department of Electrical Engineering, National United University, Taiwan. Dr Wu is a senior member of IEEE.

Meng-Jen Chen was born in Taiwan, ROC. He received his diploma in electrical engineering from the National Kaohsiung Institute of Technology, Taiwan, and his PhD degree in Electrical Engineering from the University of Strathclyde, Glasgow, UK, in 1984 and 1992, respectively. Since 1992, he has been an associate professor in the Department of Electrical Engineering, National Kaohsiung University of Applied Sciences, Taiwan. His research interests include wind energy conversion systems, photovoltaic power systems, distributed power systems, and smart microgrid.

Jin-Yuan Lin received his BS and MS degrees in Automatic Control Engineering from Feng Chia University, Taiwan, in 1981 and 1984, respectively, and his PhD degree in 2002 from National Tsing Hua University, Taiwan. He has been involved in activities in both academic and industrial areas since 1986. At present, he is an associate professor at the Department of Electrical Engineering, National United University, Taiwan. His main research interests are in the areas of mechatronics control, systematic integration, power system, and micro-controller design.

Wen-Shiush Chen was born in Miao-Li, Taiwan, on 23 June 1957. He received his BS and MS degrees in electrical engineering from National Cheng Kung University in 1980 and 1983, respectively. He earned his PhD degree from the Department of Electrical Engineering, Chung Yuan Christian University, in 2008. He had been a lecturer at the Department of Electrical Engineering, National United University (NUU) during 1987–2008. In 2008, he was promoted to an associate professor of the Department of Electrical Engineering of NUU. His research interests include numerical field analysis, optimization insulation designs, electrical insulation, and polluted insulators.

Wen-Liang Huang received his BS and MS degrees in electrical engineering from Chung Yuan Christian University, Taiwan, and the University of Nebraska-Lincoln, USA, respectively. Currently, he is an associate professor at the Department of Electrical Engineering, National Kaohsiung University of Applied Science, Taiwan. His research interests include green energy education, digital learning, and power system planning.



Corrective economic dispatch in a microgrid

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