2nd IEEE International Conference on Power and Energy (PECon 08), December 1-3, 2008, Johor Baharu, Malaysia

Market Oriented Reactive Power Expansion Planning using Locational Marginal Price Mohammad Esmali Falak1,3, Majid Oloomi Buygi1,3, Ali Karimpour2,3 1

Shahrood University of Technology, Shahrood, Iran 2 Ferdowsi University, Mashhad, Iran 3 East Electrical Energy Economics Research Group, Mashhad, Iran

Keywords—Reactive technique

power

expansion,

Scenario

Abstract— In this paper a new method for market oriented reactive power expansion planning is presented. Primary candidates are proposed by ISO and investors determine the optimal candidate by computing annual expansion profit of each candidate. Market oriented reactive power expansion planning causes an increase in the capacity of reactive power production of producers who have appropriate location in the network and/or submit suitable bids for reactive power production. Increase in capacity of these producers will lead to decrease in losses and system's operational cost. I. INTRODUCTION Power system expansion planning is divided into generation, transmission, distribution, and reactive power expansion planning. Capacitor optimal placement is among the accomplished works in reactive power planning. Capacitor optimal placement was paid attention to since 1960s. The issue of capacitor optimal placement was studied by J. J. Gringer in 1980s [1]-[2]. In [3], M. E. Baran formulated capacitor optimal placement as a nonlinear mixed integer problem. Different mathematical and heuristic methods have been presented to solve this nonlinear mixed integer optimization problem. A survey on reactive power planning methods was fulfilled by W. Zhang in [4]. In [4] formulation, advantage, and disadvantage of nine different methods which has been used to solve reactive planning problem are discussed. This paper introduces Reactive Locational Marginal Price (RLMP) as an instrument for market oriented reactive power expansion planning. Reactive Locational marginal price (RLMP) is defined as the cost of supplying the next MVar electric energy at a specific bus considering the cost of generation, cost of losses, and the physical aspects of the transmission system. The task of market oriented reactive power expansion planning is to determine the optimal location and capacity for reactive power expansion [5] and provide a proper environment for the competition of producers [6]-[7]. Annual profit is used as criterion for determining the optimal expansion plan. Effective factors on the profit of a specified reactive power producer are a) submitted bid of the producer for

1-4244-2405-4/08/$20.00 ©2008 IEEE

reactive power, and b) location of the reactive power producer in the network. Market oriented reactive power expansion planning expands reactive power of candidates who have appropriate location in the network, or submit suitable bids for reactive energy production. Expansion the capacity of these candidates will lead to decrease the operational cost of the system. II. OVERVIEW In the proposed method, first ISO proposes some buses as candidates for reactive power expansion planning. Busses with high RLMP are proper candidates for reactive expansion Annual Profit-Bid curve is drawn for each candidate to determine the maximum annual profit of each candidate. Consider the Annual Profit-Bid curve of a candidate, the reactive power corresponding to the maximum annual profit specifies the optimal amount of reactive expansion planning at this candidate location. The bid corresponding to the maximum annual profit specifies its optimal bid. In order to consider the uncertainty in competitor's bid in expansion planning, probable occurring scenarios for competitor's bid are identified. For each candidate max annual profit, which is named Annual Expansion Profit (AEP), is computed under different scenarios. The optimal candidate is selected using minimax regret and expected cost criteria. The rest of the paper is organized as follows. In section III simultaneous scheduling of active and reactive power is modeled and RLMP is defined. Determining proper candidate locations for reactive power expansion is discussed in section IV. In section V a method for computing AEP is presented. In section VI the optimal expansion candidate is selected considering the uncertainty in competitors’ bids. The presented method is applied to a 8-bus test system in section VII. Conclusion in section VIII closes the paper. III. ACTIVE & REACTIVE POWER SCHEDULING Simultaneous scheduling of active and reactive power is modeled by an optimization problem. The objective function is the total cost of system operation. Load flow equations, transmission line limitations, power production restrictions, and power consumption restrictions are constraints of this optimization problem. The problem is modeled as follows:

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2nd IEEE International Conference on Power and Energy (PECon 08), December 1-3, 2008, Johor Baharu, Malaysia

ng

Min: J (PG , PD , Q G , Q D ) =

¦ j=1

nd

+

¦C

transmission lines limits respectively. It is assumed that ISO is applied a price cap to reactive energy bid. By definition nodal price or Locational Marginal Price (LMP) is equal to the "cost of supplying next MW of load at a specific location, considering generation marginal cost, cost of transmission congestion, and losses" [8]. In this paper "cost of supplying next Mvar of load at a specific location, considering generation marginal cost, cost of transmission congestion, and losses" is defined as Reactive Locational Marginal Price (RLMP). Reactive Locational Marginal Price in bus i is calculated as follows: ∂ J (PG , PD , Q G , Q D ) (13) RLMP =

ng

C Pgj (Pgj ) +

¦

C Qgj (Q gj )

j=1

nd

Pdj (d Pj

− Pdj ) +

j=1

¦C

Qdj (d Qj

− Q dj )

(1)

j=1

S.t.: n

Pgi - Pdi =

¦V

i

Vj Yij cos(δ j - δ i + γ ij )

i∈Nb

(2)

Vj Yij sin (δ j - δ i + γ ij ) i∈Nb

(3)

j=1

n

Q gi - Q di = −

¦V

i

j=1

i

2

Pij = Vi Vj Yij cos(δ j - δ i + γ ij ) - Vi Yij cos(γ ij ) ij∈Nl (4)

2

ij∈Nl (5)

+ Q ij2 )

ij∈Nl (6)

+ Vi ( Yij sin (γ ij ) - B pij )

Sij =

i∈N

(7)

i∈Ng

(8)

i∈Ng

(9)

Pdi ≤ d pi

i∈Nd

(10)

Qdi ≤ d Qi

i∈Nd

(11)

Sijmin ≤ Sij ≤ Sijmax

i∈Nl

(12)

Vi min ≤ Vi ≤ Vi max Pgi

min

Q gi

≤ Pgi ≤ Pgi

min

max

≤ Q gi ≤ Q gi

max

According to (13), if reactive load of bus i increases by 1 MVar, total operational cost of the system will increase by RLMPi dollar. Therefore, reactive power expansion in busses which have high RLMP, will reduce final operational cost more than reactive power expansion in other busses. In this paper busses whose RLMP exceed the price cap are selected as candidates for reactive expansion. V. COMPUTING ANNUAL EXPANSION PROFIT

Where: C pgj : bid function of generator j for active power in $/h C Qgj : bid function of generator j for reactive power in $/h C Pdj : bid function of load j for active power in $/h C Qdj : bid function of load j for reactive power in $/h Pgj + jQ gj : power of generator j in MVA Pdj + jQ dj : power of load j in MVA d Pj , d Qj :active and reactive demanded of load j in MW and MVar

Sij = Pij + jQ ij : power flow in line ij in MVA

v i e jδi : voltage of bus i in kv Yij e

j ij

optimal point

IV. DETERMINING REACTIVE POWER EXPANSION CANDIDATES

Q ij = − Vi V j Yij sin (δ j - δ i + γ ij )

Pij2

∂ Q Di

: element ij of admittance matrix in siemens

B pij : half of parallel suceptance of line ij in siemens n, ng, nd, nl: number of buses, generating units, loads and lines respectively N, Ng, Nd, Nl: set of buses, generating units, loads and lines respectively

1st to 4th terms of equation (1) refer to the active power production cost, reactive power production cost, cost of active load curtailment, and cost of reactive load curtailment respectively. Equations (2)-(6) show the active and reactive power flow equations. Equations (7)(12) show bus voltages limits, active and reactive power production limits, active and reactive load limits and

From the viewpoint of investors, the best candidate location for investment is the one in which- despite the submitting high price in tender- expanded capacity of reactive power is fully dispatched. In order to identify the optimal candidate for reactive power expansion, the profit of reactive power expansion at each candidate location should be determined. To this end, at first optimal amount of producing reactive power is determined for each candidate and then its annual profit due to reactive power expansion is calculated. In order to determine the optimal amount of producing reactive power of each candidate, its Annual Profit-Bid curve is drawn and its max is determined. The algorithm of drawing the Annual Profit-Bid curve for candidate i is as follows: • Add a new unlimited reactive generator to candidate i location, while there is no new reactive generator at other candidate locations and existing generators have their own limitations in producing reactive power. • Increase the bid of new reactive power generator from zero by small steps. Compute dispatched reactive power in each step using optimization problem (1)(12). Increase the bid until its dispatched reactive power vanishes. • Compute annual profit in each step. Annual profit is equal to revenue minus investment cost. The annual profit of new reactive generator due to the selling of reactive power is given by: (14) AP = BID × Q × ξ − Q × C

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Where BID is the reactive power production bid. Q is the dispatched reactive power.

ξ

is a coefficient for

2nd IEEE International Conference on Power and Energy (PECon 08), December 1-3, 2008, Johor Baharu, Malaysia

TABLE I PARAMETERS OF TRANSMISSION LINES OF THE 8-BUS NETWORK Line From Bus To X Limit No. No. Bus No. (ohm) (MVA) 1 1 2 0.03 280 2 1 4 0.03 140 3 1 5 0.0065 380 4 2 3 0.01 120 5 3 4 0.03 230 6 4 5 0.03 200 7 5 6 0.02 300 8 6 1 0.025 250 9 7 4 0.015 250 10 7 8 0.022 340 11 8 3 0.018 240

converting hourly peak load profit to annual profit and C is the annual investment cost per 1 Mvar expansion ($/Mvar). • Plot bid and annual profit of different steps to obtain Annual Profit-Bid curve. The reactive power corresponding to the maximum annual profit specifies the optimal amount of reactive power expansion. The bid corresponding to the maximum annual profit specifies the optimal bid and maximizes the expansion profit. Hence annual expansion profit is given by:

AEP = Qopt × ( BIDopt × ξ − C )

(15)

Bus No. 1 3 4 5 6 7

where AEP is the annual expansion profit, BIDopt is the optimal bid for reactive power selling, and Qopt is the optimal amount of reactive power expansion. The candidate which has maximum AEP is selected as the final plan for expansion.

TABLE II - GENERATION DATA OF THE 8-BUS NETWORK Min Bid Bid Max Min Max MW Mvar $/MWh $/Mvarh MW Mvar 0 280 0 84 20 4 0 520 0 300 25 5 0 250 0 150 20 4 0 500 0 150 10 2 0 400 0 200 20 4 0 200 0 60 20 4

VI. SELECTING THE FINAL EXPANSION PLAN

Name

Due to uncertainty in bid of competitors, obtaining the exact Annual Profit-Bid curve for a generator is not possible. It is since uncertainty in bids of competitors causes uncertainty in the amount of dispatched reactive power and consequently uncertainty in annual expansion profit. In order to take into account uncertainty in bids of competitors reactive power expansion, strategic probable occurring scenarios for bids of competitors are identified. For each expansion candidate max AEP is computed under different scenarios. Minimax regret and expected cost criteria are used to find the minimum risk expansion plan.

L1 L2 L3 L4 L5 L6 L7 L8

VII. CASE STUDY Consider the 8-bus system shown in Fig 1. Line parameters, generation data, and load data at peak load of planning horizon are given in tables I, II, and III respectively. It is assumed that ISO is applied price cap $7/MVarh to reactive energy bid. RLMPs are computed by solving the optimization problem described in (1)-(12). Function Fmincon of programming language Matlab is

Fig. 1. Test system: eight bus network

TABLE III - LOAD DATA OF THE 8-BUS NETWORK Bus Max Bid Max Bid No. MW $/MWh Mvar $/Mvarh 1 0 0 90 30 2 300 35 90 32 3 300 28 90 35 4 250 35 75 28 5 0 0 75 35 6 250 30 75 20 7 0 0 75 25 8 300 25 90 15

used to solve the optimization problem. Fmincon uses a sequential quadratic programming method to solve the problem. Sequential quadratic programming solves a quadratic programming subproblem at each iteration. Table IV shows the RLMP of each bus. Table IV shows that RLMPs of buses 2, 7, and 8 are greater than reactive energy bid cap. Hence, these busses are selected as candidate locations for reactive power expansion. In order to consider uncertainties in bids of competitors the network is divided into two parts. Busses 1, 2, 5, and 6 are located in part A, and 3, 4, 7 and 8 ones in part B. This classification may have different criteria such as distance of busses to each other, uniform management, etc. Now predominated scenarios are identified. Bids of competitors changes in different predominated scenarios as given in table V. At the next step, in order to determine the optimal location for reactive power expansion, the Annual Profit-Bid curve is obtained for each expansion candidate in each scenario using the presented method. Figs 2, 3, and 4 show Annual Profit-Bid curve for each expansion candidates in the base case scenario, scenario 5. Table VI shows the optimal amount of reactive power production, optimal bid, and max production limit before expansion ( Q max ) for each expansion plan in scenario 5. Table VII shows the value of AEP for each expansion candidate (plan) in different scenarios. Average of AEP over different scenarios is given in last row of table VII. Regret of each plan in different scenarios is given in table VIII. Max regret of each plan over different scenarios is given in last row of table VIII. AEP averages of plans and

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2nd IEEE International Conference on Power and Energy (PECon 08), December 1-3, 2008, Johor Baharu, Malaysia

their maximum regrets show that investment in plan 1 is economically preferred. According to table VI reactive power production in bus 2 should be expanded from 0 to 105 Mvar. The optimal bid for reactive energy in bus 2 is $7/Mvarh. After investment in bus 2, RLMP is computed for each bus. Table IX shows the computed RLMPs. Table IX shows that RLMPs of buses 7, and 8 are greater than reactive energy bid cap. Hence, these busses are selected as candidate locations for reactive power expansion. The optimal amount of reactive power production, optimal bid, and max production limit before expansion ( Q max ) for each expansion plan in scenario 5 is given in table X. At the next step, in order to determine the optimal candidate for reactive power expansion AEP is computed for each plan in each scenario. Table XI shows the value of AEP for each expansion plan in different scenarios. Average of AEP over different scenarios is given in last row of table XI. Regret of each plan in different scenarios is given in table XII. Max regret of each plan over different scenarios is given in last row of table XII. Average and maximum regret of AEP for different plans show that investment in plan 8 is economically preferred. According to table X reactive power production in bus 8 should be expanded from 0 to 152 Mvar. The optimal bid for reactive energy in bus 8 is $5/Mvarh. After investment in bus 2 and 8, RLMP of each bus is computed. Table XIII shows the computed RLMPs. Table XIII shows that RLMP in all busses is less than $7/Mvarh. Therefore, reactive power expansion is terminated. Fig 5 shows the RLMP for each bus in 1) base case, 2) after expansion in bus 2, and 3) after

BUS NO. RLMP

TABLE IV - RLMP OF EACH BUS IN $/MVARH 1 2 3 4 5 6 7 6.2 32 5 4 2.1 4 25

Fig. 2. Annual Profit-Bid curve for bus 2 in scenario 5 (base case)

Fig. 3. Annual Profit-Bid curve for bus 7 in scenario 5 (base case)

8 14.4

TABLE V - PREDOMINANT SCENARIOS IN FLUCTUATIONS IN BID MADE BY COMPETITORS

Scen. No 1 2 3 4 5 6 7 8 9 10 11 12 13

Group:A

Group:B

1.3×bidbase 1.3×bidbase 1.3×bidbase bidbase bidbase bidbase 0.7×bidbase 0.7×bidbase 0.7×bidbase 1.45×bidbase 1.45×bidbase 1.45×bidbase bidbase

1.3×bidbase bidbase 0.7×bidbase 1.3×bidbase bidbase 0.7×bidbase 1.3×bidbase bidbase 0.7×bidbase 1.45×bidbase bidbase 0.55×bidbase 1.45×bidbase

Scen. No 14 15 16 17 18 19 20 21 22 23 24 25

Group:A

Group:B

bidbase 0.55×bidbase 0.55×bidbase 0.55×bidbase 1.6×bidbase 1.6×bidbase 1.6×bidbase bidbase bidbase 0.4×bidbase 0.4×bidbase 0.4×bidbase

0.55×bidbase 1.45×bidbase bidbase 0.55×bidbase 1.6×bidbase bidbase 0.4×bidbase 1.6×bidbase 0.4×bidbase 1.6×bidbase bidbase 0.4×bidbase

TABLE VI - OPTIMAL BID, OPTIMAL VALUE OF REACTIVE POWER PRODUCTION, MAX LIMIT BEFORE EXPANSION IN SCENARIO 5

BIDopt ($/MVarh)

Qopt

(MVar)

Qmax (MVar)

Plan1 (Expansion in Bus 2)

Plan2 (Expansion in Bus 7)

Plan3 (Expansion in Bus 8)

7

7

5

104.1

93.62

70.3

0

60

0

Fig. 4. Annual Profit-Bid curve for bus 8 in scenario 5 (base case) TABLE VII - ANNUAL EXPANSION PROFIT FOR PLANS 1, 2, AND 3 IN EACH SCENARIO IN M$* Scen. Plan Plan Plan Scen. Plan Plan Plan No 1 2 3 No 1 2 3 1 4.307 1.383 5.557 14 4.288 1.383 2.892 2 4.291 1.383 3.029 15 4.277 1.321 6.284 3 5.015 1.383 3.029 16 4.280 1.321 2.892 4 4.295 1.321 5.614 17 4.288 1.322 2.895 5 4.287 1.383 2.892 18 6.893 3.611 6.216 6 4.288 1.383 2.892 19 5.114 1.383 3.029 7 4.280 1.321 5.617 20 5.941 1.383 3.029 8 4.283 1.321 2.892 21 6.868 3.778 6.284 9 4.288 1.321 2.892 22 4.288 4.985 2.892 10 6.488 1.383 6.216 23 4.277 4.270 6.285 11 5.020 1.383 3.029 24 4.280 4.276 2.893 12 5.940 1.383 3.029 25 4.288 4.288 2.900 Av** 13 4.301 1.321 6.284 4.806 2.040 4.059 * M$: Million, ** Av.: average over different scenarios

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2nd IEEE International Conference on Power and Energy (PECon 08), December 1-3, 2008, Johor Baharu, Malaysia

TABLE VIII - REGRET OF PLANS 1, 2, AND 3 IN DIFFERENT SCENARIOS IN M$ Scen. No 1 2 3 4 5 6 7 8 9 10 11 12 13

Plan 1 1.249 0 0 1.319 0 0 1.336 0 0 0 0 0 1.982

Plan 2 4.173 2.907 3.632 4.293 2.903 2.904 4.295 2.961 2.966 5.104 3.636 4.556 4.962

Plan 3 0 1.261 1.986 0 1.394 1.395 0 1.390 1.395 0.271 1.990 2.910 0

Scen. No 14 15 16 17 18 19 20 21 22 23 24 25 MR*

Plan 1 0 2.007 0 0 0 0 0 0 0.697 2.008 0 0 2.008

Plan 2 2.904 4.962 2.959 2.965 3.282 3.730 4.557 3.090 0 2.014 0.003 0 5.104

expansion in bus 2 and 8. Fig 5 shows competition increases after expansion since price profile become flat after expansion. Table XIV shows the annual operation cost of the system in 1) base case, 2) after expansion in bus 2, and 3) after expansion in bus 2 and 8. As table XIV shows due to the reactive power expansion in bus 2 and 8, annual operation cost of the system reduces.

Plan 3 1.395 0 1.387 1.392 0.676 2.084 2.911 0.584 2.092 0 1.386 1.387 2.911

VIII. CONCLUSION

* MR: Maximum regret TABLE IX – RLMPS AFTER REACTIVE EXPANSION IN BUS 2 IN $/MVARH

1 4.3

2 7

3 5

4 4

5 3.6

6 4

7 18.2

8 14.6

TABLE X - OPTIMAL BID, OPTIMAL VALUE OF REACTIVE POWER PRODUCTION, MAX LIMIT BEFORE EXPANSION IN SCENARIO 5 AFTER REACTIVE EXPANSION IN BUS 2

BIDopt ($/MVarh)

Qopt

(MVar)

Qmax (MVar)

Plan2 (Expansion in Bus 7)

Plan3 (Expansion in Bus 8)

7

5

93.62

152

60

0

TABLE XI - ANNUAL EXPANSION PROFIT FOR PLANS 2, AND 3 INEACH SCENARIO IN M$ Scen.No Plan 2 Plan 3 Scen.No Plan 2 Plan 3 1 14 1.383 5.557 1.206 1.182 2 15 1.382 3.576 1.321 6.283 3 16 1.208 1.883 1.321 2.892 4 17 1.321 5.616 1.294 1.666 5 18 1.383 2.892 3.611 6.216 6 19 1.380 2.381 1.383 3.575 7 20 1.321 5.617 1.002 1.296 8 21 1.321 2.892 3.778 6.283 9 22 1.321 2.282 1.206 1.179 10 23 1.383 6.216 3.778 6.284 11 24 1.382 3.574 1.322 2.893 12 25 1.188 1.709 1.211 1.185 13 Ave.* 1.321 6.283 1.589 3.657 * Ave. : average over different scenarios TABLE XII - REGRET OF PLANS 2, AND 3 IN DIFFERENT SCENARIOS IN M$ Scen.No Plan 2 Plan 3 Scen.No Plan 2 Plan 3 1 14 4.173 0 0 0.024 2 15 2.194 0 4.962 0 3 16 0.675 0 1.571 0 4 17 4.294 0 0.372 0 5 18 1.509 0 2.605 0 6 19 1.000 0 2.191 0 7 20 4.295 0 0.293 0 8 21 1.571 0 2.505 0 9 22 0.960 0 0 0.026 10 23 4.833 0 2.506 0 11 24 2.192 0 1.571 0 12 25 0.520 0 0 0.025 13 MR** 4.962 0 4.962 0.026 ** MR: Maximum regret TABLE XIII RLMPS AFTER REACTIVE EXPANSION IN BUS 2 AND 8 IN $/MVARH 1 2 3 4 5 6 7 8 BUS NO. 5.3 7 5 4 2.9 4 4.7 5 RLMP

40 RLMP in $/MVarh

BUS NO. RLMP

In this paper a new method for market oriented reactive power expansion planning is presented. The plan which has maximum annual expansion profit is selected as the final plan. The proposed method provides a flat price profile and hence encourages competition among producers. It also decreases the total operational cost of the system.

20 8 0

7 6 5

1 2 Different cases

4 3 3

Bus No

2 1

Fig. 5- RLMP for each bus in 1) base case, 2) after expansion in bus 2, and 3) after expansion in bus 2 & 8. TABLE XIV - ANNUAL COST OF SYSTEM IN M$ Before expansion After expansion in After expansion in bus 2 bus 2&8 260.412578 239.945062 237.086564

IX. REFERENCES [1] Grainger, J., and Civanlar, s., “Volt/var control on distribution systems with lateral branches using shunt capacitors and voltage regulators—Parts I, II and III,” IEEE Trans. Power Apparatus and Systems, Vol. PAS-104, No. 11, pp. 3278–3297, Nov. 1985. [2] Grainger, J., and Lee, S., “Capacity release by shunt capacitor placement on distribution feeders: A new voltage-dependent model,” IEEE Trans. Power Apparatus and Systems, Vol. PAS-101, No. 15, pp. 1236–1244, May 1982. [3] Baran, M., and Wu, F., “Optimal sizing of capacitors placed on a radial distribution system,” IEEE Trans. Power Delivery, Vol. 4, No. 1, pp. 735–743, Jan. 1989. [4] Zhang, W., and Tolbert, L., “survey of reactive power planning methods,” IEEE Power Engineering Society General meeting, Vol. 2, June 2005. [5] Esmali Falak, M., Oloomi Buygi, M., “Market Oriented Reactive Power Expansion Planning,” in

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2nd IEEE International Conference on Power and Energy (PECon 08), December 1-3, 2008, Johor Baharu, Malaysia

proceedings of 5th International Conference on European Electricity Market, May 2005. [6] Oloomi Buygi, M., Balzer, G., Modir Shanechi, H., and Shahidehpour, M., “Market based transmission expansion planning,” IEEE Trans. PWRS, Vol. 19, No. 4, pp. 2060-2067, Nov. 2004. [7] Oloomi Buygi, M., Modir Shanechi, H., Balzer, G., Shahidehpour, M., and Pariz, N., “Network planning in unbundled power systems,” IEEE Trans. PWRS, Vol. 21, No. 3, pp. 1379-1387, Agu. 2006. [8] Shahidehpour, M., Yamin, H., and Li, Z., Market operations in electric power systems: Forecasting, scheduling and risk management, John Wiley and Sons, New York, 2002.

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