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

Keywords—Reactive power expansion, Scenario

technique

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

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:

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

1-4244-2405-4/08/$20.00 ©2008 IEEE 1323

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

Min: ∑∑==

+=ng

1jgjQgj

ng

1jgjPgjDG )(QC)(PC),(J DG Q,Q,PP

∑∑==

−+−+nd

1jdjQjQdj

nd

1jdjPjPdj )Q(dC)P(dC (1)

S.t.:

)(cosYVVPP ijijijj

n

1jidigi γδδ +=∑

=

-- i∈Nb (2)

)(sinYVVQQ ijijijj

n

1jidigi γδδ +−= ∑

=

-- i∈Nb (3)

)(cosYV-)(cosYVVP ijij2

iijijijjiij γγδδ += - ij∈Nl (4)

)(sinYVVQ ijijijjiij γδδ +−= -

)B-)(sinY(V pijijij2

i γ+ ij∈Nl (5)

)QPS 2ij

2ijij += ij∈Nl (6)

maxiimini VVV ≤≤ i∈N (7) max

gigimin

gi PPP ≤≤ i∈Ng (8) max

gigimin

gi QQQ ≤≤ i∈Ng (9)

pidi dP ≤ i∈Nd (10)

Qidi dQ ≤ i∈Nd (11) max

ijijmin

ij SSS ≤≤ i∈Nl (12)

Where: pgjC : bid function of generator j for active power in $/h

QgjC : bid function of generator j for reactive power in $/h

PdjC : bid function of load j for active power in $/h

QdjC : bid function of load j for reactive power in $/h

gjgj jQP + : power of generator j in MVA

djdj jQP + : power of load j in MVA

QjPj d,d :active and reactive demanded of load j in MW and

MVar

ijijij jQPS += : power flow in line ij in MVA

iji ev δ : voltage of bus i in kv

ijjγij eY : element ij of admittance matrix in siemens

pijB : 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

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:

pointoptimalQ),(J

RLMPDi

DGi ∂

∂= DG Q,Q,PP (13)

IV. DETERMINING REACTIVE POWER EXPANSION CANDIDATES

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 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:

CQQBIDAP ×−××= ξ (14) Where BID is the reactive power production bid. Q is the dispatched reactive power. ξ is a coefficient for

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

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 CBIDQ optopt −××= ξ (15)

where AEP is the annual expansion profit, optBID is the

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

VI. SELECTING THE FINAL EXPANSION PLAN 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.

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 I PARAMETERS OF TRANSMISSION LINES OF THE 8-BUS NETWORK

Line No.

From Bus No.

To Bus No.

X (ohm)

Limit(MVA)

1 1 2 0.03 2802 1 4 0.03 1403 1 5 0.0065 3804 2 3 0.01 1205 3 4 0.03 2306 4 5 0.03 2007 5 6 0.02 3008 6 1 0.025 2509 7 4 0.015 250

10 7 8 0.022 34011 8 3 0.018 240

TABLE II - GENERATION DATA OF THE 8-BUS NETWORK

BusNo.

MinMW

MaxMW

MinMvar

Max Mvar

Bid $/MWh

Bid$/Mvarh

1 0 280 0 84 20 43 0 520 0 300 25 54 0 250 0 150 20 45 0 500 0 150 10 26 0 400 0 200 20 47 0 200 0 60 20 4

TABLE III - LOAD DATA OF THE 8-BUS NETWORK

Name Bus No.

Max MW

Bid $/MWh

Max Mvar

Bid$/Mvarh

L1 1 0 0 90 30L2 2 300 35 90 32L3 3 300 28 90 35L4 4 250 35 75 28L5 5 0 0 75 35L6 6 250 30 75 20L7 7 0 0 75 25L8 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 ( maxQ ) 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 ( maxQ ) 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

TABLE IV - RLMP OF EACH BUS IN $/MVARH BUS NO. 1 2 3 4 5 6 7 8 RLMP 6.2 32 5 4 2.1 4 25 14.4

TABLE V - PREDOMINANT SCENARIOS IN FLUCTUATIONS IN BID MADE

BY COMPETITORS Group:BGroup:A Scen.

No Group:B Group:A Scen.

No 0.55×bidbasebidbase 14 1.3×bidbase 1.3×bidbase 1 1.45×bidbase0.55×bidbase 15 bidbase 1.3×bidbase 2

bidbase0.55×bidbase 16 0.7×bidbase 1.3×bidbase 3 0.55×bidbase0.55×bidbase 17 1.3×bidbase bidbase 4 1.6×bidbase1.6×bidbase 18 bidbase bidbase 5

bidbase1.6×bidbase 19 0.7×bidbase bidbase 6 0.4×bidbase1.6×bidbase 20 1.3×bidbase 0.7×bidbase 7 1.6×bidbasebidbase 21 bidbase 0.7×bidbase 8 0.4×bidbasebidbase 22 0.7×bidbase 0.7×bidbase 9 1.6×bidbase0.4×bidbase 23 1.45×bidbase 1.45×bidbase 10

bidbase0.4×bidbase 24 bidbase 1.45×bidbase 11 0.4×bidbase0.4×bidbase 25 0.55×bidbase 1.45×bidbase 12

1.45×bidbase bidbase 13

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

Plan1 (Expansion in Bus 2)

Plan2 (Expansion in Bus 7)

Plan3 (Expansion in Bus 8)

optBID ($/MVarh) 7 7 5

optQ (MVar) 104.1 93.62 70.3

maxQ (MVar) 0 60 0

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)

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$* Plan

3Plan

2Plan

1 Scen.No

Plan 3

Plan 2

Plan 1

Scen.No

2.8921.3834.288 14 5.557 1.3834.30716.2841.3214.277 15 3.029 1.3834.29122.8921.3214.280 16 3.029 1.3835.01532.8951.3224.288 17 5.614 1.3214.29546.2163.6116.893 18 2.892 1.3834.28753.0291.3835.114 19 2.892 1.3834.28863.0291.3835.941 20 5.617 1.3214.28076.2843.7786.868 21 2.892 1.3214.28382.8924.9854.288 22 2.892 1.3214.28896.2854.2704.277 23 6.216 1.3836.488102.8934.2764.280 24 3.029 1.3835.020112.9004.2884.288 25 3.029 1.3835.940124.059 2.040 4.806 Av** 6.284 1.321 4.301 13

* 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$ Plan

3Plan

2 Plan

1 Scen.No

Plan 3

Plan 2

Plan 1

Scen.No

1.395 2.904 0 14 0 4.173 1.249 1 0 4.962 2.007 15 1.261 2.907 0 2

1.387 2.959 0 16 1.986 3.632 0 3 1.392 2.965 0 17 0 4.293 1.319 4 0.676 3.282 0 18 1.394 2.903 0 5 2.084 3.730 0 19 1.395 2.904 0 6 2.911 4.557 0 20 0 4.295 1.336 7 0.584 3.090 0 21 1.390 2.961 0 8 2.092 0 0.697 22 1.395 2.966 0 9

0 2.014 2.008 23 0.271 5.104 0 10 1.386 0.003 0 24 1.990 3.636 0 11 1.387 0 0 25 2.910 4.556 0 12 2.911 5.104 2.008 MR* 0 4.962 1.982 13

* MR: Maximum regret

TABLE IX – RLMPS AFTER REACTIVE EXPANSION IN BUS 2 IN $/MVARH BUS NO. 1 2 3 4 5 6 7 8 RLMP 4.3 7 5 4 3.6 4 18.2 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

Plan2 (Expansion in Bus 7)

Plan3 (Expansion in Bus 8)

optBID ($/MVarh) 7 5

optQ (MVar) 93.62 152

maxQ (MVar) 60 0

TABLE XI - ANNUAL EXPANSION PROFIT FOR PLANS 2, AND 3 INEACH

SCENARIO IN M$ Plan 3Plan 2 Scen.No Plan 3 Plan 2 Scen.No 1.1821.206 14 5.557 1.383 1 6.2831.321 15 3.576 1.382 2 2.8921.321 16 1.883 1.208 3 1.666 1.294 17 5.616 1.321 4 6.2163.611 18 2.892 1.383 5 3.5751.383 19 2.381 1.380 6 1.2961.002 20 5.617 1.321 7 6.2833.778 21 2.892 1.321 8 1.1791.206 22 2.282 1.321 9 6.2843.778 23 6.216 1.383 10 2.8931.322 24 3.574 1.382 11 1.185 1.211 25 1.709 1.188 12 3.657 1.589 Ave.* 6.283 1.321 13

* Ave. : average over different scenarios

TABLE XII - REGRET OF PLANS 2, AND 3 IN DIFFERENT SCENARIOS IN M$ Plan 3Plan 2 Scen.No Plan 3 Plan 2 Scen.No 0.024 0 14 0 4.173 1

0 4.962 15 0 2.194 2 0 1.571 16 0 0.675 3 0 0.372 17 0 4.294 4 0 2.605 18 0 1.509 5 0 2.191 19 0 1.000 6 0 0.293 20 0 4.295 7 0 2.505 21 0 1.571 8

0.026 0 22 0 0.960 9 0 2.506 23 0 4.833 10 0 1.571 24 0 2.192 11

0.025 0 25 0 0.520 12 0.026 4.962 MR** 0 4.962 13

** MR: Maximum regret

TABLE XIII RLMPS AFTER REACTIVE EXPANSION IN BUS 2 AND 8 IN $/MVARH BUS NO. 1 2 3 4 5 6 7 8 RLMP 5.3 7 5 4 2.9 4 4.7 5

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.

VIII. CONCLUSION 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.

12

34

56

78

1

2

3

0

20

40

Bus NoDifferent cases

RLM

P in

$/M

Var

h

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$ After expansion in

bus 2&8After expansion in

bus 2 Before expansion

237.086564239.945062 260.412578

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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