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Optimal spatial and temporal demand side management in a power system comprising renewable energy sources
Renewable Energy 108 (2017) 533e547
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Renewable Energy journal homepage: www.elsevier.com/locate/renene
Optimal spatial and temporal demand side management in a power system comprising renewable energy sources Dimitrije Kotur*, Zeljko Ðurisi c Department of Power Systems, School of Electrical Engineering, University of Belgrade, Belgrade, Serbia
a r t i c l e i n f o
a b s t r a c t
Article history: Received 1 September 2016 Received in revised form 24 January 2017 Accepted 22 February 2017 Available online 23 February 2017
The increase in installed capacity of renewable energy sources (RES) has a positive effect on the development of smart grids and demand side management (DSM). The reason for this is the intermittent nature of renewable energy, which is directly related to the problem of balancing the production and consumption of power within the power system. By using the DSM, the power consumption in the system comprising RES can be easier adjusted to the power production. The paper proposes an improved concept of DSM through the spatial and temporal DSM. The optimal spatial and temporal DSM aims at determining the power diagram of each individual load bus in order to achieve the optimal state in the whole system. The optimal state of the system can be quantified through the minimum daily energy losses or minimum daily operating costs. A mathematical definition of the optimal spatial and temporal DSM problem is presented as well as the algorithm for its solution. The proposed methodology has been tested by three test networks. The results confirm the overall system performance improvements that include: reduction of energy losses in the system, reduction of the operating costs and the increase of the voltage quality within the system. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Renewable energy sources Demand side management Power system losses Power plants operating costs Interior point method
1. Introduction Environmental protection requirements and limited reserves of fossil fuels have led to the expansion of renewable energy sources (RES) [1,2]. However, the emergence of renewable energy in the power systems has led to new problems associated with power balancing between production and consumption [3]. The reason for this is the intermittent nature of RES. In order to successfully maintain power balance between production and consumption, any increase of the installed capacities of intermittent RES must be accompanied by the corresponding changes in management strategy of other power plants within power system, primarily conventional power plants [4]. Application of smart grids and demand side management (DSM) represent possible solution for a large-scale integration of RES [5]. With DSM included, the consumption ceases to be a passive element of the system, which opens up to the operator additional options for power system control. In this paper, an analysis of the possibilities offered by load shifting in the process of integration of
* Corresponding author. E-mail address: [email protected] (D. Kotur). http://dx.doi.org/10.1016/j.renene.2017.02.070 0960-1481/© 2017 Elsevier Ltd. All rights reserved.
renewable energy into a power system is carried out. Load shifting implies shifting of one part of the load from one hour to another within a specified time interval under condition that the total energy taken within that interval remains constant. Depending on whether a device can or cannot participate in the load shifting process, the devices can be load deferrable or load non-deferrable. The load non-deferrable type includes all devices whose control would interrupt the comfort of human living or jeopardize the work processes of industrial consumers. Therefore, the system operator does not have the freedom of controlling them directly. For households, these include lighting, televisions, computers, etc. The deferrable load is electric power demand that can be served at any time within certain time span. This includes devices whose power can be time shifted without significant disruption of the comfort of living, like washing machines, storage heaters, refrigerators, dishwashers, etc. The management of these devices is described in the literature [6e12]. DSM can be done directly or indirectly. In a direct DSM, distribution system operator directly controls operation of deferrable devices. In an indirect DSM, consumers are encouraged by tariff differences to consume more electricity during low load hours [13]. In this case, the categorization of devices to deferrable and non-deferrable is unnecessary because, due to the tariff
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policy, consumers can also change load diagrams of devices categorized non-deferrable. These include cookers, microwave ovens, etc. Besides households, industrial and commercial consumers are also very important as regards DSM. The possibility of DSM within these categories was analyzed in the literature [14e18]. Integration of RES through DSM is nowadays the subject of considerable research work. The basic idea is to shift load from the hours when production of renewable energy is low to the hours when this production is large. The analyses of DSM in microgrids and households comprising distributed RES were carried out in Refs. [19e21]. These analyses were aimed at reducing dependence of these systems upon external network. Although these analyses do not include operation of the entire system, they provide a good insight in how consumption may be balanced with production of intermittent RES by applying DSM. The effect of DSM on integration of Wind Power Plants (WPP) into power system was analyzed in Ref. [22]. Similar work was done in Ref. [23], where these analyses were done on examples from China, with the emphasis on heat consumers. The optimal dimensioning of various types of RES to be used, together with the systems for energy storage and DSM, for the purpose of reducing the usage of fossil fuel power plants in Ontario, Canada was analyzed in Ref. [24]. A similar analysis was performed in Ref. [25], however, the analyses are in this case performed on the example of a residential building. In Refs. [26,27] an overview of the existing methodologies for integration of renewable energy with the DSM is presented. In the cited literature, the optimal DSM is used in order to maintain, at the system level, power balance between production and consumption in systems comprising intermittent energy sources. The temporal component of DSM has been considered only. RES, as predominantly distributed energy sources, are dispersed throughout the system. The available types of RES, in general, have different variations in time at different geographic locations, therefore, when operation of the system is being planned, in addition to the intermittent nature of RES, it is necessary to consider their dispersed locations and the corresponding spatial variation of production. In this paper, the spatial component of DSM is introduced into the optimization problems of minimizing the energy losses in the system with integrated RES, optimal economic dispatching of the conventional power plants and the optimal voltage control in power network. When the optimal spatial and temporal DSM is applied, during a large-scale production of RES, the consumption should not be forced in all network buses, which is assumed by DSM involving only the temporal coordinate, but only in those buses that are electrically the nearest to the RES. In this way, the energy is produced and consumed locally, thereby reducing power flows in the system. This offers many benefits: reduction of power system losses, transmission capacity release, increases of system stability, low variation of voltage magnitudes, etc. However, since it is usually impossible to completely align production with consumption of RES at the local level, the spatial and temporal optimization of DSM needs to be treated at the level of the entire power system. The time interval for all analyzes is 24 h, but this interval may be varied. The paper is organized as follows: the methodology of spatial and temporal DSM in a power system comprising RES is defined in Section 2. The defined optimization problem is solved by using Interior Point Method in Section 3. Proposed methodology and the algorithm has been tested by three test networks, including one distribution and one transmission. It is shown that the proposed methodology can have practical applicability in the day ahead operational planning of the system.
2. The optimization problem definition In this Section, the optimization process aimed at determining the values of the control variables that will provide an optimal state of a power system is described. The optimal state is defined either through the minimum daily operating costs of power plants or minimum daily energy losses. In conventional power systems, system management is only possible in generation buses, which is done through the control of their active powers and voltages. With the advent of DSM and emergence of modern power electronic devices, primarily FACTS devices [28,29], load buses have also become an active part of the system management. The goal of optimization defined in this paper is to determine the optimal diagrams of production and consumption, as well as voltage diagrams, which indirectly includes the optimal control of reactive power. Every optimization problem can be written in the following form:
min f ðxÞ;
(2.1)
gðxÞ ¼ 0;
(2.2)
hl hðxÞ hu :
(2.3)
Equation (2.1) defines the objective function, (2.2) equality constraints, (2.3) inequality constraints, and x is the vector of unknown variables. In the optimization problem defined in this paper, vector x is defined by the set of equations (2.4)e(2.7):
x ¼ ½P P ¼ ½ P1 U ¼ ½ U1
Q T ;
U P2
U2
::: :::
(2.4) Pi Ui
::: :::
Pn1 Un1
P n T ; Un T ;
Q ¼ ½ Q1 Q2 ::: Qi ::: Qn1 Qn T ; Qn ¼ 0:
(2.5) (2.6) (2.7)
In equations (2.5)e(2.7), Pi defines the vector of unknown average hourly power injection into bus i, Ui and Qi are vectors of the voltage magnitudes and the voltage angles in bus i. The system has n buses and the voltage angles of the last bus are taken for the reference, therefore, Qn ¼ 0. In this paper, DSM is done for the time period of one day, with hourly resolution. Therefore, each vector has 24 elements which correspond to the average hourly values of the specified variables. Vector of power injection in each bus i can be obtained by subtracting the vectors of power generation (PiG ) and power consumption (PiL ):
iT h G P L G L P ¼ P1G P1L P2G P2L ::: PiG PiL ::: Pn1 : n1 Pn Pn
(2.8)
In Equation (2.8), it is formally mathematically assumed that in each bus both production and consumption can exist. In the case that a power plant is connected to bus i or there is a distributed production from RES, power generation vector can be different from zero, otherwise its value is a zero vector. The load diagram consists of two parts e deferrable and nondeferrable. Fig. 1 shows an example of the daily load diagram with its deferrable and non-deferrable part. According to Fig. 1, load in bus i within an arbitrary hour t can be represented as the sum of its deferrable and non-deferrable parts:
Ðurisic / Renewable Energy 108 (2017) 533e547 D. Kotur, Z.
def
nondef
L Pi;t ¼ Pi;t þ Pi;t
; i ¼ 1; 2; …n; t ¼ 1; 2; …; 24:
hour t (t ¼ 1,..,24) for each bus i (i ¼ 1, .., n) the following equation can be defined (2.13):
(2.9)
def Vector Pi;t defines hourly average values of deferrable load in bus i, while the vector Pinondef represents hourly average values of non-deferrable load. In case that the load is not connected to bus i, L is equal to zero. The final form of the unknown power injection Pi;t vector is defined by the equation:
h P ¼ P1G P1def P1nondef
:::
def
PiG Pi
nondef
Pi
:::
minWloss ¼ min
24 X n X
Pi;t $Dt:
(2.11)
In Equation (2.11), i represents index of the bus in the system, Operational costs of a conventional power plant depend on the power generation of the plant and they are usually modeled by a quadratic function [30]. Therefore, the criterion of minimum daily operating costs of all power plants in the analyzed system can formally be defined by mathematical relation (2.12) in which ak, bk and ck represent cost coefficients of the unit connected to bus k.
2 G G ak $ Pk;t þ bk $Pk;t þ ck :
Uk;t $Ui;t $ Gi;k $cos Qi;t Qk;t þ Bi;k $sin Qi;t Qk;t ¼ 0: (2.13)
iT
nondef
Pn
:
(2.10)
In Equation (2.13), Gi;k and Bi;k represent, respectively, conductance and susceptance at position (i,k) of the network admittance matrix. In addition to the equations of power flows, it is possible to define two more categories of equality constraints. In the analyzed system, for each hydropower plant, the daily amount of available water is defined, which also defines an available daily amount of hydroelectric energy Wihydro. This forms another type of equality constraints which define that the energy produced daily by an impoundable hydropower plant must have certain value in order to avoid overflow. These constraints can mathematically be written in the following form:
Wihydro
Dt ¼ 1h and t ordinal number of the hour in the analyzed day.
24 X n X
n X k¼1
def
t¼1 i¼1
min Price ¼ min
Pi;t
PnG Pn
It is necessary to define the vectors of unknown voltage magnitudes and voltage angles. For each bus i ¼ 1, 2, ..., n, the vector Ui containing 24 elements representing the hourly average voltage magnitudes is defined. In a similar way, the vector of unknown voltage angles Qi is defined, with the difference that the last bus is considered as the reference bus, therefore, the vector Qn is equal to zero. The objective function of the defined optimization problem may be the minimum daily energy losses or minimum daily operating costs of conventional power plants. Minimization of losses in the system is equivalent to the minimization of differences between total production and total consumption, so this criterion can be formally defined by Equation (2.11), in which all powers (both production and consumption) are treated as algebraic variables:
535
24 X
hydro Pi;t $Dt ¼ 0;
(2.14)
t¼1 hydro where Pi;t represents average hourly value of power production by i-th hydropower plant, Dt ¼ 1h. With DSM, only the shape of load diagram is changed, with no change of the total consumed energy. In this way, the last category of equality constraints are defined, which is formally written by Equation (2.15):
Wideferrable
24 X
def Pi;t $Dt ¼ 0;
(2.15)
t¼1 deferrable
(2.12)
t¼1 k¼1
In the defined optimization problem, Equations (2.11) and (2.12), there are several categories of equality constraints. The most important of these are the equations of power flow. With the help of FACTS devices it is possible to control the voltage in load buses, therefore, it is enough to define equality constraints only for the equations of active power. From the aspect of active power, at any
where Wi represents deferrable energy of load in bus i, i ¼ 1, …, n, Dt ¼ 1h. For a full definition of the optimization problem, it is necessary to define the inequality constraints. For each generation bus i (the bus in which a conventional power plant is connected), the limit of the minimum and maximum power output is defined by Equation (2.16). It is also necessary to define the scope of permissible voltage magnitudes for each bus i (i ¼ 1,...,n), which is defined by Equation (2.17).
PiG;min PiG PiG;max ;
(2.16)
Uimin Ui Uimax :
(2.17)
In Equation (2.17), it was assumed that the different values of voltage magnitudes in different buses can be assigned. Usually, these differences are made between generation and load buses.
3. The methodology for solution of the optimization problem
Fig. 1. Example of load diagram, with deferrable and non e deferrable parts.
All methods for solution of an optimization problem can be divided into two groups: conventional methods and heuristic
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methods. With conventional methods, a minimum of the criterion function can be found by an iterative process of finding the point at which the gradient of this function is equal to zero. These include Gradient Method [31], Newton Method [32], Interior Point Method [33,34], etc. The main advantages of these methods are high speed of convergence of the iterative process and the solution found is exact. The conventional methods differ in the way of implementation of the inequality constraints. Heuristic methods find the optimal solution through different searching mechanisms of the entire set of the possible solutions. The best known are Particle Swarm Algorithm [35] and Genetic Algorithm (GA) [36]. These methods are easy to program and they also consider inequality constraints easily. On the other hand, these methods do not calculate the gradient of the criterion function, therefore, it is not possible to ascertain whether it is really the right optimum, nor how far is it from the true optimum. The convergence time of these methods is directly dependent on the number of variables in the optimization problem, and their application in systems with a large number of variables is often unacceptable because of the unacceptably large time for convergence. In this paper, the Interior Point Method (IPM) is used [33]. Like all conventional methods, the minimum is obtained by equating the partial derivatives of the criterion functions to zero. The equality constraints are easily considered by using the Method of Lagrange Multipliers [37]. The IPM differs from other conventional methods in the way it considers inequality constraints. The idea is to transform all inequality constraints into equality constraints by adding slack variables, whose value must be greater or equal to 0. In this way, a new formulation of the optimization problem (2.1)e(2.3) is obtained:
min f ðxÞ;
(3.1)
gðxÞ ¼ 0;
(3.2)
hðxÞ hl sl ¼ 0;
(3.3)
hðxÞ þ hu su ¼ 0;
(3.4)
su ; sl 0:
(3.5)
0 are adopted, while for voltage magnitudes the initial values are equal to 1 p.u. (per unit). The initial powers of conventional power plants are determined as the arithmetic mean of the minimum and maximum output power, while the initial powers of load buses correspond to the values before application of DSM. Initial values of the variables (zero iteration) can formally be expressed as follows:
3 2 03 P0 P 0 x ¼ 4 U 5 ¼ 4 1 5: Q0 0 2
0
(3.10)
Vector P 0 can formally be expressed by equations (3.11)e(3.13):
P 0 ¼ P G;0 P L;0 ; P G;0 ¼
(3.11)
P1G;min þP1G;max 2
h P L;0 ¼ P1L;0
P2L;0
:::
:::
PiG;min þPiG;max 2
PiL;0
:::
T PnG;min þPnG;max 2
:::
PnL;0
iT
;
(3.12)
:
(3.13)
where PiL;0 represents the vector of average hourly load power in bus i (i ¼ 1,2,..n) before application of DSM, while P G;0 is the vector of initial values of power production. The dimensions of vector P G;0 are n t, where n represents the number of buses in a system and t is the number of hours of the analyzed time horizon. 4. The 3-bus test system In this Section, the proposed methodology has been tested by the example of a test network comprising three buses. The main objective of this test is to show, and quantitatively demonstrate, the influence of the spatial and temporal DSM on the simplest network topology. Topological scheme of the proposed network is shown in Fig. 2. The network parameters are shown in Table 1. All values are given in per-unit system. The system consists of 2 load buses and 1 generation bus. It is assumed that the load diagrams of both buses are equal, and that they had 30% of the deferrable load within each hour. In Fig. 3 both load
In order to apply the Method of Lagrange Multipliers, it is necessary to eliminate the inequality constraints defined by Equation (3.5). This is solved by adding the barrier function m$ðlnðsl Þ þ lnðsu ÞÞ to the objective function (3.1), resulting in a new optimization problem that contains only equality constraints:
minðf ðxÞ m$ðlnðsl Þ þ lnðsu ÞÞÞ;
(3.6)
gðxÞ ¼ 0;
(3.7)
hðxÞ hl sl ¼ 0;
(3.8)
hðxÞ þ hu su ¼ 0:
(3.9)
It is possible to apply the Method of Lagrange Multipliers [37] to the problem defined by Equations (3.6)e(3.9). By substituting equations (2.11)e(2.17) into equations (3.1)e(3.5), the proposed mathematical model for the spatial and temporal DSM in the analyzed power system comprising RES is obtained. This mathematical problem is solved iteratively, by using the methodology described in Appendix A [33]. In order to fully define the iterative process, it is necessary to define initial values of the variables. For angles, the initial values of
Fig. 2. Three bus test system.
Table 1 Parameters of the 3-bus test system network. From bus
To bus
R [p.u.]
X [p.u.]
B/2 [p.u.]
1 2 1
3 3 2
20/484 10/484 10/484
80/484 40/484 40/484
0 0 0
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Fig. 3. Demand in the second and third bus having deferrable and non e deferrable parts.
diagrams are shown with their deferrable and non-deferrable parts. In both load buses, besides consumption, there is a production of the RES, shown in Fig. 4. It is assumed that in bus 2 production of photovoltaic systems dominate, so the production in this bus during the day is higher. On the other hand, it is assumed that in bus 3, production of wind turbines are dominant, so its production during the night is higher than that of bus 2. The methodology developed so far is applied to this system in order to determine the optimal production diagram of bus 1 (containing the only conventional power plant in the system), optimal load diagrams in buses 2 and 3 and optimal voltage diagrams of all buses of the system. Both optimization criteria are
tested, equations (2.11) and (2.12). To determine the operating costs of a power plant, it is necessary to define the function of operating costs depending on the power output of the power plant. This function is shown by equation (4.1):
Cost ¼ a þ b$Pg þ c$Pg2 :
(4.1)
Values of the coefficients (expressed in MU e Monetary Unit) in equation (4.1) are: a ¼ 561 MU, b ¼ 792 MU and c ¼ 15:62 MU [38]. In addition, for each bus the allowable voltage range is defined. The scope of the generation buses is from 0.95 p.u. to 1.05 p.u, and of the load buses from 0.9 p.u. to 1.1 p.u.
Fig. 4. Productions from the RES in the second and third bus.
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At first, the objective functions before application of DSM are calculated for two cases e (i) the system with the RES and (ii) the system without the RES. Table 2 shows the obtained results. After this, the optimal spatial and temporal DSM is performed as described in the previous sections. According to the defined methodology, equations (2.11) and (2.12), the algorithm for optimal DSM also integrates the optimal economic dispatching of the conventional power plants, and the optimal voltage control. The obtained results for both objective functions are the same, i.e. the obtained state leads to the minimum daily energy losses, as well as the minimum daily operating costs. The optimal load diagrams are shown in Fig. 5. Diagrams in the first row include both spatial and temporal DSM, while the second row shows diagrams of DSM with only temporal coordinate. By analyzing only the temporal DSM, it can be concluded that the load should be shifted to the hours wherein production of the RES is high. However, the results obtained in this test system show that in this case, the optimal state is not achieved. In order to achieve the specified optimization criterion, it is necessary to introduce the spatial coordinate of DSM, which will enable a spatial redistribution of production from the RES. The diagrams of Fig. 5, show that, in the optimal state, load in each bus follows the production of RES in the corresponding bus. Table 3 shows the results of the optimal DSM, whereby in one case the spatial coordinate of DSM is considered while in the other it is not. The results presented in Table 3 show that in the case of only temporal DSM, energy losses in the analyzed test system are reduced by 2.68%, while the optimal operating costs have decreased by 0.173% compared to the case without the DSM. In the case of the optimal spatial and temporal DSM, energy losses are reduced by as much as 5.037%, while the operating costs have decreased 0.2% compared to the case without DSM. These results
Table 3 The results of the optimization of the 3-bus test system with DSM.
Daily energy losses [p.u.] Daily operating costs of power plant in the slack bus [MU]
System with temporal DSM
System with both spatial and temporal DSM
0.4792 37407
0.4676 37397
have confirmed the improvements obtained by introducing spatial coordinate of DSM. In addition, the optimal spatial and temporal DSM leads to smaller fluctuations of the values of voltages, which can be seen in Fig. 6. The value of 30% of deferrable load is relatively large, therefore, another analysis is performed. In this analysis, the percentage of the deferrable load was varied, and for each of the assumed values the optimization was carried out. Table 4 shows the results of these analyzes. From Table 4 it can be concluded that the largest profits are achieved by changing the percentage of deferrable load from 0 to 10%. The reason is that both losses and operating costs are a quadratic function of the power value, therefore, the first cutting off of the peaks achieves the greatest relative reduction in both energy losses and operating costs. The previously defined optimization problem has also been solved by using GA. The results coincide with those obtained in the previous analyses, with the difference that the convergence time of GA is about 50 min. Compared to the IPM convergence time of about 1 s, it can be concluded that in this case, IPM has a greater practical use than GA.
5. The 11-bus transmission test system Table 2 The results of analysis for the 3-bus test system without DSM.
Daily energy losses [p.u.] Daily operating costs [MU] of power plant in the slack bus
System without RES
System with RES
0.8847 46351
0.4924 37472
In this Section, the proposed methodology has been tested by the case of the real 400 kV transmission network in Serbia containing 11 buses (8 load buses and 3 generation buses). The main purpose of this test is to show and quantitatively demonstrate the importance of the spatial and temporal DSM applied to a real transmission system. Fig. 7 shows topological scheme of the test network.
Fig. 5. Results of the spatial and temporal DSM for the 3-bus test system.
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Fig. 6. Production from bus 1 and voltages in all buses.
Table 4 Energy losses and operating costs with different values of deferrable load. Deferrable load
Energy losses [p.u.]
Operating costs [MU]
0% 10% 20% 30%
0.4924 0.4740 0.4696 0.4676
37472 37437 37419 37397
The test network parameters are shown in Table 5, in the perunit system. The base value of power and voltage are: SB ¼ 100 MVA and UB ¼ 400 kV. Fig. 8 shows diagrams of the load buses (with their deferrable and non-deferrable loads). In this case, it was also assumed that 30% of load energy is deferrable. The defined scope of allowable voltage values for load buses is 0.9e1.1 p.u. Fig. 9 shows diagrams of production from the RES. It is assumed that in buses 3 and 8 PV power plants are connected, while in buses 7 and 10 WPPs are connected. These diagrams are based on the literature data concerning the solar and wind energy potential in Serbia [39,40]. Coal-fired power plants are connected to buses 1 and 6, an impoundable hydropower plant is connected to bus 5. Each power plant is characterized by the minimum and maximum power outputs, as well as by the coefficients of an operating cost function [38]. These values are given in Table 6. For the impoundable hydropower plant, the total daily available energy is defined by equation (2.14). The minimum and maximum allowable voltage value limits in the generation buses are 0.95 and 1.05 p.u. Table 5 Parameters of the 11-bus system network.
Fig. 7. The 11-bus test system: Serbian 400 kV transmission network.
From bus
To bus
R [p.u.]
X [p.u.]
B/2 [p.u.]
4 5 6 6 7 1 2 1 3 3 1 10
5 6 7 8 8 8 3 2 9 4 10 11
0.00153 0.00283 0.0009 0.00159 0.00043 0.00089 0.00286 0.00187 0.00233 0.00181 0.00084 0.00153
0.0176 0.03282 0.01054 0.01824 0.0049 0.01038 0.02847 0.0215 0.02585 0.02106 0.00961 0.01704
0.4471 0.8306 0.2605 0.4682 0.1187 0.2625 0.7378 0.5448 0.6699 0.5328 0.9726 0.4336
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Fig. 8. Demand diagrams of the load buses of the 11-bus test system.
Fig. 9. Production from the RES in analyzed 11-bus test system.
Table 6 Basic characteristics of generation buses. Nodes
Pmin [p.u.]
Pmax [p.u.]
Wavailable [p.u.]
a [MU]
b [MU]
c [MU]
1. 5. 6.
13 0 4
18 12 7
/ 100 /
561 0 78
792 0 997
15.62 0 48.72
Once the network and system parameters have been defined, three analyses were carried out. In the first two analyzes, it was assumed that the system did not possess the ability of the spatial and temporal DSM. In the first case, no presence of RES was assumed, whereas in the second case, the RES were integrated into the system. In each of these analyzes, the optimal economic
dispatching of the aggregates within the system was carried out, as well as the optimal management of the voltages in all buses. The results of these analyzes are summarized in Table 7. After that, the optimal spatial and temporal DSM has been carried out, including in the analysis the two objective functions. Fig. 10 shows the optimal diagrams obtained for the objective function of minimum daily energy losses. Table 7 The results of analysis of the 11-bus system without spatial and temporal DSM.
Daily energy losses [p.u.] Daily operating costs [MU]
System without RES
System with RES
5.8787 648 840
4.8459 543 570
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Fig. 10. Optimal load diagrams for the minimum daily energy losses in the analyzed test system.
The obtained results confirm that in the case of a large-scale production of the RES, the load should be shifted only in those buses that are electrically close to the corresponding RES. Therefore, it is obtained that the load of the buses containing RES follows the production from these RES. In this way, the energy is produced and used locally, thereby reducing power flows in the transmission min ¼ network. In the optimal state, the daily energy losses are WWloss 4:7474 p.u. This value is 2.03% lower than the value of the losses when there is no spatial and temporal DSM. In addition, the spatial and temporal DSM reduces the variation of voltage profiles of all buses, thus achieving a better quality of the voltage in the system. Fig. 11 shows the optimal production of power plants in the system along with the voltage profiles of all system buses. It can be seen that the voltage magnitudes in all system buses are practically identical with very small temporal variations. Also, it can be concluded that the variation of production in the conventional power plants is small, which positively affects the technical
performance and operating costs of the power plants. The analyzed network contains branches of the same voltage level (400 kV). If the analysis is spread to the parts of the network at lower voltage levels, additional effects can be achieved by optimizing the tap-changer operation of the interconnection power transformers, which would reduce the required power of FACTS devices and power losses in the transmission system. In order to verify the results and benefits of the proposed IPM method, GA is also used for the calculation of minimum daily energy losses. The obtained result is 5.06 p.u., and the convergence time is around 4 days. Compared to IPM, whose convergence time is around 4 s, GA is both less accurate and impractical compared to the proposed IPM. Therefore, the remaining analyses are done only by using IPM. In the second analysis, the objective function was the minimum daily operating costs. By applying the developed methodology and optimization criteria (2.12), the optimal power diagrams of the load buses are shown in Fig. 12.
Fig. 11. Optimal power production diagrams with optimal voltage profiles for the minimum daily energy losses.
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Fig. 12. Optimal load diagrams for the minimum system operating costs.
Load diagrams in this case are only slightly different from the diagrams of Fig. 10. This was to be expected because an increase in the losses leads to an increase in the demand for energy from power plants. Because of that, the criterion of minimum daily operating costs tends to reduce the gross consumption in the system. The resulting operating costs are 541 550 MU which is 0.37% less than in the system without the spatial and temporal DSM. Fig. 13 shows the optimal production of all conventional power plants in the system, along with the voltage profiles of all buses when the objective function is the minimum of daily operating costs of the conventional power plants. The prediction of production diagrams from RES is essential for a practical implementation of the proposed methodology of spatial and temporal DSM. In the previous analysis, it is assumed that the production of RES is entirely predictable. Many different models for the prediction of production of WPP and PV systems have been developed. Some of these models are presented in the literature [41e50]. Each of them makes errors, so the proposed spatial and temporal DSM would have some deviations compared to a day ahead forecasting. These deviations can be reduced through the
Fig. 14. The 16-bus distribution test system.
Fig. 13. Optimal power production diagrams and optimal voltage profiles when objective function is the minimum of daily operating costs.
Ðurisic / Renewable Energy 108 (2017) 533e547 D. Kotur, Z. Table 8 Parameters of the 16-bus system network. From bus
To bus
R [p.u.]
X [p.u.]
1 2 2 2 3 3 7 7 13 13 4 9 9 14
2 3 7 13 4 6 9 8 12 14 5 10 11 15
0.05 0.075 0.11 0.11 0.09 0.08 0.08 0.11 0.09 0.08 0.04 0.11 0.08 0.04
0.8 0.1 0.11 0.11 0.18 0.11 0.11 0.11 0.12 0.11 0.04 0.11 0.11 0.04
543
intraday correction of the calculated diagrams [46e51]. A day ahead prediction of deferrable and non-deferrable load is much more precise [52] and it is expected that it would not bring great error in the calculations. 6. The 16-bus distribution test system In this Section, the proposed methodology has been tested by the example of a 16 bus distribution network. This network represents IEEE 16 bus test system [53], in which power transformer is added in order to interconnect the entire system. The main purpose of this test is to show and quantitatively demonstrate the importance of the proposed spatial and temporal DSM applied to a distribution system. Fig. 14 shows topological scheme of the test network. The network parameters are shown in Table 8 [53]. The base values of
Fig. 15. Demand diagrams of the load buses of the 16-bus distribution test system before DSM.
Fig. 16. Production from the RES in the analyzed 16-bus distribution test system.
544
Ðurisic / Renewable Energy 108 (2017) 533e547 D. Kotur, Z.
Table 9 The results of analysis for the 16-bus test system.
Daily energy losses [MWh] Percentage reduction of daily energy losses
System without RES and DSM
System with RES and without DSM
System with RES and temporal DSM
System with RES and spatial and temporal DSM
11.91 /
8.67 0
8.4 3.21%
8.25 5.09%
Fig. 17. Optimal load diagrams in the analyzed 16-bus test system.
power and voltage are: SB ¼ 100 MVA and UB ¼ 11 kV. Fig. 15 shows load diagrams before DSM. In this case, it is also assumed that 30% of load energy is deferrable. Fig. 16 shows diagrams of production from the RES. It is assumed that PV power plants are connected to buses 6 and 11, and in bus 15 WPPs are connected. Once the network and system parameters have been defined, four analyses were carried out. In the first two analyzes, it was assumed that the system did not possess the ability of DSM. In the first case, no presence of RES was assumed, whereas in the second case, the RES were integrated into the system. In the third case, only temporal DSM was implemented, while in the fourth case, the optimal spatial and temporal DSM was carried out. The objective function was minimum daily energy losses. The results are shown in Table 9. Fig. 17 shows the optimal load diagrams obtained after implementation of the spatial and temporal DSM. The results obtained in the case of the distribution network confirm all the conclusions illustrated in the case of the transmission network. In addition, at the connection point of the distribution network to the transmission system (bus 0), variation of the diagram of active power taken from the external network is considerably reduced, from 47.12% to 11.62%, Fig. 18. In addition, the spatial and temporal DSM can improve the quality of the voltage, which is also presented in Fig. 18 where the voltage diagram of bus 11 (the weakest in the analyzed network) is shown. Implementation of the spatial and temporal DSM significantly reduces voltage variations in all consumer buses. Further improvement could be obtained by including optimization in the tap changer operation of power transformers.
Fig. 18. External grid active power diagrams and voltage diagrams in the most critical bus.
7. Conclusions The paper introduces the concept of spatial and temporal DSM. The spatial and temporal DSM represents a new methodology that allows the system operator to consider the intermittent nature of renewable energy sources, as well as their dispersed allocation. The developed mathematical model, based on Interior Point Method, allows solving highly complex optimization problems of the spatial and temporal DSM in real networks having a large number of buses and the equality and inequality constraints. From the three test examples performed, the following positive effects of the spatial and temporal DSM are identified:
Ðurisic / Renewable Energy 108 (2017) 533e547 D. Kotur, Z.
1. It is shown that both spatial and temporal DSM improves the performance of the entire power systems, in terms of both energy losses and operating costs. Also, it can be concluded that in the case when production from the RES is large, load should be shifted in the buses that are electrically close to the RES. 2. The spatial and temporal DSM reduces the peak loads of some elements in the power system (power transformers, power lines, etc). This has many benefits: the service lifetime of these elements is increased, increased transmission capacity of power lines, etc. 3. The spatial and temporal DSM reduces the voltage variations of load buses. Beside power quality, these effects reduce tap changing in the regulating power transformers which extends their service lifetime. 4. Positive effects of the temporal and spatial DSM are higher when the installed capacity of the RES grows. 5. When the objective function is minimum daily operating costs, the spatial component of DSM has a greater effect on a power system having longer transmission lines and loaded networks. In the case of relatively low power losses, the significance of the space coordinate is reduced. 6. The positive effects of DSM are the greatest when the percentage value of deferrable load changes from 0% to 10%. After that, further increase of deferrable load would improve the positive effects to a lesser extent. The analyses carried out in this paper prove that the improvements can be obtained by considering the spatial component of DSM. In this way, a significant step is made in the research area having the ultimate purpose of developing a methodology that will enable successful integration of renewable energy in a power system. The proposed spatial and temporal DSM algorithm can practically be implemented through a coordinated management of the TSO and DSO. By using the proposed methodology, the TSO would define the optimal total load diagram of each bus in the transmission system, according to which DSO would use smart grid and techniques of direct or/and indirect DSM in order to spatially distribute the load in all distribution system buses. If DSM is implemented by using real time pricing, the use of the proposed methodology implies that the electricity price in the same hour would be different in different buses of the transmission and distribution systems.
545
Lm ¼ f ðxÞ mðlnðsl Þ þ lnðsu ÞÞ lT gðxÞ pTl ðhðxÞ hl sl Þ pTu ð hðxÞ þ hu su Þ: (A-5) In equation (A-5), l, pTl , pTu respectively, represent the Lagrange’s coefficients which take into account the equality constraints in equations (A-2), (A-3), (A-4). The criteria of reaching the optimal value are obtained by solving the following system of equations:
Vsl Lm ¼ m$S1 l $e þ pl ¼ 0;
(A-6)
Vsu Lm ¼ m$S1 u $e þ pu ¼ 0;
(A-7)
Vpl Lm ¼ hðxÞ þ hl þ sl ¼ 0;
(A-8)
Vpu Lm ¼ hðxÞ hu þ su ¼ 0;
(A-9)
Vl Lm ¼ gðxÞ ¼ 0;
(A-10)
Vx Lm ¼ Vf ðxÞ VgðxÞ$lT VhðxÞ$ pTl pTu :
(A-11)
For obtaining the numerical solution of the system (A-6) - (A11), Newton’s iterative procedure is applied by using the following matrix equation (A-12):
2 6 6 6 6 6 6 6 4
m$S2 l
0 I m$S2 0 0 u I 0 0 0 I 0 0 0 0 0 0 VhðxÞT 2 3 Vsl Lm 6 Vsu Lm 7 6 7 6 Vpl Lm 7 7 ¼ 6 6 Vpu Lm 7: 6 7 4 Vl Lm 5 Vx L m
0 I 0 0 0 VhðxÞT
0 0 0 0 0 VgðxÞT
3 2 3 0 Dsl 7 6 7 6 Dsu 7 0 7 7 Dpl 7 hðxÞ 7 6 7 7$6 Dpu 7 hðxÞ 7 6 7 7 6 VgðxÞ 5 4 Dl 5 Dx V2 L m x
(A-12) In equation (A-12),
V2x Lm
is calculated as:
V2x Lm ¼ V2x f ðxÞ V2x gðxÞ$lT V2x hðxÞ$ pTl pTu
Acknowledgement This research was partially supported by the Transmission System Operator of Serbia, project Smarter grid, and European Commission, project EmBuild (Horizon2020).
Since the calculation time of each iteration drastically increases with increasing size of the matrix, the problem can be simplified by solving the system (A-14):
Appendix A Let equations (A-1)e(A-4), define the optimization problem.
minðf ðxÞ m$ðlnðsl Þ þ lnðsu ÞÞÞ;
(A-1)
gðxÞ ¼ 0;
(A-2)
hðxÞ hl sl ¼ 0;
(A-3)
hðxÞ þ hu su ¼ 0:
(A-4)
0 VgðxÞT
VgðxÞ V L Dl ¼ l m ; $ x Dx Hd
(A-14)
Where the matrices Hd and x are calculated as follows:
2 $Vx hðxÞ; Hd ¼ V2x Lm þ m$Vx hðxÞT $ S2 l þ Su h
In order to eliminate the equality constraints, the Method of Lagrange Multipliers is applied. Then the optimization problem is reduced to finding minimum of the function defined by (A-5):
(A-13)
(A-15)
2 x ¼ Vx Lm þ Vx hðxÞT $ m$ S2 u $Vpu $Lm Sl $Vpu $Lm þ Vsl Lm
i Vsu Lm :
(A-16) After solving the system (A-14), a change of other variables may be obtained by equations (A-17) and (A-18):
Ðurisic / Renewable Energy 108 (2017) 533e547 D. Kotur, Z.
546
Dsl ¼ VhðxÞ$Dx Vpl Lm Dsu ¼ VhðxÞ$Dx Vpu Lm ;
(A-17)
2 Dpl ¼ m$S2 l $Dsl Vsl Lm Dpu ¼ m$Su $Dsu Vsu Lm :
(A-18)
Values of the variables in the next iteration are calculated according to equations (A-19) - (A-20):
skþ1 ¼ skl þ akp $Dskl ; skþ1 ¼ sku þ akp $Dsku ; xkþ1 ¼ xk þ akp $Dxk ; u l (A-19) kþ1 k pkþ1 ¼ pkl þ akd $pkl ; pkþ1 ¼ pku þ akd $pku ; l ¼ l þ akd $Dl: u l
(A-20) The coefficients akp and akd are calculated as:
(
akp
¼ min 1; g min
Dski 30
ski
Dski
(
) ;
akd
¼ min 1; g min
Dpki 30
pki
)
Dpki
(A-21) Coefficient m is calculated in each iteration according to formula (A-22)
mkþ1 ¼
$ sTl $pl þ sTu $pu : 2$n þ m
s
(A-22)
The condition of convergence is defined by equation (A-23):
Dxk k Dl ε:
(A-23)
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Contents lists available at ScienceDirect
Renewable Energy journal homepage: www.elsevier.com/locate/renene
Optimal spatial and temporal demand side management in a power system comprising renewable energy sources Dimitrije Kotur*, Zeljko Ðurisi c Department of Power Systems, School of Electrical Engineering, University of Belgrade, Belgrade, Serbia
a r t i c l e i n f o
a b s t r a c t
Article history: Received 1 September 2016 Received in revised form 24 January 2017 Accepted 22 February 2017 Available online 23 February 2017
The increase in installed capacity of renewable energy sources (RES) has a positive effect on the development of smart grids and demand side management (DSM). The reason for this is the intermittent nature of renewable energy, which is directly related to the problem of balancing the production and consumption of power within the power system. By using the DSM, the power consumption in the system comprising RES can be easier adjusted to the power production. The paper proposes an improved concept of DSM through the spatial and temporal DSM. The optimal spatial and temporal DSM aims at determining the power diagram of each individual load bus in order to achieve the optimal state in the whole system. The optimal state of the system can be quantified through the minimum daily energy losses or minimum daily operating costs. A mathematical definition of the optimal spatial and temporal DSM problem is presented as well as the algorithm for its solution. The proposed methodology has been tested by three test networks. The results confirm the overall system performance improvements that include: reduction of energy losses in the system, reduction of the operating costs and the increase of the voltage quality within the system. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Renewable energy sources Demand side management Power system losses Power plants operating costs Interior point method
1. Introduction Environmental protection requirements and limited reserves of fossil fuels have led to the expansion of renewable energy sources (RES) [1,2]. However, the emergence of renewable energy in the power systems has led to new problems associated with power balancing between production and consumption [3]. The reason for this is the intermittent nature of RES. In order to successfully maintain power balance between production and consumption, any increase of the installed capacities of intermittent RES must be accompanied by the corresponding changes in management strategy of other power plants within power system, primarily conventional power plants [4]. Application of smart grids and demand side management (DSM) represent possible solution for a large-scale integration of RES [5]. With DSM included, the consumption ceases to be a passive element of the system, which opens up to the operator additional options for power system control. In this paper, an analysis of the possibilities offered by load shifting in the process of integration of
* Corresponding author. E-mail address: [email protected] (D. Kotur). http://dx.doi.org/10.1016/j.renene.2017.02.070 0960-1481/© 2017 Elsevier Ltd. All rights reserved.
renewable energy into a power system is carried out. Load shifting implies shifting of one part of the load from one hour to another within a specified time interval under condition that the total energy taken within that interval remains constant. Depending on whether a device can or cannot participate in the load shifting process, the devices can be load deferrable or load non-deferrable. The load non-deferrable type includes all devices whose control would interrupt the comfort of human living or jeopardize the work processes of industrial consumers. Therefore, the system operator does not have the freedom of controlling them directly. For households, these include lighting, televisions, computers, etc. The deferrable load is electric power demand that can be served at any time within certain time span. This includes devices whose power can be time shifted without significant disruption of the comfort of living, like washing machines, storage heaters, refrigerators, dishwashers, etc. The management of these devices is described in the literature [6e12]. DSM can be done directly or indirectly. In a direct DSM, distribution system operator directly controls operation of deferrable devices. In an indirect DSM, consumers are encouraged by tariff differences to consume more electricity during low load hours [13]. In this case, the categorization of devices to deferrable and non-deferrable is unnecessary because, due to the tariff
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Ðurisic / Renewable Energy 108 (2017) 533e547 D. Kotur, Z.
policy, consumers can also change load diagrams of devices categorized non-deferrable. These include cookers, microwave ovens, etc. Besides households, industrial and commercial consumers are also very important as regards DSM. The possibility of DSM within these categories was analyzed in the literature [14e18]. Integration of RES through DSM is nowadays the subject of considerable research work. The basic idea is to shift load from the hours when production of renewable energy is low to the hours when this production is large. The analyses of DSM in microgrids and households comprising distributed RES were carried out in Refs. [19e21]. These analyses were aimed at reducing dependence of these systems upon external network. Although these analyses do not include operation of the entire system, they provide a good insight in how consumption may be balanced with production of intermittent RES by applying DSM. The effect of DSM on integration of Wind Power Plants (WPP) into power system was analyzed in Ref. [22]. Similar work was done in Ref. [23], where these analyses were done on examples from China, with the emphasis on heat consumers. The optimal dimensioning of various types of RES to be used, together with the systems for energy storage and DSM, for the purpose of reducing the usage of fossil fuel power plants in Ontario, Canada was analyzed in Ref. [24]. A similar analysis was performed in Ref. [25], however, the analyses are in this case performed on the example of a residential building. In Refs. [26,27] an overview of the existing methodologies for integration of renewable energy with the DSM is presented. In the cited literature, the optimal DSM is used in order to maintain, at the system level, power balance between production and consumption in systems comprising intermittent energy sources. The temporal component of DSM has been considered only. RES, as predominantly distributed energy sources, are dispersed throughout the system. The available types of RES, in general, have different variations in time at different geographic locations, therefore, when operation of the system is being planned, in addition to the intermittent nature of RES, it is necessary to consider their dispersed locations and the corresponding spatial variation of production. In this paper, the spatial component of DSM is introduced into the optimization problems of minimizing the energy losses in the system with integrated RES, optimal economic dispatching of the conventional power plants and the optimal voltage control in power network. When the optimal spatial and temporal DSM is applied, during a large-scale production of RES, the consumption should not be forced in all network buses, which is assumed by DSM involving only the temporal coordinate, but only in those buses that are electrically the nearest to the RES. In this way, the energy is produced and consumed locally, thereby reducing power flows in the system. This offers many benefits: reduction of power system losses, transmission capacity release, increases of system stability, low variation of voltage magnitudes, etc. However, since it is usually impossible to completely align production with consumption of RES at the local level, the spatial and temporal optimization of DSM needs to be treated at the level of the entire power system. The time interval for all analyzes is 24 h, but this interval may be varied. The paper is organized as follows: the methodology of spatial and temporal DSM in a power system comprising RES is defined in Section 2. The defined optimization problem is solved by using Interior Point Method in Section 3. Proposed methodology and the algorithm has been tested by three test networks, including one distribution and one transmission. It is shown that the proposed methodology can have practical applicability in the day ahead operational planning of the system.
2. The optimization problem definition In this Section, the optimization process aimed at determining the values of the control variables that will provide an optimal state of a power system is described. The optimal state is defined either through the minimum daily operating costs of power plants or minimum daily energy losses. In conventional power systems, system management is only possible in generation buses, which is done through the control of their active powers and voltages. With the advent of DSM and emergence of modern power electronic devices, primarily FACTS devices [28,29], load buses have also become an active part of the system management. The goal of optimization defined in this paper is to determine the optimal diagrams of production and consumption, as well as voltage diagrams, which indirectly includes the optimal control of reactive power. Every optimization problem can be written in the following form:
min f ðxÞ;
(2.1)
gðxÞ ¼ 0;
(2.2)
hl hðxÞ hu :
(2.3)
Equation (2.1) defines the objective function, (2.2) equality constraints, (2.3) inequality constraints, and x is the vector of unknown variables. In the optimization problem defined in this paper, vector x is defined by the set of equations (2.4)e(2.7):
x ¼ ½P P ¼ ½ P1 U ¼ ½ U1
Q T ;
U P2
U2
::: :::
(2.4) Pi Ui
::: :::
Pn1 Un1
P n T ; Un T ;
Q ¼ ½ Q1 Q2 ::: Qi ::: Qn1 Qn T ; Qn ¼ 0:
(2.5) (2.6) (2.7)
In equations (2.5)e(2.7), Pi defines the vector of unknown average hourly power injection into bus i, Ui and Qi are vectors of the voltage magnitudes and the voltage angles in bus i. The system has n buses and the voltage angles of the last bus are taken for the reference, therefore, Qn ¼ 0. In this paper, DSM is done for the time period of one day, with hourly resolution. Therefore, each vector has 24 elements which correspond to the average hourly values of the specified variables. Vector of power injection in each bus i can be obtained by subtracting the vectors of power generation (PiG ) and power consumption (PiL ):
iT h G P L G L P ¼ P1G P1L P2G P2L ::: PiG PiL ::: Pn1 : n1 Pn Pn
(2.8)
In Equation (2.8), it is formally mathematically assumed that in each bus both production and consumption can exist. In the case that a power plant is connected to bus i or there is a distributed production from RES, power generation vector can be different from zero, otherwise its value is a zero vector. The load diagram consists of two parts e deferrable and nondeferrable. Fig. 1 shows an example of the daily load diagram with its deferrable and non-deferrable part. According to Fig. 1, load in bus i within an arbitrary hour t can be represented as the sum of its deferrable and non-deferrable parts:
Ðurisic / Renewable Energy 108 (2017) 533e547 D. Kotur, Z.
def
nondef
L Pi;t ¼ Pi;t þ Pi;t
; i ¼ 1; 2; …n; t ¼ 1; 2; …; 24:
hour t (t ¼ 1,..,24) for each bus i (i ¼ 1, .., n) the following equation can be defined (2.13):
(2.9)
def Vector Pi;t defines hourly average values of deferrable load in bus i, while the vector Pinondef represents hourly average values of non-deferrable load. In case that the load is not connected to bus i, L is equal to zero. The final form of the unknown power injection Pi;t vector is defined by the equation:
h P ¼ P1G P1def P1nondef
:::
def
PiG Pi
nondef
Pi
:::
minWloss ¼ min
24 X n X
Pi;t $Dt:
(2.11)
In Equation (2.11), i represents index of the bus in the system, Operational costs of a conventional power plant depend on the power generation of the plant and they are usually modeled by a quadratic function [30]. Therefore, the criterion of minimum daily operating costs of all power plants in the analyzed system can formally be defined by mathematical relation (2.12) in which ak, bk and ck represent cost coefficients of the unit connected to bus k.
2 G G ak $ Pk;t þ bk $Pk;t þ ck :
Uk;t $Ui;t $ Gi;k $cos Qi;t Qk;t þ Bi;k $sin Qi;t Qk;t ¼ 0: (2.13)
iT
nondef
Pn
:
(2.10)
In Equation (2.13), Gi;k and Bi;k represent, respectively, conductance and susceptance at position (i,k) of the network admittance matrix. In addition to the equations of power flows, it is possible to define two more categories of equality constraints. In the analyzed system, for each hydropower plant, the daily amount of available water is defined, which also defines an available daily amount of hydroelectric energy Wihydro. This forms another type of equality constraints which define that the energy produced daily by an impoundable hydropower plant must have certain value in order to avoid overflow. These constraints can mathematically be written in the following form:
Wihydro
Dt ¼ 1h and t ordinal number of the hour in the analyzed day.
24 X n X
n X k¼1
def
t¼1 i¼1
min Price ¼ min
Pi;t
PnG Pn
It is necessary to define the vectors of unknown voltage magnitudes and voltage angles. For each bus i ¼ 1, 2, ..., n, the vector Ui containing 24 elements representing the hourly average voltage magnitudes is defined. In a similar way, the vector of unknown voltage angles Qi is defined, with the difference that the last bus is considered as the reference bus, therefore, the vector Qn is equal to zero. The objective function of the defined optimization problem may be the minimum daily energy losses or minimum daily operating costs of conventional power plants. Minimization of losses in the system is equivalent to the minimization of differences between total production and total consumption, so this criterion can be formally defined by Equation (2.11), in which all powers (both production and consumption) are treated as algebraic variables:
535
24 X
hydro Pi;t $Dt ¼ 0;
(2.14)
t¼1 hydro where Pi;t represents average hourly value of power production by i-th hydropower plant, Dt ¼ 1h. With DSM, only the shape of load diagram is changed, with no change of the total consumed energy. In this way, the last category of equality constraints are defined, which is formally written by Equation (2.15):
Wideferrable
24 X
def Pi;t $Dt ¼ 0;
(2.15)
t¼1 deferrable
(2.12)
t¼1 k¼1
In the defined optimization problem, Equations (2.11) and (2.12), there are several categories of equality constraints. The most important of these are the equations of power flow. With the help of FACTS devices it is possible to control the voltage in load buses, therefore, it is enough to define equality constraints only for the equations of active power. From the aspect of active power, at any
where Wi represents deferrable energy of load in bus i, i ¼ 1, …, n, Dt ¼ 1h. For a full definition of the optimization problem, it is necessary to define the inequality constraints. For each generation bus i (the bus in which a conventional power plant is connected), the limit of the minimum and maximum power output is defined by Equation (2.16). It is also necessary to define the scope of permissible voltage magnitudes for each bus i (i ¼ 1,...,n), which is defined by Equation (2.17).
PiG;min PiG PiG;max ;
(2.16)
Uimin Ui Uimax :
(2.17)
In Equation (2.17), it was assumed that the different values of voltage magnitudes in different buses can be assigned. Usually, these differences are made between generation and load buses.
3. The methodology for solution of the optimization problem
Fig. 1. Example of load diagram, with deferrable and non e deferrable parts.
All methods for solution of an optimization problem can be divided into two groups: conventional methods and heuristic
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methods. With conventional methods, a minimum of the criterion function can be found by an iterative process of finding the point at which the gradient of this function is equal to zero. These include Gradient Method [31], Newton Method [32], Interior Point Method [33,34], etc. The main advantages of these methods are high speed of convergence of the iterative process and the solution found is exact. The conventional methods differ in the way of implementation of the inequality constraints. Heuristic methods find the optimal solution through different searching mechanisms of the entire set of the possible solutions. The best known are Particle Swarm Algorithm [35] and Genetic Algorithm (GA) [36]. These methods are easy to program and they also consider inequality constraints easily. On the other hand, these methods do not calculate the gradient of the criterion function, therefore, it is not possible to ascertain whether it is really the right optimum, nor how far is it from the true optimum. The convergence time of these methods is directly dependent on the number of variables in the optimization problem, and their application in systems with a large number of variables is often unacceptable because of the unacceptably large time for convergence. In this paper, the Interior Point Method (IPM) is used [33]. Like all conventional methods, the minimum is obtained by equating the partial derivatives of the criterion functions to zero. The equality constraints are easily considered by using the Method of Lagrange Multipliers [37]. The IPM differs from other conventional methods in the way it considers inequality constraints. The idea is to transform all inequality constraints into equality constraints by adding slack variables, whose value must be greater or equal to 0. In this way, a new formulation of the optimization problem (2.1)e(2.3) is obtained:
min f ðxÞ;
(3.1)
gðxÞ ¼ 0;
(3.2)
hðxÞ hl sl ¼ 0;
(3.3)
hðxÞ þ hu su ¼ 0;
(3.4)
su ; sl 0:
(3.5)
0 are adopted, while for voltage magnitudes the initial values are equal to 1 p.u. (per unit). The initial powers of conventional power plants are determined as the arithmetic mean of the minimum and maximum output power, while the initial powers of load buses correspond to the values before application of DSM. Initial values of the variables (zero iteration) can formally be expressed as follows:
3 2 03 P0 P 0 x ¼ 4 U 5 ¼ 4 1 5: Q0 0 2
0
(3.10)
Vector P 0 can formally be expressed by equations (3.11)e(3.13):
P 0 ¼ P G;0 P L;0 ; P G;0 ¼
(3.11)
P1G;min þP1G;max 2
h P L;0 ¼ P1L;0
P2L;0
:::
:::
PiG;min þPiG;max 2
PiL;0
:::
T PnG;min þPnG;max 2
:::
PnL;0
iT
;
(3.12)
:
(3.13)
where PiL;0 represents the vector of average hourly load power in bus i (i ¼ 1,2,..n) before application of DSM, while P G;0 is the vector of initial values of power production. The dimensions of vector P G;0 are n t, where n represents the number of buses in a system and t is the number of hours of the analyzed time horizon. 4. The 3-bus test system In this Section, the proposed methodology has been tested by the example of a test network comprising three buses. The main objective of this test is to show, and quantitatively demonstrate, the influence of the spatial and temporal DSM on the simplest network topology. Topological scheme of the proposed network is shown in Fig. 2. The network parameters are shown in Table 1. All values are given in per-unit system. The system consists of 2 load buses and 1 generation bus. It is assumed that the load diagrams of both buses are equal, and that they had 30% of the deferrable load within each hour. In Fig. 3 both load
In order to apply the Method of Lagrange Multipliers, it is necessary to eliminate the inequality constraints defined by Equation (3.5). This is solved by adding the barrier function m$ðlnðsl Þ þ lnðsu ÞÞ to the objective function (3.1), resulting in a new optimization problem that contains only equality constraints:
minðf ðxÞ m$ðlnðsl Þ þ lnðsu ÞÞÞ;
(3.6)
gðxÞ ¼ 0;
(3.7)
hðxÞ hl sl ¼ 0;
(3.8)
hðxÞ þ hu su ¼ 0:
(3.9)
It is possible to apply the Method of Lagrange Multipliers [37] to the problem defined by Equations (3.6)e(3.9). By substituting equations (2.11)e(2.17) into equations (3.1)e(3.5), the proposed mathematical model for the spatial and temporal DSM in the analyzed power system comprising RES is obtained. This mathematical problem is solved iteratively, by using the methodology described in Appendix A [33]. In order to fully define the iterative process, it is necessary to define initial values of the variables. For angles, the initial values of
Fig. 2. Three bus test system.
Table 1 Parameters of the 3-bus test system network. From bus
To bus
R [p.u.]
X [p.u.]
B/2 [p.u.]
1 2 1
3 3 2
20/484 10/484 10/484
80/484 40/484 40/484
0 0 0
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Fig. 3. Demand in the second and third bus having deferrable and non e deferrable parts.
diagrams are shown with their deferrable and non-deferrable parts. In both load buses, besides consumption, there is a production of the RES, shown in Fig. 4. It is assumed that in bus 2 production of photovoltaic systems dominate, so the production in this bus during the day is higher. On the other hand, it is assumed that in bus 3, production of wind turbines are dominant, so its production during the night is higher than that of bus 2. The methodology developed so far is applied to this system in order to determine the optimal production diagram of bus 1 (containing the only conventional power plant in the system), optimal load diagrams in buses 2 and 3 and optimal voltage diagrams of all buses of the system. Both optimization criteria are
tested, equations (2.11) and (2.12). To determine the operating costs of a power plant, it is necessary to define the function of operating costs depending on the power output of the power plant. This function is shown by equation (4.1):
Cost ¼ a þ b$Pg þ c$Pg2 :
(4.1)
Values of the coefficients (expressed in MU e Monetary Unit) in equation (4.1) are: a ¼ 561 MU, b ¼ 792 MU and c ¼ 15:62 MU [38]. In addition, for each bus the allowable voltage range is defined. The scope of the generation buses is from 0.95 p.u. to 1.05 p.u, and of the load buses from 0.9 p.u. to 1.1 p.u.
Fig. 4. Productions from the RES in the second and third bus.
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At first, the objective functions before application of DSM are calculated for two cases e (i) the system with the RES and (ii) the system without the RES. Table 2 shows the obtained results. After this, the optimal spatial and temporal DSM is performed as described in the previous sections. According to the defined methodology, equations (2.11) and (2.12), the algorithm for optimal DSM also integrates the optimal economic dispatching of the conventional power plants, and the optimal voltage control. The obtained results for both objective functions are the same, i.e. the obtained state leads to the minimum daily energy losses, as well as the minimum daily operating costs. The optimal load diagrams are shown in Fig. 5. Diagrams in the first row include both spatial and temporal DSM, while the second row shows diagrams of DSM with only temporal coordinate. By analyzing only the temporal DSM, it can be concluded that the load should be shifted to the hours wherein production of the RES is high. However, the results obtained in this test system show that in this case, the optimal state is not achieved. In order to achieve the specified optimization criterion, it is necessary to introduce the spatial coordinate of DSM, which will enable a spatial redistribution of production from the RES. The diagrams of Fig. 5, show that, in the optimal state, load in each bus follows the production of RES in the corresponding bus. Table 3 shows the results of the optimal DSM, whereby in one case the spatial coordinate of DSM is considered while in the other it is not. The results presented in Table 3 show that in the case of only temporal DSM, energy losses in the analyzed test system are reduced by 2.68%, while the optimal operating costs have decreased by 0.173% compared to the case without the DSM. In the case of the optimal spatial and temporal DSM, energy losses are reduced by as much as 5.037%, while the operating costs have decreased 0.2% compared to the case without DSM. These results
Table 3 The results of the optimization of the 3-bus test system with DSM.
Daily energy losses [p.u.] Daily operating costs of power plant in the slack bus [MU]
System with temporal DSM
System with both spatial and temporal DSM
0.4792 37407
0.4676 37397
have confirmed the improvements obtained by introducing spatial coordinate of DSM. In addition, the optimal spatial and temporal DSM leads to smaller fluctuations of the values of voltages, which can be seen in Fig. 6. The value of 30% of deferrable load is relatively large, therefore, another analysis is performed. In this analysis, the percentage of the deferrable load was varied, and for each of the assumed values the optimization was carried out. Table 4 shows the results of these analyzes. From Table 4 it can be concluded that the largest profits are achieved by changing the percentage of deferrable load from 0 to 10%. The reason is that both losses and operating costs are a quadratic function of the power value, therefore, the first cutting off of the peaks achieves the greatest relative reduction in both energy losses and operating costs. The previously defined optimization problem has also been solved by using GA. The results coincide with those obtained in the previous analyses, with the difference that the convergence time of GA is about 50 min. Compared to the IPM convergence time of about 1 s, it can be concluded that in this case, IPM has a greater practical use than GA.
5. The 11-bus transmission test system Table 2 The results of analysis for the 3-bus test system without DSM.
Daily energy losses [p.u.] Daily operating costs [MU] of power plant in the slack bus
System without RES
System with RES
0.8847 46351
0.4924 37472
In this Section, the proposed methodology has been tested by the case of the real 400 kV transmission network in Serbia containing 11 buses (8 load buses and 3 generation buses). The main purpose of this test is to show and quantitatively demonstrate the importance of the spatial and temporal DSM applied to a real transmission system. Fig. 7 shows topological scheme of the test network.
Fig. 5. Results of the spatial and temporal DSM for the 3-bus test system.
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Fig. 6. Production from bus 1 and voltages in all buses.
Table 4 Energy losses and operating costs with different values of deferrable load. Deferrable load
Energy losses [p.u.]
Operating costs [MU]
0% 10% 20% 30%
0.4924 0.4740 0.4696 0.4676
37472 37437 37419 37397
The test network parameters are shown in Table 5, in the perunit system. The base value of power and voltage are: SB ¼ 100 MVA and UB ¼ 400 kV. Fig. 8 shows diagrams of the load buses (with their deferrable and non-deferrable loads). In this case, it was also assumed that 30% of load energy is deferrable. The defined scope of allowable voltage values for load buses is 0.9e1.1 p.u. Fig. 9 shows diagrams of production from the RES. It is assumed that in buses 3 and 8 PV power plants are connected, while in buses 7 and 10 WPPs are connected. These diagrams are based on the literature data concerning the solar and wind energy potential in Serbia [39,40]. Coal-fired power plants are connected to buses 1 and 6, an impoundable hydropower plant is connected to bus 5. Each power plant is characterized by the minimum and maximum power outputs, as well as by the coefficients of an operating cost function [38]. These values are given in Table 6. For the impoundable hydropower plant, the total daily available energy is defined by equation (2.14). The minimum and maximum allowable voltage value limits in the generation buses are 0.95 and 1.05 p.u. Table 5 Parameters of the 11-bus system network.
Fig. 7. The 11-bus test system: Serbian 400 kV transmission network.
From bus
To bus
R [p.u.]
X [p.u.]
B/2 [p.u.]
4 5 6 6 7 1 2 1 3 3 1 10
5 6 7 8 8 8 3 2 9 4 10 11
0.00153 0.00283 0.0009 0.00159 0.00043 0.00089 0.00286 0.00187 0.00233 0.00181 0.00084 0.00153
0.0176 0.03282 0.01054 0.01824 0.0049 0.01038 0.02847 0.0215 0.02585 0.02106 0.00961 0.01704
0.4471 0.8306 0.2605 0.4682 0.1187 0.2625 0.7378 0.5448 0.6699 0.5328 0.9726 0.4336
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Fig. 8. Demand diagrams of the load buses of the 11-bus test system.
Fig. 9. Production from the RES in analyzed 11-bus test system.
Table 6 Basic characteristics of generation buses. Nodes
Pmin [p.u.]
Pmax [p.u.]
Wavailable [p.u.]
a [MU]
b [MU]
c [MU]
1. 5. 6.
13 0 4
18 12 7
/ 100 /
561 0 78
792 0 997
15.62 0 48.72
Once the network and system parameters have been defined, three analyses were carried out. In the first two analyzes, it was assumed that the system did not possess the ability of the spatial and temporal DSM. In the first case, no presence of RES was assumed, whereas in the second case, the RES were integrated into the system. In each of these analyzes, the optimal economic
dispatching of the aggregates within the system was carried out, as well as the optimal management of the voltages in all buses. The results of these analyzes are summarized in Table 7. After that, the optimal spatial and temporal DSM has been carried out, including in the analysis the two objective functions. Fig. 10 shows the optimal diagrams obtained for the objective function of minimum daily energy losses. Table 7 The results of analysis of the 11-bus system without spatial and temporal DSM.
Daily energy losses [p.u.] Daily operating costs [MU]
System without RES
System with RES
5.8787 648 840
4.8459 543 570
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Fig. 10. Optimal load diagrams for the minimum daily energy losses in the analyzed test system.
The obtained results confirm that in the case of a large-scale production of the RES, the load should be shifted only in those buses that are electrically close to the corresponding RES. Therefore, it is obtained that the load of the buses containing RES follows the production from these RES. In this way, the energy is produced and used locally, thereby reducing power flows in the transmission min ¼ network. In the optimal state, the daily energy losses are WWloss 4:7474 p.u. This value is 2.03% lower than the value of the losses when there is no spatial and temporal DSM. In addition, the spatial and temporal DSM reduces the variation of voltage profiles of all buses, thus achieving a better quality of the voltage in the system. Fig. 11 shows the optimal production of power plants in the system along with the voltage profiles of all system buses. It can be seen that the voltage magnitudes in all system buses are practically identical with very small temporal variations. Also, it can be concluded that the variation of production in the conventional power plants is small, which positively affects the technical
performance and operating costs of the power plants. The analyzed network contains branches of the same voltage level (400 kV). If the analysis is spread to the parts of the network at lower voltage levels, additional effects can be achieved by optimizing the tap-changer operation of the interconnection power transformers, which would reduce the required power of FACTS devices and power losses in the transmission system. In order to verify the results and benefits of the proposed IPM method, GA is also used for the calculation of minimum daily energy losses. The obtained result is 5.06 p.u., and the convergence time is around 4 days. Compared to IPM, whose convergence time is around 4 s, GA is both less accurate and impractical compared to the proposed IPM. Therefore, the remaining analyses are done only by using IPM. In the second analysis, the objective function was the minimum daily operating costs. By applying the developed methodology and optimization criteria (2.12), the optimal power diagrams of the load buses are shown in Fig. 12.
Fig. 11. Optimal power production diagrams with optimal voltage profiles for the minimum daily energy losses.
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Fig. 12. Optimal load diagrams for the minimum system operating costs.
Load diagrams in this case are only slightly different from the diagrams of Fig. 10. This was to be expected because an increase in the losses leads to an increase in the demand for energy from power plants. Because of that, the criterion of minimum daily operating costs tends to reduce the gross consumption in the system. The resulting operating costs are 541 550 MU which is 0.37% less than in the system without the spatial and temporal DSM. Fig. 13 shows the optimal production of all conventional power plants in the system, along with the voltage profiles of all buses when the objective function is the minimum of daily operating costs of the conventional power plants. The prediction of production diagrams from RES is essential for a practical implementation of the proposed methodology of spatial and temporal DSM. In the previous analysis, it is assumed that the production of RES is entirely predictable. Many different models for the prediction of production of WPP and PV systems have been developed. Some of these models are presented in the literature [41e50]. Each of them makes errors, so the proposed spatial and temporal DSM would have some deviations compared to a day ahead forecasting. These deviations can be reduced through the
Fig. 14. The 16-bus distribution test system.
Fig. 13. Optimal power production diagrams and optimal voltage profiles when objective function is the minimum of daily operating costs.
Ðurisic / Renewable Energy 108 (2017) 533e547 D. Kotur, Z. Table 8 Parameters of the 16-bus system network. From bus
To bus
R [p.u.]
X [p.u.]
1 2 2 2 3 3 7 7 13 13 4 9 9 14
2 3 7 13 4 6 9 8 12 14 5 10 11 15
0.05 0.075 0.11 0.11 0.09 0.08 0.08 0.11 0.09 0.08 0.04 0.11 0.08 0.04
0.8 0.1 0.11 0.11 0.18 0.11 0.11 0.11 0.12 0.11 0.04 0.11 0.11 0.04
543
intraday correction of the calculated diagrams [46e51]. A day ahead prediction of deferrable and non-deferrable load is much more precise [52] and it is expected that it would not bring great error in the calculations. 6. The 16-bus distribution test system In this Section, the proposed methodology has been tested by the example of a 16 bus distribution network. This network represents IEEE 16 bus test system [53], in which power transformer is added in order to interconnect the entire system. The main purpose of this test is to show and quantitatively demonstrate the importance of the proposed spatial and temporal DSM applied to a distribution system. Fig. 14 shows topological scheme of the test network. The network parameters are shown in Table 8 [53]. The base values of
Fig. 15. Demand diagrams of the load buses of the 16-bus distribution test system before DSM.
Fig. 16. Production from the RES in the analyzed 16-bus distribution test system.
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Table 9 The results of analysis for the 16-bus test system.
Daily energy losses [MWh] Percentage reduction of daily energy losses
System without RES and DSM
System with RES and without DSM
System with RES and temporal DSM
System with RES and spatial and temporal DSM
11.91 /
8.67 0
8.4 3.21%
8.25 5.09%
Fig. 17. Optimal load diagrams in the analyzed 16-bus test system.
power and voltage are: SB ¼ 100 MVA and UB ¼ 11 kV. Fig. 15 shows load diagrams before DSM. In this case, it is also assumed that 30% of load energy is deferrable. Fig. 16 shows diagrams of production from the RES. It is assumed that PV power plants are connected to buses 6 and 11, and in bus 15 WPPs are connected. Once the network and system parameters have been defined, four analyses were carried out. In the first two analyzes, it was assumed that the system did not possess the ability of DSM. In the first case, no presence of RES was assumed, whereas in the second case, the RES were integrated into the system. In the third case, only temporal DSM was implemented, while in the fourth case, the optimal spatial and temporal DSM was carried out. The objective function was minimum daily energy losses. The results are shown in Table 9. Fig. 17 shows the optimal load diagrams obtained after implementation of the spatial and temporal DSM. The results obtained in the case of the distribution network confirm all the conclusions illustrated in the case of the transmission network. In addition, at the connection point of the distribution network to the transmission system (bus 0), variation of the diagram of active power taken from the external network is considerably reduced, from 47.12% to 11.62%, Fig. 18. In addition, the spatial and temporal DSM can improve the quality of the voltage, which is also presented in Fig. 18 where the voltage diagram of bus 11 (the weakest in the analyzed network) is shown. Implementation of the spatial and temporal DSM significantly reduces voltage variations in all consumer buses. Further improvement could be obtained by including optimization in the tap changer operation of power transformers.
Fig. 18. External grid active power diagrams and voltage diagrams in the most critical bus.
7. Conclusions The paper introduces the concept of spatial and temporal DSM. The spatial and temporal DSM represents a new methodology that allows the system operator to consider the intermittent nature of renewable energy sources, as well as their dispersed allocation. The developed mathematical model, based on Interior Point Method, allows solving highly complex optimization problems of the spatial and temporal DSM in real networks having a large number of buses and the equality and inequality constraints. From the three test examples performed, the following positive effects of the spatial and temporal DSM are identified:
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1. It is shown that both spatial and temporal DSM improves the performance of the entire power systems, in terms of both energy losses and operating costs. Also, it can be concluded that in the case when production from the RES is large, load should be shifted in the buses that are electrically close to the RES. 2. The spatial and temporal DSM reduces the peak loads of some elements in the power system (power transformers, power lines, etc). This has many benefits: the service lifetime of these elements is increased, increased transmission capacity of power lines, etc. 3. The spatial and temporal DSM reduces the voltage variations of load buses. Beside power quality, these effects reduce tap changing in the regulating power transformers which extends their service lifetime. 4. Positive effects of the temporal and spatial DSM are higher when the installed capacity of the RES grows. 5. When the objective function is minimum daily operating costs, the spatial component of DSM has a greater effect on a power system having longer transmission lines and loaded networks. In the case of relatively low power losses, the significance of the space coordinate is reduced. 6. The positive effects of DSM are the greatest when the percentage value of deferrable load changes from 0% to 10%. After that, further increase of deferrable load would improve the positive effects to a lesser extent. The analyses carried out in this paper prove that the improvements can be obtained by considering the spatial component of DSM. In this way, a significant step is made in the research area having the ultimate purpose of developing a methodology that will enable successful integration of renewable energy in a power system. The proposed spatial and temporal DSM algorithm can practically be implemented through a coordinated management of the TSO and DSO. By using the proposed methodology, the TSO would define the optimal total load diagram of each bus in the transmission system, according to which DSO would use smart grid and techniques of direct or/and indirect DSM in order to spatially distribute the load in all distribution system buses. If DSM is implemented by using real time pricing, the use of the proposed methodology implies that the electricity price in the same hour would be different in different buses of the transmission and distribution systems.
545
Lm ¼ f ðxÞ mðlnðsl Þ þ lnðsu ÞÞ lT gðxÞ pTl ðhðxÞ hl sl Þ pTu ð hðxÞ þ hu su Þ: (A-5) In equation (A-5), l, pTl , pTu respectively, represent the Lagrange’s coefficients which take into account the equality constraints in equations (A-2), (A-3), (A-4). The criteria of reaching the optimal value are obtained by solving the following system of equations:
Vsl Lm ¼ m$S1 l $e þ pl ¼ 0;
(A-6)
Vsu Lm ¼ m$S1 u $e þ pu ¼ 0;
(A-7)
Vpl Lm ¼ hðxÞ þ hl þ sl ¼ 0;
(A-8)
Vpu Lm ¼ hðxÞ hu þ su ¼ 0;
(A-9)
Vl Lm ¼ gðxÞ ¼ 0;
(A-10)
Vx Lm ¼ Vf ðxÞ VgðxÞ$lT VhðxÞ$ pTl pTu :
(A-11)
For obtaining the numerical solution of the system (A-6) - (A11), Newton’s iterative procedure is applied by using the following matrix equation (A-12):
2 6 6 6 6 6 6 6 4
m$S2 l
0 I m$S2 0 0 u I 0 0 0 I 0 0 0 0 0 0 VhðxÞT 2 3 Vsl Lm 6 Vsu Lm 7 6 7 6 Vpl Lm 7 7 ¼ 6 6 Vpu Lm 7: 6 7 4 Vl Lm 5 Vx L m
0 I 0 0 0 VhðxÞT
0 0 0 0 0 VgðxÞT
3 2 3 0 Dsl 7 6 7 6 Dsu 7 0 7 7 Dpl 7 hðxÞ 7 6 7 7$6 Dpu 7 hðxÞ 7 6 7 7 6 VgðxÞ 5 4 Dl 5 Dx V2 L m x
(A-12) In equation (A-12),
V2x Lm
is calculated as:
V2x Lm ¼ V2x f ðxÞ V2x gðxÞ$lT V2x hðxÞ$ pTl pTu
Acknowledgement This research was partially supported by the Transmission System Operator of Serbia, project Smarter grid, and European Commission, project EmBuild (Horizon2020).
Since the calculation time of each iteration drastically increases with increasing size of the matrix, the problem can be simplified by solving the system (A-14):
Appendix A Let equations (A-1)e(A-4), define the optimization problem.
minðf ðxÞ m$ðlnðsl Þ þ lnðsu ÞÞÞ;
(A-1)
gðxÞ ¼ 0;
(A-2)
hðxÞ hl sl ¼ 0;
(A-3)
hðxÞ þ hu su ¼ 0:
(A-4)
0 VgðxÞT
VgðxÞ V L Dl ¼ l m ; $ x Dx Hd
(A-14)
Where the matrices Hd and x are calculated as follows:
2 $Vx hðxÞ; Hd ¼ V2x Lm þ m$Vx hðxÞT $ S2 l þ Su h
In order to eliminate the equality constraints, the Method of Lagrange Multipliers is applied. Then the optimization problem is reduced to finding minimum of the function defined by (A-5):
(A-13)
(A-15)
2 x ¼ Vx Lm þ Vx hðxÞT $ m$ S2 u $Vpu $Lm Sl $Vpu $Lm þ Vsl Lm
i Vsu Lm :
(A-16) After solving the system (A-14), a change of other variables may be obtained by equations (A-17) and (A-18):
Ðurisic / Renewable Energy 108 (2017) 533e547 D. Kotur, Z.
546
Dsl ¼ VhðxÞ$Dx Vpl Lm Dsu ¼ VhðxÞ$Dx Vpu Lm ;
(A-17)
2 Dpl ¼ m$S2 l $Dsl Vsl Lm Dpu ¼ m$Su $Dsu Vsu Lm :
(A-18)
Values of the variables in the next iteration are calculated according to equations (A-19) - (A-20):
skþ1 ¼ skl þ akp $Dskl ; skþ1 ¼ sku þ akp $Dsku ; xkþ1 ¼ xk þ akp $Dxk ; u l (A-19) kþ1 k pkþ1 ¼ pkl þ akd $pkl ; pkþ1 ¼ pku þ akd $pku ; l ¼ l þ akd $Dl: u l
(A-20) The coefficients akp and akd are calculated as:
(
akp
¼ min 1; g min
Dski 30
ski
Dski
(
) ;
akd
¼ min 1; g min
Dpki 30
pki
)
Dpki
(A-21) Coefficient m is calculated in each iteration according to formula (A-22)
mkþ1 ¼
$ sTl $pl þ sTu $pu : 2$n þ m
s
(A-22)
The condition of convergence is defined by equation (A-23):
Dxk k Dl ε:
(A-23)
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