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Rationalization of operation of an industrial network
Electric Power Systems Research 78 (2008) 1664–1671
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Rationalization of operation of an industrial network J. Nahman ∗ , D. Salamon 1 , Z. Stojkovic´ 2 , J. Mikulovic´ 1 Faculty of Electrical Engineering, University of Belgrade, Bulevar Kralja Aleksandra 73, 11000 Belgrade, Serbia
a r t i c l e
i n f o
Article history: Received 10 April 2007 Received in revised form 2 December 2007 Accepted 24 February 2008 Available online 8 April 2008 Keywords: Industrial networks Measurement data processing Capacitor banks allocation Harmonics resonance conditions Load phase shifting Optimization
a b s t r a c t The paper deals with the rationalization of the electric power supply of an industrial network. The relevant indices characterizing daily load diagrams are defined, and measurement data are statistically processed to select the representative days for the optimization of reactive power compensation. Mathematical models for the optimization of compensation and load shifting are provided and efficiently processed by applying the Hooke and Jeeves’ discrete search approach. Optimal allocation of capacitor banks at available network load points and their sizes have been determined to maximize the annual cost saving related to real and reactive energy consumption charges. The optimal solution recommended is analyzed for harmonics resonance conditions. The optimal phase shifts of daily load diagrams of feeders supplying various groups of loads are determined potentially minimizing the peak of the aggregated load and associated charges. © 2008 Elsevier B.V. All rights reserved.
1. Introduction This paper analyzes the possibilities of reducing the cost of the electric energy supply to the coal separation and draining plant (CSDP) and industrial heating plant (IHP) being parts of the industrial complex of an open pit mine. This mine provides the coal for a large electrical power plant and for the coal market, in some lesser extent. The total installed capacity of electrical energy consumers of CSDP and IHP is 18 MVA and 10 MVA, respectively. This consumption is fed via two separate 35 kV overhead line Al./St. 70/12 mm2 /mm2 feeders supplied at 35 kV bus of a 110 kV/35 kV, 31.5 MVA transformer. The electrical energy delivery costs of two plants under consideration consist of the charges for consumed real and reactive electrical energy as well as of the monthly real power peak charges billed to cover the installed capacity cost of the source power system. The energy consumption and load peaks are measured at the 35-kV bus of the source transformer station. Both plants under consideration, but particularly the CSDP, include a great number of induction motors operating with relatively low power factors, which directly affects the cost of reactive energy
consumption as well as, to a less degree, the power losses and consequently, real energy consumption. The CSDP will be of the major concern in this paper. In the time this analysis was performed, with a very restricted information and data acquisition and measurement system available, it was not possible to properly observe the operation of the subject plants and to assess the possibilities of decreasing the electrical energy related operation cost. The text that follows gives a brief description of activities undertaken, approaches applied and results obtained by a task group, which could be, maybe, of some interest for a broader readership. 2. Identification of current-operating conditions The schemes of networks supplying the CSDP and IHP are displayed in Figs. 1 and 2, respectively. To get a clear picture of the load conditions, hourly measurements at bus 35 kV of real and reactive power carried by the feeders have been conducted in the 3-month period: October–December. The following indices are adopted as representative for each day: Daily real power peak value Pmax = maxi (Pi ),
∗ Corresponding author. Tel.: +381 11 337 0104; fax: +381 11 324 8681. E-mail addresses: [email protected] (J. Nahman), [email protected] (D. ´ [email protected] Salamon), [email protected] (Z. Stojkovic), ´ (J. Mikulovic). 1 Tel.: +381 11 3218 341, fax: +381 11 3248 681. 2 Tel.: +381 11 3218 375, fax: +381 11 3248 681. 0378-7796/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2008.02.006
i = 1, . . . , 24
(1)
Daily mean real power 1 Pi 24 24
Pav =
i=1
(2)
J. Nahman et al. / Electric Power Systems Research 78 (2008) 1664–1671
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Table 1 Daily load indices for CSDP Index X
X¯
92−1
92−1 /X (%)
Pmax (MW) Pav (MW) Pq (MW)2 Qmax (MVAr) Qav (MVAr) Qq (MVAr)2
4.8989 2.8206 9.2496 4.6056 2.7185 8.5234
0.4973 0.2414 1.2746 0.5250 0.2368 1.2229
10.1 8.6 13.7 11.4 8.7 14.3
Daily mean reactive power 1 Qi 24 24
Qav =
(5)
i=1
Daily mean square reactive power 1 2 Qi 24 24
Qq =
(6)
i=1
Indices Pmax and Qmax are both relevant for the feeder loading capacity. The mean real and reactive powers are closely related to voltage drops, feeder power factor and reactive power consumption. The mean square real and reactive powers are correlated with power losses. The indices listed above were determined for all days during the period under observation and statistically processed to obtain an idea on the homogeneity of the sample studied. Consider index X which value for day i equals Xi . The mean value of this index for the sample of days is
Fig. 1. One-line diagram of CSDP power supply.
Daily mean square real power
1 Xi 92 92
X¯ =
1 2 Pi 24 24
Pq =
(3)
i=1
(7)
i=1
Standard deviation
Daily reactive power maximum Qmax = maxi (Qi ) ,
i = 1, . . . , 24
92−1 = (4)
92 i=1
(X¯ − Xi )
2
(8)
92 − 1
provides a measure of the dispersion of X values around sample mean. The lower is the standard deviation the higher is the credibility of sample mean as a representative measure of quantity X. Table 1 shows the results obtained for the daily load indices for CSDP. As can be observed from the data in Table 1, the relative deviation from the sample mean is very low for all indices. It means that the load diagrams of CSDP for various days within the period under investigation slightly differ from one another. This fact makes it possible to base the analysis of the plant operation upon a single day, say the day whose indices are closest to the sample means. For CSDP this day was 9 October characterized with indices presented in Table 2. Average power factor for CSDP as determined from the average data for real and reactive power at feeder source node, given Table 2 Mean values of CSDP load indices on October 9th
Fig. 2. One-line diagram of IHP power supply.
Index X
X¯
Percentage declination from sample mean (%)
Pmax (MW) Pav (MW) Pq (MW)2 Qmax (MVAr) Qav (MVAr) Qq (MVAr)2
5.0000 2.7833 8.7958 4.8000 2.6792 8.1854
2.06 −1.32 −4.91 4.22 −1.45 −3.97
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J. Nahman et al. / Electric Power Systems Research 78 (2008) 1664–1671 Table 4 Mean values of IHP load indices on 6 November
Table 3 Daily load indices for IHP supply Index X
X¯
92−1
92−1 /X (%)
Index X
X¯
Percentage declination from sample mean (%)
Pmax (MW) Pav (MW) Pq (MW)2 Qmax (MVAr) Qav (MVAr) Qq (MVAr)2
4.921 3.4858 12.9052 3.5244 2.2899 5.5494
0.7129 0.5858 3.8243 0.5661 0.2431 1.0665
14.5 16.8 29.6 16.1 10.6 19.2
Pmax (MW) Pav (MW) Pq (MW)2 Qmax (MVAr) Qav (MVAr) Qq (MVAr)2
4.7000 3.6875 13.8046 3.4000 2.3375 5.5971
−4.49 5.79 6.97 −3.53 2.08 0.86
in Table 1, is 0.72, which is the same result as obtained for the selected representative day. This value indicated that the installation of capacitor banks for reactive power compensation should be seriously analyzed as a measure to improve this situation and reduce cost. The results obtained by processing the data acquired for IHP feeder during the 3-month period are presented in Table 3. From the data listed in Table 3, we see again that the discrepancies from sample means are relatively moderate and that these means can be taken as representative for load indices, with a somewhat higher tolerance in this case. Thus, we conclude, as for the CSDP feeder, that the whole time period could also be reasonably well represented by a single day whose means of indices most closely mach with sample means. This representative day is 6 November with data given in Table 4. The average power factor calculated using the sample mean of real and reactive powers equals 0.83. For the selected represen-
tative day this factor is 0.84. These values are higher than for the CSDP feeder as a partial compensation of reactive power by shunt capacitors has been applied in IHP. In case of more considerable relative standard deviations for a sample of days, this sample should be partitioned into two or more samples built of more closely matching daily diagrams. In this case, two or more characteristic days would be taken as representative for loads, each for the corresponding new sample of days. Table 5 gives the data for the network supplying the CSDP, which is the major object of consideration in this paper. The description of the elements forming network branches is quoted including the data on the consumption and available capacitor banks at their down stream nodes that are numbered the same as the corresponding branches. Cables 6 kV are of various types. All impedances are referred to the 6 kV side.
Table 5 CSDP network data Branch/node number 1
2
3
4 5 6 7 8 9 10 11 12 13 14 15 a b c
Branch type
Branch impedance ()
Overhead line 35 kV Al/St, 70/12 mm2 /mm2 , L = 3 km Four parallel connected transformers 35 kV/6 kV, 4 MVA, x = 6%, LNL = 5.5 kWb , LT = 38.5 kWc Two parallel 6 kV cables, 35 mm2 , L = 700 m 6 kV cable, 95 mm2 , L = 700 m 6 kV cable, 95 mm2 , L = 100 m 6 kV cable, 150 mm2 , L = 200 m 6 kV cable, 150 mm2 , L = 200 m 6 kV cable, 120 mm2 , L = 210 m 6 kV cable, 150 mm2 , L = 400 m 6 kV cable, 95 mm2 , L = 1100 m 6 kV cable, 95 mm2 , L = 460 m 6 kV cable, 95 mm2 , L = 460 m 6 kV cable, 95 mm2 , L = 550 m 6 kV cable, 95 mm2 , L = 870 m 6 kV cable, 95 mm2 , L = 760 m
0.038 + j0.030
28
0.025 + j0.180
425
1420
2178
0.181 + j0.037
135
410
729
0.168 + j0.066
75
320
636
0.024 + j0.009
425
1900
2351
0.033 + j0.018
670
1500
2351
0.53 + j0.045
200
630
900
0.039 + j0.019
25
400
1519
0.065 + j0.035
200
630
739
0.264 + j0.105
300
945
1109
0.110 + j0.044
0
1260
1670
0.110 + j0.044
0
315
418
0.132 + j0.052
0
630
835
0.209 + j0.083
0
630
835
0.168 + j0.066
0
1260
1670
Fed via transformer 35 kV/6 kV, 400 kVA. No load losses. Total load losses.
Installed capacitor banks (kVAr)
Installed motor load (kW) 315a
Total installed power (kVA) 384
J. Nahman et al. / Electric Power Systems Research 78 (2008) 1664–1671
3. Mathematical model of radial networks To analyze the effects of compensation of reactive power upon the operating costs it is necessary to model the relevant network for power flow and power losses calculation for various locations and sizes of capacitor banks. The network feeding the CSDP consumers is radial, branching and supplied from one side. The operation of such networks can be modeled using the connection matrix describing the topological structure of the network [1,2]. Consider a radial network containing n branches. Generally, each branch supplies a load point at its down stream end. The elements of the n by n Boolean upper triangular connection matrix [m] are
1 if branch i supplies branch k 0 otherwise
mik =
(9)
It is implied that mii = 1. The real and reactive components of currents flowing through network branches equal 1 U 1 [ii ] = −[m][q] U [ir ] = [m][p]
(10)
[p] = [r]([ir2 ] + [ir2 ])
(11)
[x]([ir2 ] + [ir2 ])
(12)
[q] =
with [r] and [x] being diagonal matrices of resistances and reactances of network branches. Elements pjk and qjk are losses in branch j at hour k. The corresponding real and reactive energy losses are [Wp ] = [p]t
(13)
[Wq ] = [q]t
(14)
with t being 1 h in this application. Total real and reactive energy losses per day are obtainable by summing the elements of matrix [Wp ] and the elements of matrix [Wq ], respectively. The consumption of real and reactive power during hour k determined at source 35 kV bus can be calculated using the following expressions: pk =
n
pjk +
j=1
qk =
n
n
pjk
(15)
qjk
(16)
j=1
qjk +
j=1
n j=1
Voltage drops at load points are [u] = [m]t ([r] + j[x])([ir ] + j[ii ])
(17)
Element ujk of n by 24 matrix [u] is the voltage drop at node j during hour k. Voltage at node j during this period is ujk = Us − ujk
with Us designating the voltage at feeder source bus. Voltage Us is continuously maintained closely to the rated value by the source transformer regulator (load tap changer). The usage of the model described depends on the type of load models adopted in the analysis. If the constant load current model is adopted, the network model gives the solutions in a single run. Otherwise, the solution is obtained by using a simple iteration procedure. For the first run, rated voltages are assumed for all load points as in the previous case. Based upon the results obtained for load point voltages in the preceding run, new real and imaginary load currents are determined for all loads and time periods in accordance with the adopted functional correlation of load currents and voltage magnitudes. These currents are input variables for the next run. Such an iteration procedure is repeated until a prescribed stopping rule is met. In the present application, because of the lacking of some more detailed data on loads during the whole study time period, the constant current load model was used. The loads for different consumers were determined based upon the assessment of their relative share in the total consumption Sj pjk = pk n
S j=1 j
Elements pkj and qkj of n by 24 matrices [p] and [q] are real and reactive power load at node k during time interval j (during hour j, in this application), respectively. U is the rated voltage. The elements of n by 24-dimensional matrices [ir ] and [ii ] are real and imaginary components of phase currents flowing through network branches √ during 24-h period multiplied by 3. Real and reactive power losses in network branches can be determined as
(18)
1667
Sj
qjk = qk n
(19)
S j=1 j
where Sj denotes the assessed consumption of consumer j. Consumption was assessed by inspection of available measured data and typical working conditions and power factors. The measured real and reactive powers at source bus pk and qk include power losses. Therefore expressions (19) give higher loads at load points than actual. To rectify this, the following approach was used: for the regime with peak real power the load consumption at load points was determined for the representative day using Eq. (19) by inserting the corresponding measured values for pk and qk . Using these data, pk and qk were then determined by applying Eqs. (15) and (16). For CSDP feeder 5.118 MW was calculated for the real power and 5.089 MVAr for the reactive power. The actually measured values during daily peak were 5.00 MW and 4.80 MVAr. Based upon these data, real powers pjk in Eq. (19) were corrected by multiplying them by 5.00/5.118 = 0.978 and reactive powers by multiplying them by 4.80/5.089 = 0.943. These correction factors were applied to loads for all time segments (hours) during the 24-h period. The pk and qk values calculated with corrected loads were found to agree very well with measurement data for all time segments, with highest discrepancies being less than 1%. 4. Optimal compensation of reactive power A comprehensive survey of solution techniques for capacitor optimal allocation and sizing used in the past was given in [3]. Analytical approach formulated as a mixed integer programming problem was used in [4,5]. Some heuristic strategies are also presented [6] as well as several approaches based upon various search methods including simulated annealing [7]. In this application the Hooke and Jeeves’ search method was applied [8,9] that made it possible to simultaneously determine, through the same search procedure, both the optimal locations and sizes of capacitor banks. The compensation optimization approach will be detailed outlined for the CSDP network. Only available for installation of capacitor banks are nodes 11–15. The bank options available on Serbian market are listed in Table 6. The real market prices in authors’ country are given that include installation costs.
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J. Nahman et al. / Electric Power Systems Research 78 (2008) 1664–1671
Table 6 Installation options Option index
Capacity (kVAr)
Price (D )a
1 2 3 4 5
0 120 240 360 480
0 1485 1785 2475 3705
a
1 D = US$ 1.48.
The optimization goal is to maximize the savings achievable by applying compensation which may result from reducing particularly the reactive energy consumption as well as the power losses that decrease real energy consumption. The objective function that should be maximized is F = 365 [(Wp − Wpn )cp + (Wq − Wqn )cq ] − g
Ci
(20)
i
By Wp the real energy losses are denoted and Wq is the reactive energy consumption. These both variables are for network CSDP representative day, for present state. Index n designates new values achieved after installation of capacitor banks. By cp and cq the relevant charges per unit of energy are denoted. The last term in Eq. (20) gives the annualized cost of capacitor banks with index i including all nodes with capacitor banks and g being the capital recovery rate N
g=
d(1 + d) N
(1 + d) − 1
(21)
In Eq. (21) d is the annual discount rate while N is the technical life of capacitors expressed by number of years. The control variables in the optimization problem are components of the 1 by 5 vector v with element vi being the index of the option from Table 6 applied at node i.In the case under consideration the following data were relevant: Wp = 0.925 MWh, Wq = 62.8 MVArh, cp = 11.1 D /MWh, cq = 0.93 D /MVArh, d = 0.08 and N = 10. As stated before, the optimization problem was solved by applying the Hooke and Jeeves’ discrete search method. Fig. 3 displays the block diagram of the calculation flow. The search algorithm is constituted of local search moves and pattern moves in the direction of potentially better solutions. Acceleration factor ˛ > 1 speeds up the search. The coordinates of points M are the elements of vector v. The best solution found in previous calculation steps is stored and used at each step for comparison with new solution. The stopping rule adopted was the number of generations of base point M in the first block. The calculations performed in this application indicated a fast convergence to the best solution. With ˛ = 2 and vi = 1, ∀i, as the initial solution, the exact solution was found after five generations of base point M. This result was checked by variation of the initial point and stopping rule extension. It is noteworthy to stress that only 46 calculations of objective function out of 55 = 3125 possible solutions sufficed to find the best solution, which shows the efficiency of the method in the considered case. The voltage magnitude upper limits were not violated during the entire search at any load point at any time period. If it were the case at a load point during a time period, the compensation at this point would be decreased for k: 120 kVAr during this period, with k being an integer and 120 kVAr denoting the size of a capacitor bank segment. The data for the best solution found are given in Table 7. The annual savings provided by the best solution equal D 17,383. As the total cost for banks of capacitors is D 18,525, the investment for compensation will be paid off after approximately 1 year.
Fig. 3. Block diagram of the search flow.
The cost analysis performed leads to the conclusion that actual energy tariffs and prices in the authors’ country, applied in this study, should be seriously reconsidered for consistence. There is a remarkable discrepancy between the reactive energy charges and capacitor prices. It is important to note that the optimization approach applied in the paper can be used in general case with no restrictions on available capacitor bank locations. If this were the case for CSDP network, the control variables would be option numbers for capacitor banks at all network load points. The objective function would remain the same as in the considered case.
Table 7 Optimal solution for compensation for CSDP feeder Node number
Option
Capacity (kVAr)
Cost (D )
11 12 13 14 15
5 5 5 5 5
480 480 480 480 480
3705 3705 3705 3705 3705
J. Nahman et al. / Electric Power Systems Research 78 (2008) 1664–1671
5. Analysis of resonance conditions The optimal solution found for the location and sizes of capacitor banks was tested for resonance conditions. The resonance conditions can be clearly identified by calculating the driving point impedances at network nodes for the characteristic harmonics. For networks with insulated neutral supplying prevailing induction motor and thermal loads, which is the network type analyzed in this paper, most pronounced are the fifth and the seventh harmonics [10]. Therefore, the analysis of the resonance conditions was performed for these two harmonics, for all CSDP network nodes, as a considerable amount of compensation was recommended for this network. The single line network model is used as the fifth and the seventh harmonics build the symmetrical negative and positive systems, respectively, that allows such a network presentation. All network parameters are referred to the 6 kV side. The shunt admittance of load node i has the following general form [11]: yi = gi − j
bi + bMi h
(22)
(23)
qi
(24)
Ur2
bMi =
1 XMi
(25)
with h denoting the harmonic order, pi and qi being the real and reactive no motor load at node i for minimum load conditions and XMi designating the motor blocked rotor reactance at fundamental frequency f = 50 Hz. Minimum network total load is 2.5 MVA, which was proportionally distributed among load points using Eq. (19). Ci and ci denote the capacity of the attached capacitor bank, if any, and one half of the capacity of the branches connected to node i, respectively. For cables 2fc = 110 S/km and for the 35 kV overhead line feeder 2fc = 105 S/km, both values referred to the 6 kV side. By exception, for source node we have y0 = −j
1 + jc1 2fh Xh
(26)
with X denoting the reactance of the source network for fundamental frequency as viewed at the 35 kV bus, which equals 0.072 , referred to the 6 kV side. The impedance of network branch supplying node i equals zi = ri + jxi h
(28)
with [A] being the network incidence matrix. By [yb ] the diagonal matrix of network series branch admittances is denoted with elements being inversed branch impedances given in Eq. (27). In the second step the complete network [Y] matrix is composed as [Y ] = [Y1 ] + [Yn ]
Node
Zdri for h = 5
Zdri for h = 7
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.680 10.194 0.797 6.613 9.271 0.469 0.704 3.354 6.902 3.359 1.702 1.044 10.049 3.504 3.628 1.143
1.400 17.382 2.505 12.245 16.407 1.671 2.305 7.467 12.553 7.401 4.732 3.451 16.595 7.746 8.166 3.905
[Z] = [Y ]−1
(30)
Diagonal elements of [Z] are driving point impedances viewed at network nodes if the load admittances are taken into account at all load points including the load point injecting the harmonics into the network. The driving point impedance for a load point i relevant for the analysis of harmonic penetration is obtainable by excluding, for this node, its motor admittance. This admittance is, then, series connected to the impedance obtained by former exclusion. If the driving point impedance for node i obtained from [Z] is zdi , then the relevant driving point impedance for this node will be
zrdi =
bM 1 +j i zdi h
−1
+ jXMi h
(31)
For source node the relevant impedance equals
zdr0 =
1 1 +j zd0 Xh
−1
+ jXh
(32)
The calculations listed above were simply conducted using MATLAB software by appropriately applying the elementwise operations with matrices. The results of this analysis are presented in Table 8. As can be seen from Table 8, the driving point impedances are low for both harmonics under consideration, at all network nodes. Thus, it can be concluded that the capacitor banks whose installation was recommended for economy reasons, will not cause high-harmonic voltage deformations.
(27)
with ri and xi being the resistance and reactance for fundamental frequency. The Y-parameter matrix of the network in Fig. 1 is formed in two steps. In the first step the matrix due to only series branch admittances is formed [Y1 ] = [A][yb ][AT ]
Table 8 Driving point impedances for network nodes for the harmonics of the fifth and seventh order
network [Z] matrix is now + j(ci + Ci )2fh
where p gi = 2i Ur bi =
1669
6. Load phase shifting to minimize monthly real power peak A means to reduce the charges paid for real power peaks is phase shifting of loads supplied at the same source node. Consider n feeders supplied at the source node. Their consumption at a series of T time instants covering a month period, say hours, can be represented by n by T matrix [A] with elements Aij being real power consumption of feeder i at time instant j. The monthly peak equals pm = maxj
n Aij
(33)
i=1
(29)
where [Yn ] is a diagonal matrix with elements being the shunt admittances at network nodes given in Eqs. (22) and (26). The
as the sum of matrix [A] column j gives total real power consumption at source node at time instant j. The highest of this sum is the monthly peak.
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J. Nahman et al. / Electric Power Systems Research 78 (2008) 1664–1671
Table 9 Optimal phase shifts of feeder CSDP daily load diagram Month
Optimal d1 (h)
Pm for d1 = 0 h (MW)
Pm for d1 = 1 h (MW)
Third column − fourth column (MW)
October November December
1 1 1
11.60 10.40 10.30
10.60 9.70 9.50
1.00 0.70 0.80
The minimization of the real power monthly peak can be dealt with as an optimization problem. The control variables are load phase shifts di and the objective is to minimize performance measure pm . One among the feeders, arbitrary chosen, should be adopted as referent feeder whose phase shift is zero. The choice of the referent feeder does not affect the result of optimization, as the phase shifts among feeder loads are only relevant. For the optimization the Hooke and Jeeves’ approach can again be used with di being the coordinates of possible solution points. In the calculation procedure the following relationships should be accounted for For di ≥ 0 if j + di ≤ T
then Aij = Aij+di
else Aij = Aij+di −T
(34)
then Aij = Aij+di
else Aij = AiT +j+di
(35)
For di < 0 if j + di > 0
On the left-hand side of equations in (34) and (35) are new, modified row i elements of matrix [A] after the load diagram of feeder i is shifted for di time units. On the right-hand side of equations are elements of the initial [A] matrix being the input parameter for the optimization program. The maximum allowable positive and negative phase shifts, in absolute terms, should be specified for all feeder loads. The above-described optimization procedure was conducted for CSDP and IHP feeders being under consideration. The maximum allowable shift in absolute terms was 2 h with 1 h steps. In this case, as there are only two feeders, only five phase shifts options are available. Hence, the optimal shift could be simply obtainable by enumeration. However, the number of phase shifts that should be considered in the case of, say, n feeders when there are m available shift options for each feeder, the number of situations that should be explored would be mn−1 , which could be a time consuming task for enumeration. The optimization in the case under consideration was performed for the 3 months during which the loads of feeders were monitored. It was found that by shifting the load diagram of CSDP feeder to lead 1 h before the IHP feeder diagram a considerable reduction of monthly real power peaks are achieved in all the 3 months observed. The results obtained are presented in Table 9. The mean peak reduction per month is 0.833 MW that, for a monthly peak charge of 4000 D /MW, gives 3464 D /month cost saving. However, the covered sample of months is limited and a prolonged observation of load diagrams throughout at least the whole-year period would be necessary before drawing a final conclusion on load shifting. The analysis of daily diagrams for the 3 months that were observed has shown that the load shift found to be optimal concerning monthly peaks would lead to an increase of daily peaks for 24 out of 92 days, which means in 26% cases. The mean daily peak reduction is 0.3522 MW with standard deviation 0.6588 being greater than the mean. These results support the conclusion that a prolonged analysis of shifting would be recommendable. The operations of the two plants under consideration are not functionally correlated which allows the phase shifting of some activities. The inspection of daily load diagrams reveals that the
maximum daily peaks for both plant networks appear between 7 and 8 pm in most cases. From the consultations with engineers controlling the heath production processes it is concluded that there would be no obstacles to reduce some activities of the IHP during the indicated critical period and postpone them for 1 h. As illustration, Fig. 4 displays the aggregated daily load diagram of feeders under consideration for 12 November before and after phase shifting.
7. Conclusions The paper presented some results of the analysis of operation of an industrial plant network and investigated possible rationalization measures. Indices are defined for characterization of daily loads. These indices, processed by statistical methods, made it possible to investigate the homogeneity of recorded data and to identify representative daily load diagrams best matching the mean values of indices of the observed load samples. It was established that such a representative daily diagram for whole 3-month period studied exists for both feeders under consideration. Based upon the representative load diagram and upon an appropriate feederoperating model the optimization of reactive energy compensation is performed by taking into account the available locations and capacitor banks sizes, which was detailed demonstrated for the network supplying a plant with a considerable portion of induction motors load. It was shown that large savings of cost can be achieved by compensation. The optimal solution found has been checked and approved safe for resonance conditions. A general mathematical model was developed to optimize the phase shifts of loads supplied by the source substation in order to reduce the charges on monthly real power peaks. It was applied, for demonstration, to assess the gains potentially achievable for the two feeders supplying the considered industrial networks. For both the reactive energy compensation and load diagram phase-shifting optimizations the Hooke and Jeeves’ discrete search approach was successfully applied. All necessary information on the corresponding mathematical models, calculation flow and results obtained is presented.
Fig. 4. Aggregated load diagrams of feeders on November 12 before (- - -) and after (—) shifting.
J. Nahman et al. / Electric Power Systems Research 78 (2008) 1664–1671
References [1] M. Papadopoulos, N.D. Hatziargyriou, M.E. Papadakis, Graphic aided interactive analysis of radial distribution systems, IEEE Trans. Power Deliv. 2 (October) (1987) 1297–1302. [2] J. Nahman, D. Peric, Distribution system performance evaluation accounting for data uncertainty, IEEE Trans. Power Deliv. 18 (July) (2003) 694–700. [3] H.N. Ng, M.M.A. Salama, A.Y. Chikhani, Classification of capacitor allocation techniques, IEEE Trans. Power Deliv. 15 (January) (2000) 387–392. [4] M.E. Baran, F.F. Wu, Optimal capacitor placement on radial distribution systems, IEEE Trans. Power Deliv. 4 (January) (1989) 725–734. [5] M.E. Baran, F.F. Wu, Optimal sizing of capacitors placed on a radial distribution system, IEEE Trans. Power Deliv. 4 (January) (1989) 735–743. [6] S.F. Mekhamer, M.E. El-Havary, S.A. Soliman, M.A. Moustafa, M.M. Mansour, New heuristic strategies for reactive power compensation of radial distribution feeders, IEEE Trans. Power Deliv. 17 (October) (2002) 1128–1135. [7] H.D. Chiang, J.C. Wang, O. Cockings, H.D. Chin, Optimal capacitor placement in distribution systems. Parts I and II, IEEE Trans. Power Deliv. 5 (April) (1990) 634–649. [8] G.K. Walsh, Methods of Optimization, J. Wiley & Sons, London, 1975. [9] J. Nahman, D. Tubic, Optimal sparing strategy for a group of substations, IEEE Trans. Power Deliv. 6 (October) (1991) 1469–1475. [10] J. Burke, Power Distribution Engineering—Fundamentals and Applications, M. Dekker Inc., New York, 1994. [11] J. Arrilaga, D.A. Bradley, P.S. Bodger, Power System Harmonics, J. Wiley & Sons, New York, 1985. Jovan Nahman was born in Belgrade, Yugoslavia. He received his Dip. Eng. Grade in electric power engineering from the Electrical Engineering Faculty, University of
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Belgrade, in 1960, and TechD from the same University in 1969. From 1960 to 2001 he was with the Faculty of Electrical Engineering, Belgrade, as a Professor at the Power System Department. Currently he is engaged as a freelance consultant for electric power systems. Dragutin Salamon was born in Vukovar, Croatia, in 1949. He received Dipl Eng, MSc and TechD degrees in electric power engineering from the Electrical Engineering Faculty, University of Belgrade, Belgrade, Serbia and Montenegro (former Yugoslavia), in 1973, 1978 and 1992, respectively. In 1973, he joined the Faculty of Electrical Engineering, University of Belgrade, where he is employed as associated professor at the Power System Department. His special area of interest and research are high-voltage substations grounding systems and power system measurements. Zlatan Stojkovi´c (1960) graduated in 1984, took his M.Sc. degree in 1991 and his PhD degree in 1995 at the Faculty of Electrical Engineering in Belgrade/Yugoslavia. Since 1993 he with the Faculty of Electrical Engineering, University of Belgrade. He is at present professor teaching computer-aided design in power systems. Under a scholarship granted by the Alexander von Humboldt Foundation, Bonn, Germany, he was a Postdoctoral Fellow at the Institute of Electric Energy Systems and High Voltage Technology, University of Karlsruhe, Germany. Jovan Mikulovi´c was born in Zajecar, Serbia, in 1968. He received the Dipl Eng and MSc degrees in electrical engineering from the University of Belgrade, Belgrade, Serbia, in 1994 and 2001, respectively. In 1995, he joined the Faculty of Electrical Engineering, Belgrade. From 2000 to 2002 he was with “C.E.M. s.r.l” Company, Gorizia, Italy. Presently, he is a research and teaching assistant at the Faculty of Electrical Engineering, University of Belgrade. His area of research are high-voltage technology and power system measurements.
Contents lists available at ScienceDirect
Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr
Rationalization of operation of an industrial network J. Nahman ∗ , D. Salamon 1 , Z. Stojkovic´ 2 , J. Mikulovic´ 1 Faculty of Electrical Engineering, University of Belgrade, Bulevar Kralja Aleksandra 73, 11000 Belgrade, Serbia
a r t i c l e
i n f o
Article history: Received 10 April 2007 Received in revised form 2 December 2007 Accepted 24 February 2008 Available online 8 April 2008 Keywords: Industrial networks Measurement data processing Capacitor banks allocation Harmonics resonance conditions Load phase shifting Optimization
a b s t r a c t The paper deals with the rationalization of the electric power supply of an industrial network. The relevant indices characterizing daily load diagrams are defined, and measurement data are statistically processed to select the representative days for the optimization of reactive power compensation. Mathematical models for the optimization of compensation and load shifting are provided and efficiently processed by applying the Hooke and Jeeves’ discrete search approach. Optimal allocation of capacitor banks at available network load points and their sizes have been determined to maximize the annual cost saving related to real and reactive energy consumption charges. The optimal solution recommended is analyzed for harmonics resonance conditions. The optimal phase shifts of daily load diagrams of feeders supplying various groups of loads are determined potentially minimizing the peak of the aggregated load and associated charges. © 2008 Elsevier B.V. All rights reserved.
1. Introduction This paper analyzes the possibilities of reducing the cost of the electric energy supply to the coal separation and draining plant (CSDP) and industrial heating plant (IHP) being parts of the industrial complex of an open pit mine. This mine provides the coal for a large electrical power plant and for the coal market, in some lesser extent. The total installed capacity of electrical energy consumers of CSDP and IHP is 18 MVA and 10 MVA, respectively. This consumption is fed via two separate 35 kV overhead line Al./St. 70/12 mm2 /mm2 feeders supplied at 35 kV bus of a 110 kV/35 kV, 31.5 MVA transformer. The electrical energy delivery costs of two plants under consideration consist of the charges for consumed real and reactive electrical energy as well as of the monthly real power peak charges billed to cover the installed capacity cost of the source power system. The energy consumption and load peaks are measured at the 35-kV bus of the source transformer station. Both plants under consideration, but particularly the CSDP, include a great number of induction motors operating with relatively low power factors, which directly affects the cost of reactive energy
consumption as well as, to a less degree, the power losses and consequently, real energy consumption. The CSDP will be of the major concern in this paper. In the time this analysis was performed, with a very restricted information and data acquisition and measurement system available, it was not possible to properly observe the operation of the subject plants and to assess the possibilities of decreasing the electrical energy related operation cost. The text that follows gives a brief description of activities undertaken, approaches applied and results obtained by a task group, which could be, maybe, of some interest for a broader readership. 2. Identification of current-operating conditions The schemes of networks supplying the CSDP and IHP are displayed in Figs. 1 and 2, respectively. To get a clear picture of the load conditions, hourly measurements at bus 35 kV of real and reactive power carried by the feeders have been conducted in the 3-month period: October–December. The following indices are adopted as representative for each day: Daily real power peak value Pmax = maxi (Pi ),
∗ Corresponding author. Tel.: +381 11 337 0104; fax: +381 11 324 8681. E-mail addresses: [email protected] (J. Nahman), [email protected] (D. ´ [email protected] Salamon), [email protected] (Z. Stojkovic), ´ (J. Mikulovic). 1 Tel.: +381 11 3218 341, fax: +381 11 3248 681. 2 Tel.: +381 11 3218 375, fax: +381 11 3248 681. 0378-7796/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2008.02.006
i = 1, . . . , 24
(1)
Daily mean real power 1 Pi 24 24
Pav =
i=1
(2)
J. Nahman et al. / Electric Power Systems Research 78 (2008) 1664–1671
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Table 1 Daily load indices for CSDP Index X
X¯
92−1
92−1 /X (%)
Pmax (MW) Pav (MW) Pq (MW)2 Qmax (MVAr) Qav (MVAr) Qq (MVAr)2
4.8989 2.8206 9.2496 4.6056 2.7185 8.5234
0.4973 0.2414 1.2746 0.5250 0.2368 1.2229
10.1 8.6 13.7 11.4 8.7 14.3
Daily mean reactive power 1 Qi 24 24
Qav =
(5)
i=1
Daily mean square reactive power 1 2 Qi 24 24
Qq =
(6)
i=1
Indices Pmax and Qmax are both relevant for the feeder loading capacity. The mean real and reactive powers are closely related to voltage drops, feeder power factor and reactive power consumption. The mean square real and reactive powers are correlated with power losses. The indices listed above were determined for all days during the period under observation and statistically processed to obtain an idea on the homogeneity of the sample studied. Consider index X which value for day i equals Xi . The mean value of this index for the sample of days is
Fig. 1. One-line diagram of CSDP power supply.
Daily mean square real power
1 Xi 92 92
X¯ =
1 2 Pi 24 24
Pq =
(3)
i=1
(7)
i=1
Standard deviation
Daily reactive power maximum Qmax = maxi (Qi ) ,
i = 1, . . . , 24
92−1 = (4)
92 i=1
(X¯ − Xi )
2
(8)
92 − 1
provides a measure of the dispersion of X values around sample mean. The lower is the standard deviation the higher is the credibility of sample mean as a representative measure of quantity X. Table 1 shows the results obtained for the daily load indices for CSDP. As can be observed from the data in Table 1, the relative deviation from the sample mean is very low for all indices. It means that the load diagrams of CSDP for various days within the period under investigation slightly differ from one another. This fact makes it possible to base the analysis of the plant operation upon a single day, say the day whose indices are closest to the sample means. For CSDP this day was 9 October characterized with indices presented in Table 2. Average power factor for CSDP as determined from the average data for real and reactive power at feeder source node, given Table 2 Mean values of CSDP load indices on October 9th
Fig. 2. One-line diagram of IHP power supply.
Index X
X¯
Percentage declination from sample mean (%)
Pmax (MW) Pav (MW) Pq (MW)2 Qmax (MVAr) Qav (MVAr) Qq (MVAr)2
5.0000 2.7833 8.7958 4.8000 2.6792 8.1854
2.06 −1.32 −4.91 4.22 −1.45 −3.97
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J. Nahman et al. / Electric Power Systems Research 78 (2008) 1664–1671 Table 4 Mean values of IHP load indices on 6 November
Table 3 Daily load indices for IHP supply Index X
X¯
92−1
92−1 /X (%)
Index X
X¯
Percentage declination from sample mean (%)
Pmax (MW) Pav (MW) Pq (MW)2 Qmax (MVAr) Qav (MVAr) Qq (MVAr)2
4.921 3.4858 12.9052 3.5244 2.2899 5.5494
0.7129 0.5858 3.8243 0.5661 0.2431 1.0665
14.5 16.8 29.6 16.1 10.6 19.2
Pmax (MW) Pav (MW) Pq (MW)2 Qmax (MVAr) Qav (MVAr) Qq (MVAr)2
4.7000 3.6875 13.8046 3.4000 2.3375 5.5971
−4.49 5.79 6.97 −3.53 2.08 0.86
in Table 1, is 0.72, which is the same result as obtained for the selected representative day. This value indicated that the installation of capacitor banks for reactive power compensation should be seriously analyzed as a measure to improve this situation and reduce cost. The results obtained by processing the data acquired for IHP feeder during the 3-month period are presented in Table 3. From the data listed in Table 3, we see again that the discrepancies from sample means are relatively moderate and that these means can be taken as representative for load indices, with a somewhat higher tolerance in this case. Thus, we conclude, as for the CSDP feeder, that the whole time period could also be reasonably well represented by a single day whose means of indices most closely mach with sample means. This representative day is 6 November with data given in Table 4. The average power factor calculated using the sample mean of real and reactive powers equals 0.83. For the selected represen-
tative day this factor is 0.84. These values are higher than for the CSDP feeder as a partial compensation of reactive power by shunt capacitors has been applied in IHP. In case of more considerable relative standard deviations for a sample of days, this sample should be partitioned into two or more samples built of more closely matching daily diagrams. In this case, two or more characteristic days would be taken as representative for loads, each for the corresponding new sample of days. Table 5 gives the data for the network supplying the CSDP, which is the major object of consideration in this paper. The description of the elements forming network branches is quoted including the data on the consumption and available capacitor banks at their down stream nodes that are numbered the same as the corresponding branches. Cables 6 kV are of various types. All impedances are referred to the 6 kV side.
Table 5 CSDP network data Branch/node number 1
2
3
4 5 6 7 8 9 10 11 12 13 14 15 a b c
Branch type
Branch impedance ()
Overhead line 35 kV Al/St, 70/12 mm2 /mm2 , L = 3 km Four parallel connected transformers 35 kV/6 kV, 4 MVA, x = 6%, LNL = 5.5 kWb , LT = 38.5 kWc Two parallel 6 kV cables, 35 mm2 , L = 700 m 6 kV cable, 95 mm2 , L = 700 m 6 kV cable, 95 mm2 , L = 100 m 6 kV cable, 150 mm2 , L = 200 m 6 kV cable, 150 mm2 , L = 200 m 6 kV cable, 120 mm2 , L = 210 m 6 kV cable, 150 mm2 , L = 400 m 6 kV cable, 95 mm2 , L = 1100 m 6 kV cable, 95 mm2 , L = 460 m 6 kV cable, 95 mm2 , L = 460 m 6 kV cable, 95 mm2 , L = 550 m 6 kV cable, 95 mm2 , L = 870 m 6 kV cable, 95 mm2 , L = 760 m
0.038 + j0.030
28
0.025 + j0.180
425
1420
2178
0.181 + j0.037
135
410
729
0.168 + j0.066
75
320
636
0.024 + j0.009
425
1900
2351
0.033 + j0.018
670
1500
2351
0.53 + j0.045
200
630
900
0.039 + j0.019
25
400
1519
0.065 + j0.035
200
630
739
0.264 + j0.105
300
945
1109
0.110 + j0.044
0
1260
1670
0.110 + j0.044
0
315
418
0.132 + j0.052
0
630
835
0.209 + j0.083
0
630
835
0.168 + j0.066
0
1260
1670
Fed via transformer 35 kV/6 kV, 400 kVA. No load losses. Total load losses.
Installed capacitor banks (kVAr)
Installed motor load (kW) 315a
Total installed power (kVA) 384
J. Nahman et al. / Electric Power Systems Research 78 (2008) 1664–1671
3. Mathematical model of radial networks To analyze the effects of compensation of reactive power upon the operating costs it is necessary to model the relevant network for power flow and power losses calculation for various locations and sizes of capacitor banks. The network feeding the CSDP consumers is radial, branching and supplied from one side. The operation of such networks can be modeled using the connection matrix describing the topological structure of the network [1,2]. Consider a radial network containing n branches. Generally, each branch supplies a load point at its down stream end. The elements of the n by n Boolean upper triangular connection matrix [m] are
1 if branch i supplies branch k 0 otherwise
mik =
(9)
It is implied that mii = 1. The real and reactive components of currents flowing through network branches equal 1 U 1 [ii ] = −[m][q] U [ir ] = [m][p]
(10)
[p] = [r]([ir2 ] + [ir2 ])
(11)
[x]([ir2 ] + [ir2 ])
(12)
[q] =
with [r] and [x] being diagonal matrices of resistances and reactances of network branches. Elements pjk and qjk are losses in branch j at hour k. The corresponding real and reactive energy losses are [Wp ] = [p]t
(13)
[Wq ] = [q]t
(14)
with t being 1 h in this application. Total real and reactive energy losses per day are obtainable by summing the elements of matrix [Wp ] and the elements of matrix [Wq ], respectively. The consumption of real and reactive power during hour k determined at source 35 kV bus can be calculated using the following expressions: pk =
n
pjk +
j=1
qk =
n
n
pjk
(15)
qjk
(16)
j=1
qjk +
j=1
n j=1
Voltage drops at load points are [u] = [m]t ([r] + j[x])([ir ] + j[ii ])
(17)
Element ujk of n by 24 matrix [u] is the voltage drop at node j during hour k. Voltage at node j during this period is ujk = Us − ujk
with Us designating the voltage at feeder source bus. Voltage Us is continuously maintained closely to the rated value by the source transformer regulator (load tap changer). The usage of the model described depends on the type of load models adopted in the analysis. If the constant load current model is adopted, the network model gives the solutions in a single run. Otherwise, the solution is obtained by using a simple iteration procedure. For the first run, rated voltages are assumed for all load points as in the previous case. Based upon the results obtained for load point voltages in the preceding run, new real and imaginary load currents are determined for all loads and time periods in accordance with the adopted functional correlation of load currents and voltage magnitudes. These currents are input variables for the next run. Such an iteration procedure is repeated until a prescribed stopping rule is met. In the present application, because of the lacking of some more detailed data on loads during the whole study time period, the constant current load model was used. The loads for different consumers were determined based upon the assessment of their relative share in the total consumption Sj pjk = pk n
S j=1 j
Elements pkj and qkj of n by 24 matrices [p] and [q] are real and reactive power load at node k during time interval j (during hour j, in this application), respectively. U is the rated voltage. The elements of n by 24-dimensional matrices [ir ] and [ii ] are real and imaginary components of phase currents flowing through network branches √ during 24-h period multiplied by 3. Real and reactive power losses in network branches can be determined as
(18)
1667
Sj
qjk = qk n
(19)
S j=1 j
where Sj denotes the assessed consumption of consumer j. Consumption was assessed by inspection of available measured data and typical working conditions and power factors. The measured real and reactive powers at source bus pk and qk include power losses. Therefore expressions (19) give higher loads at load points than actual. To rectify this, the following approach was used: for the regime with peak real power the load consumption at load points was determined for the representative day using Eq. (19) by inserting the corresponding measured values for pk and qk . Using these data, pk and qk were then determined by applying Eqs. (15) and (16). For CSDP feeder 5.118 MW was calculated for the real power and 5.089 MVAr for the reactive power. The actually measured values during daily peak were 5.00 MW and 4.80 MVAr. Based upon these data, real powers pjk in Eq. (19) were corrected by multiplying them by 5.00/5.118 = 0.978 and reactive powers by multiplying them by 4.80/5.089 = 0.943. These correction factors were applied to loads for all time segments (hours) during the 24-h period. The pk and qk values calculated with corrected loads were found to agree very well with measurement data for all time segments, with highest discrepancies being less than 1%. 4. Optimal compensation of reactive power A comprehensive survey of solution techniques for capacitor optimal allocation and sizing used in the past was given in [3]. Analytical approach formulated as a mixed integer programming problem was used in [4,5]. Some heuristic strategies are also presented [6] as well as several approaches based upon various search methods including simulated annealing [7]. In this application the Hooke and Jeeves’ search method was applied [8,9] that made it possible to simultaneously determine, through the same search procedure, both the optimal locations and sizes of capacitor banks. The compensation optimization approach will be detailed outlined for the CSDP network. Only available for installation of capacitor banks are nodes 11–15. The bank options available on Serbian market are listed in Table 6. The real market prices in authors’ country are given that include installation costs.
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J. Nahman et al. / Electric Power Systems Research 78 (2008) 1664–1671
Table 6 Installation options Option index
Capacity (kVAr)
Price (D )a
1 2 3 4 5
0 120 240 360 480
0 1485 1785 2475 3705
a
1 D = US$ 1.48.
The optimization goal is to maximize the savings achievable by applying compensation which may result from reducing particularly the reactive energy consumption as well as the power losses that decrease real energy consumption. The objective function that should be maximized is F = 365 [(Wp − Wpn )cp + (Wq − Wqn )cq ] − g
Ci
(20)
i
By Wp the real energy losses are denoted and Wq is the reactive energy consumption. These both variables are for network CSDP representative day, for present state. Index n designates new values achieved after installation of capacitor banks. By cp and cq the relevant charges per unit of energy are denoted. The last term in Eq. (20) gives the annualized cost of capacitor banks with index i including all nodes with capacitor banks and g being the capital recovery rate N
g=
d(1 + d) N
(1 + d) − 1
(21)
In Eq. (21) d is the annual discount rate while N is the technical life of capacitors expressed by number of years. The control variables in the optimization problem are components of the 1 by 5 vector v with element vi being the index of the option from Table 6 applied at node i.In the case under consideration the following data were relevant: Wp = 0.925 MWh, Wq = 62.8 MVArh, cp = 11.1 D /MWh, cq = 0.93 D /MVArh, d = 0.08 and N = 10. As stated before, the optimization problem was solved by applying the Hooke and Jeeves’ discrete search method. Fig. 3 displays the block diagram of the calculation flow. The search algorithm is constituted of local search moves and pattern moves in the direction of potentially better solutions. Acceleration factor ˛ > 1 speeds up the search. The coordinates of points M are the elements of vector v. The best solution found in previous calculation steps is stored and used at each step for comparison with new solution. The stopping rule adopted was the number of generations of base point M in the first block. The calculations performed in this application indicated a fast convergence to the best solution. With ˛ = 2 and vi = 1, ∀i, as the initial solution, the exact solution was found after five generations of base point M. This result was checked by variation of the initial point and stopping rule extension. It is noteworthy to stress that only 46 calculations of objective function out of 55 = 3125 possible solutions sufficed to find the best solution, which shows the efficiency of the method in the considered case. The voltage magnitude upper limits were not violated during the entire search at any load point at any time period. If it were the case at a load point during a time period, the compensation at this point would be decreased for k: 120 kVAr during this period, with k being an integer and 120 kVAr denoting the size of a capacitor bank segment. The data for the best solution found are given in Table 7. The annual savings provided by the best solution equal D 17,383. As the total cost for banks of capacitors is D 18,525, the investment for compensation will be paid off after approximately 1 year.
Fig. 3. Block diagram of the search flow.
The cost analysis performed leads to the conclusion that actual energy tariffs and prices in the authors’ country, applied in this study, should be seriously reconsidered for consistence. There is a remarkable discrepancy between the reactive energy charges and capacitor prices. It is important to note that the optimization approach applied in the paper can be used in general case with no restrictions on available capacitor bank locations. If this were the case for CSDP network, the control variables would be option numbers for capacitor banks at all network load points. The objective function would remain the same as in the considered case.
Table 7 Optimal solution for compensation for CSDP feeder Node number
Option
Capacity (kVAr)
Cost (D )
11 12 13 14 15
5 5 5 5 5
480 480 480 480 480
3705 3705 3705 3705 3705
J. Nahman et al. / Electric Power Systems Research 78 (2008) 1664–1671
5. Analysis of resonance conditions The optimal solution found for the location and sizes of capacitor banks was tested for resonance conditions. The resonance conditions can be clearly identified by calculating the driving point impedances at network nodes for the characteristic harmonics. For networks with insulated neutral supplying prevailing induction motor and thermal loads, which is the network type analyzed in this paper, most pronounced are the fifth and the seventh harmonics [10]. Therefore, the analysis of the resonance conditions was performed for these two harmonics, for all CSDP network nodes, as a considerable amount of compensation was recommended for this network. The single line network model is used as the fifth and the seventh harmonics build the symmetrical negative and positive systems, respectively, that allows such a network presentation. All network parameters are referred to the 6 kV side. The shunt admittance of load node i has the following general form [11]: yi = gi − j
bi + bMi h
(22)
(23)
qi
(24)
Ur2
bMi =
1 XMi
(25)
with h denoting the harmonic order, pi and qi being the real and reactive no motor load at node i for minimum load conditions and XMi designating the motor blocked rotor reactance at fundamental frequency f = 50 Hz. Minimum network total load is 2.5 MVA, which was proportionally distributed among load points using Eq. (19). Ci and ci denote the capacity of the attached capacitor bank, if any, and one half of the capacity of the branches connected to node i, respectively. For cables 2fc = 110 S/km and for the 35 kV overhead line feeder 2fc = 105 S/km, both values referred to the 6 kV side. By exception, for source node we have y0 = −j
1 + jc1 2fh Xh
(26)
with X denoting the reactance of the source network for fundamental frequency as viewed at the 35 kV bus, which equals 0.072 , referred to the 6 kV side. The impedance of network branch supplying node i equals zi = ri + jxi h
(28)
with [A] being the network incidence matrix. By [yb ] the diagonal matrix of network series branch admittances is denoted with elements being inversed branch impedances given in Eq. (27). In the second step the complete network [Y] matrix is composed as [Y ] = [Y1 ] + [Yn ]
Node
Zdri for h = 5
Zdri for h = 7
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.680 10.194 0.797 6.613 9.271 0.469 0.704 3.354 6.902 3.359 1.702 1.044 10.049 3.504 3.628 1.143
1.400 17.382 2.505 12.245 16.407 1.671 2.305 7.467 12.553 7.401 4.732 3.451 16.595 7.746 8.166 3.905
[Z] = [Y ]−1
(30)
Diagonal elements of [Z] are driving point impedances viewed at network nodes if the load admittances are taken into account at all load points including the load point injecting the harmonics into the network. The driving point impedance for a load point i relevant for the analysis of harmonic penetration is obtainable by excluding, for this node, its motor admittance. This admittance is, then, series connected to the impedance obtained by former exclusion. If the driving point impedance for node i obtained from [Z] is zdi , then the relevant driving point impedance for this node will be
zrdi =
bM 1 +j i zdi h
−1
+ jXMi h
(31)
For source node the relevant impedance equals
zdr0 =
1 1 +j zd0 Xh
−1
+ jXh
(32)
The calculations listed above were simply conducted using MATLAB software by appropriately applying the elementwise operations with matrices. The results of this analysis are presented in Table 8. As can be seen from Table 8, the driving point impedances are low for both harmonics under consideration, at all network nodes. Thus, it can be concluded that the capacitor banks whose installation was recommended for economy reasons, will not cause high-harmonic voltage deformations.
(27)
with ri and xi being the resistance and reactance for fundamental frequency. The Y-parameter matrix of the network in Fig. 1 is formed in two steps. In the first step the matrix due to only series branch admittances is formed [Y1 ] = [A][yb ][AT ]
Table 8 Driving point impedances for network nodes for the harmonics of the fifth and seventh order
network [Z] matrix is now + j(ci + Ci )2fh
where p gi = 2i Ur bi =
1669
6. Load phase shifting to minimize monthly real power peak A means to reduce the charges paid for real power peaks is phase shifting of loads supplied at the same source node. Consider n feeders supplied at the source node. Their consumption at a series of T time instants covering a month period, say hours, can be represented by n by T matrix [A] with elements Aij being real power consumption of feeder i at time instant j. The monthly peak equals pm = maxj
n Aij
(33)
i=1
(29)
where [Yn ] is a diagonal matrix with elements being the shunt admittances at network nodes given in Eqs. (22) and (26). The
as the sum of matrix [A] column j gives total real power consumption at source node at time instant j. The highest of this sum is the monthly peak.
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J. Nahman et al. / Electric Power Systems Research 78 (2008) 1664–1671
Table 9 Optimal phase shifts of feeder CSDP daily load diagram Month
Optimal d1 (h)
Pm for d1 = 0 h (MW)
Pm for d1 = 1 h (MW)
Third column − fourth column (MW)
October November December
1 1 1
11.60 10.40 10.30
10.60 9.70 9.50
1.00 0.70 0.80
The minimization of the real power monthly peak can be dealt with as an optimization problem. The control variables are load phase shifts di and the objective is to minimize performance measure pm . One among the feeders, arbitrary chosen, should be adopted as referent feeder whose phase shift is zero. The choice of the referent feeder does not affect the result of optimization, as the phase shifts among feeder loads are only relevant. For the optimization the Hooke and Jeeves’ approach can again be used with di being the coordinates of possible solution points. In the calculation procedure the following relationships should be accounted for For di ≥ 0 if j + di ≤ T
then Aij = Aij+di
else Aij = Aij+di −T
(34)
then Aij = Aij+di
else Aij = AiT +j+di
(35)
For di < 0 if j + di > 0
On the left-hand side of equations in (34) and (35) are new, modified row i elements of matrix [A] after the load diagram of feeder i is shifted for di time units. On the right-hand side of equations are elements of the initial [A] matrix being the input parameter for the optimization program. The maximum allowable positive and negative phase shifts, in absolute terms, should be specified for all feeder loads. The above-described optimization procedure was conducted for CSDP and IHP feeders being under consideration. The maximum allowable shift in absolute terms was 2 h with 1 h steps. In this case, as there are only two feeders, only five phase shifts options are available. Hence, the optimal shift could be simply obtainable by enumeration. However, the number of phase shifts that should be considered in the case of, say, n feeders when there are m available shift options for each feeder, the number of situations that should be explored would be mn−1 , which could be a time consuming task for enumeration. The optimization in the case under consideration was performed for the 3 months during which the loads of feeders were monitored. It was found that by shifting the load diagram of CSDP feeder to lead 1 h before the IHP feeder diagram a considerable reduction of monthly real power peaks are achieved in all the 3 months observed. The results obtained are presented in Table 9. The mean peak reduction per month is 0.833 MW that, for a monthly peak charge of 4000 D /MW, gives 3464 D /month cost saving. However, the covered sample of months is limited and a prolonged observation of load diagrams throughout at least the whole-year period would be necessary before drawing a final conclusion on load shifting. The analysis of daily diagrams for the 3 months that were observed has shown that the load shift found to be optimal concerning monthly peaks would lead to an increase of daily peaks for 24 out of 92 days, which means in 26% cases. The mean daily peak reduction is 0.3522 MW with standard deviation 0.6588 being greater than the mean. These results support the conclusion that a prolonged analysis of shifting would be recommendable. The operations of the two plants under consideration are not functionally correlated which allows the phase shifting of some activities. The inspection of daily load diagrams reveals that the
maximum daily peaks for both plant networks appear between 7 and 8 pm in most cases. From the consultations with engineers controlling the heath production processes it is concluded that there would be no obstacles to reduce some activities of the IHP during the indicated critical period and postpone them for 1 h. As illustration, Fig. 4 displays the aggregated daily load diagram of feeders under consideration for 12 November before and after phase shifting.
7. Conclusions The paper presented some results of the analysis of operation of an industrial plant network and investigated possible rationalization measures. Indices are defined for characterization of daily loads. These indices, processed by statistical methods, made it possible to investigate the homogeneity of recorded data and to identify representative daily load diagrams best matching the mean values of indices of the observed load samples. It was established that such a representative daily diagram for whole 3-month period studied exists for both feeders under consideration. Based upon the representative load diagram and upon an appropriate feederoperating model the optimization of reactive energy compensation is performed by taking into account the available locations and capacitor banks sizes, which was detailed demonstrated for the network supplying a plant with a considerable portion of induction motors load. It was shown that large savings of cost can be achieved by compensation. The optimal solution found has been checked and approved safe for resonance conditions. A general mathematical model was developed to optimize the phase shifts of loads supplied by the source substation in order to reduce the charges on monthly real power peaks. It was applied, for demonstration, to assess the gains potentially achievable for the two feeders supplying the considered industrial networks. For both the reactive energy compensation and load diagram phase-shifting optimizations the Hooke and Jeeves’ discrete search approach was successfully applied. All necessary information on the corresponding mathematical models, calculation flow and results obtained is presented.
Fig. 4. Aggregated load diagrams of feeders on November 12 before (- - -) and after (—) shifting.
J. Nahman et al. / Electric Power Systems Research 78 (2008) 1664–1671
References [1] M. Papadopoulos, N.D. Hatziargyriou, M.E. Papadakis, Graphic aided interactive analysis of radial distribution systems, IEEE Trans. Power Deliv. 2 (October) (1987) 1297–1302. [2] J. Nahman, D. Peric, Distribution system performance evaluation accounting for data uncertainty, IEEE Trans. Power Deliv. 18 (July) (2003) 694–700. [3] H.N. Ng, M.M.A. Salama, A.Y. Chikhani, Classification of capacitor allocation techniques, IEEE Trans. Power Deliv. 15 (January) (2000) 387–392. [4] M.E. Baran, F.F. Wu, Optimal capacitor placement on radial distribution systems, IEEE Trans. Power Deliv. 4 (January) (1989) 725–734. [5] M.E. Baran, F.F. Wu, Optimal sizing of capacitors placed on a radial distribution system, IEEE Trans. Power Deliv. 4 (January) (1989) 735–743. [6] S.F. Mekhamer, M.E. El-Havary, S.A. Soliman, M.A. Moustafa, M.M. Mansour, New heuristic strategies for reactive power compensation of radial distribution feeders, IEEE Trans. Power Deliv. 17 (October) (2002) 1128–1135. [7] H.D. Chiang, J.C. Wang, O. Cockings, H.D. Chin, Optimal capacitor placement in distribution systems. Parts I and II, IEEE Trans. Power Deliv. 5 (April) (1990) 634–649. [8] G.K. Walsh, Methods of Optimization, J. Wiley & Sons, London, 1975. [9] J. Nahman, D. Tubic, Optimal sparing strategy for a group of substations, IEEE Trans. Power Deliv. 6 (October) (1991) 1469–1475. [10] J. Burke, Power Distribution Engineering—Fundamentals and Applications, M. Dekker Inc., New York, 1994. [11] J. Arrilaga, D.A. Bradley, P.S. Bodger, Power System Harmonics, J. Wiley & Sons, New York, 1985. Jovan Nahman was born in Belgrade, Yugoslavia. He received his Dip. Eng. Grade in electric power engineering from the Electrical Engineering Faculty, University of
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Belgrade, in 1960, and TechD from the same University in 1969. From 1960 to 2001 he was with the Faculty of Electrical Engineering, Belgrade, as a Professor at the Power System Department. Currently he is engaged as a freelance consultant for electric power systems. Dragutin Salamon was born in Vukovar, Croatia, in 1949. He received Dipl Eng, MSc and TechD degrees in electric power engineering from the Electrical Engineering Faculty, University of Belgrade, Belgrade, Serbia and Montenegro (former Yugoslavia), in 1973, 1978 and 1992, respectively. In 1973, he joined the Faculty of Electrical Engineering, University of Belgrade, where he is employed as associated professor at the Power System Department. His special area of interest and research are high-voltage substations grounding systems and power system measurements. Zlatan Stojkovi´c (1960) graduated in 1984, took his M.Sc. degree in 1991 and his PhD degree in 1995 at the Faculty of Electrical Engineering in Belgrade/Yugoslavia. Since 1993 he with the Faculty of Electrical Engineering, University of Belgrade. He is at present professor teaching computer-aided design in power systems. Under a scholarship granted by the Alexander von Humboldt Foundation, Bonn, Germany, he was a Postdoctoral Fellow at the Institute of Electric Energy Systems and High Voltage Technology, University of Karlsruhe, Germany. Jovan Mikulovi´c was born in Zajecar, Serbia, in 1968. He received the Dipl Eng and MSc degrees in electrical engineering from the University of Belgrade, Belgrade, Serbia, in 1994 and 2001, respectively. In 1995, he joined the Faculty of Electrical Engineering, Belgrade. From 2000 to 2002 he was with “C.E.M. s.r.l” Company, Gorizia, Italy. Presently, he is a research and teaching assistant at the Faculty of Electrical Engineering, University of Belgrade. His area of research are high-voltage technology and power system measurements.