Abstract
With the addition of new elements such as distributed generation, demand response, and price-sensitive loads, as well as grid smartness, the operation and management of distribution networks have faced additional complexities. Hence, the electricity price has a vital role for all market participants and strongly affects the demand, generation, and subsequently, the load flow analysis in the network. In this situation, there is a need for a load flow algorithm in which price is integrated into load flow equations. Therefore, this paper seeks to present a suitable algorithm called price-based load flow which considers the electricity price as a state variable similar to technical variables. In this situation, the price is added to the load flow equations as a new state variable and the effect of location on the price is considered. Therefore, it is possible to use the radial structure of the distribution network and communication between adjacent buses in order to provide an effective and appropriate method to find the price simultaneously with technical variables. The numerical study on the IEEE 33-bus distribution network shows the performance of the proposed method. Unlike the centralized market structure, the method does not need all network information to calculate distribution locational marginal price using optimal power flow. In this method, each bus can perform its computation only by information exchange with adjacent buses.
1. Introduction
The power industry is shifting from a centralized system towards a more decentralized power system, relying to a larger extent on small-scale generation from renewable energy sources (RES) and combined heat and power (CHP) units. It allows more active participation of consumers in the generation or smarter demand response management. The power system is increasingly affected by distributed energy resources (DER). DER involves distributed generation (DG), demand response (DR), and energy storage. DG is made of relatively small-scale generation capacities connected to the distribution network (medium and low voltage). DR is the second key component of a decentralized power system. Demand response does not necessarily save energy, but it often leads to shifting loads around in time. This may eliminate the need to reduce excess energy supply during periods of low demand or high supply. The management of small end-users must be automated which requires online communications. In this regard, smart metering is an important technology for DR. Consequently, it should be expected that distribution networks will undergo the transition to smart grids. Smart grids are active and dynamic electricity networks that have active end-users and function bilaterally as opposed to the traditional passive top-down (generator to consumer) network. The emergence of these networks will bring significant changes in the operation of the power distribution system [1].
In [2] aim to design a decentralized market to increase the distribution system flexibility. It investigates how decentralized flexibility markets can incentivize consumers with small-scale energy resources to efficiently manage the distribution network demand. The combination of consumer-level energy technologies and communications allows passive consumers to become prosumers, consumers who proactively manage their energy consumption, generation, and storage of energy. In a decentralized market design, obtaining flexibility from competing aggregators allows a distribution system operator (DSO) to handle local demand constraints.
Network variables derived from load flow analysis include the power and voltage at buses, the power and current flowing in feeders, and losses. If the local generation and demand at buses are not price-dependent, the voltage at buses and the current in feeders can be easily computed by the previous methods of load flow analysis.
In the theory of market, the market price is computed with the objective function of maximizing social welfare, i.e., the total profits of buyers and sellers, to find the market equilibrium point (MEP). Hence, the intersection point of supply and demand curves determines the generation and demand in the market equilibrium. As a result, the effect of network topology is not considered in the calculation of MEP.
With the advent of distributed generation and price-sensitive load (PSL), each network component generates and consumes in proportion to its price as well as technical conditions. The price first comes from the wholesale power market to the distribution network, and then the local market responds to it by utilizing DG and DR. In this case, as the network becomes more complex, distribution dispatching will have an economic role in addition to its previous technical role. In this case, the electricity price plays a critical role in decision-making.
The typical network in Figure 1 illustrates the basic idea. Obviously, the local generation and local demand at each bus depend on the price at the bus. In other words, when the price at buses decreases, the demand at buses 2 and 3 increases and the generation at bus 2 decreases. Therefore, the power and current flowing through the connecting feeder of buses 2 and 3 are increased to supply the demand at bus 3. In addition, with decreasing generation at bus 2 and increasing demand at bus 2, the injected power into the power supply point (PSP) goes higher. It leads to more power and current flow through the connecting feeder of buses 1 and 2. Consequently, as the current increases and the voltage drop in feeders increases, the voltage at buses 2 and 3 decreases. As a result, all network variables are affected by price changes, which indicate the importance of price in the load flow analysis.
Considering the price-dependent generation and demand at each bus, the net active power at the bus is an unknown variable. So, in each iteration, it is necessary that the mentioned quantities are computed according to the equations (1) and (2). is the price at bus , and and are the functions relating to the price at bus with generation and demand at bus , respectively. and are the active power generation and consumption at bus , respectively. is the net active power at bus .
Therefore, it is first necessary to calculate the price at each bus to obtain the generation and demand. On the other hand, the price at buses depends on the generation and demand in the network. So, it is observed that the power and the price are interrelated. This interrelation makes the price-based load flow more complex than the conventional load flow. Hence, the question is about what approach is appropriate to find bus prices as well as other technical variables.
As DG becomes more deployed, distribution networks take on many of the same characteristics as transmission. As a result, the nodal pricing in the transmission network can be modified to be applied in the distribution network. In [3], the nodal pricing in the distribution network is proposed to determine the right price to locate the DG resources for reducing the network losses. In this method, the optimization problem of purchasing electricity from DG and PSP was formulated, considering the power balance constraint.
To consider the effects of the transmission network, the concept of locational marginal price (LMP) has been introduced. It is one of the most effective methods of price signals that varies spatially and temporally. The LMP is computed at network locations taking into account the constraints and losses. LMPs can reduce system losses and increase efficiency. In addition, LMPs reduce the network’s need for an upgrade by promoting DERs in locations with constraints. Therefore, LMPs indicate locations where investment can be cost-effective. The LMP is defined as the incremental cost to supply the next unit of demand at a specific location (bus) taking into account the supply bids, demand offers, and network characteristics including losses and limits. The LMP generally has the following three components: the energy component, the congestion component, and the loss component. The energy component is the load weighted average of the system bus prices, which is the same at all locations. The congestion component shows the marginal cost of congestion due to feeder capacities and reserve requirements. The loss component represents the marginal cost of losses at a specific spot. In this way, a range of prices is shown that reflects the cost of providing electricity at each network location [4].
Corresponding to LMP in the transmission network, the distribution locational marginal price (DLMP) is similarly used in the distribution network. Figure 2 schematically indicates the mentioned description.
In [5] use distribution locational marginal price at each bus instead of constant MEP. The DLMP is computed by the optimal power flow (OPF), which includes the lossless DCOPF model and the ACOPF model. It is usually computed using the constrained optimization method or in more detail, the Lagrange multiplier corresponding to the power balance equation at each bus in OPF. This is performed by the optimal power flow solver of the MATPOWER package of MATLAB software.
In [6], the study and formulation of DLMP are expressed to maximize the social welfare function considering the demand response. Demand response is defined as the changes in electric usage by the end-use customers from their normal consumption patterns in response to electricity price changes over time. In other words, DR is an incentive payment designed to reduce electricity consumption during peak hours.
In [7] propose the DLMP method for electric distribution networks to alleviate the congestion influenced by electric vehicles (EVs). It states that grid congestion depends on several factors such as local grid rating and topology, penetration and distribution of EVs, and charging management procedures. In [8], the necessity of the electricity market at the distribution level has been studied to utilize microgrid (MG) potential as an active player, considering the technical constraints such as voltage and congestion. In [9] present the formulation of optimal framework for a distribution day-ahead market, and the market participants such as MGs, distributed storage (DS), DGs, and load aggregators (LAs) are modeled. In [10], due to the high resistance to inductance ratio and unbalanced characteristics of distribution networks, a linear approximated DLMP is presented. In [11] the aim is to develop a market mechanism that motivates price-making agents to trade power at DLMPs. In this design, the market rules allow agents to strategically propose price and power. These rules make a noncooperative game between the DSO and DGs. It is proved that DLMP can be computed at a generalized Nash equilibrium (GNE) of the game by developing a decentralized algorithm.
In the conventional method of calculating DLMP, the price is considered an external element in the power flow computation. In this situation, the price is often considered a variable in the objective function of the optimal power flow. Thus, it is needed to solve the optimization problem to clear the market and find DLMP. Therefore, DSO requires all values of network variables to calculate DLMPs in the central markets. Hence, a costly infrastructure is needed to analyze a large amount of data.
Furthermore in OPF, the conventional DC or AC power flow model is used, in which the loads are specified, and the generation is computed to maximize social welfare. Conventional power flow algorithms such as Gauss–Seidel and Newton–Raphson are often used for transmission networks. Therefore, their use for power distribution networks due to their radial structure and high ratio of feeder resistance to reactance may result in inaccurate results and even divergence. Thus, it seems necessary to use a power flow method with more appropriate results. Therefore, using the conventional AC or DC power flow to calculate DLMP may lead to inaccurate results in the decentralized distribution systems with radial structures.
The aim of this research is to look at the bus price in a new way in the context of the radial distribution network equations. Hence, a distributed methodology is introduced to compute the DLMPs for each bus. The method can be implemented properly for a decentralized environment based on local information. Therefore, the proposed method can be used in smart grids, even when the information is limited to adjacent buses. In other words, unlike the centralized market structure, the method does not need all network information to calculate DLMP using the OPF problem. Hence, the bus price is introduced as a new state variable in a load flow analysis. Therefore, this paper aims to use the mentioned features and express a new model of load flow equations that is useful in decentralized distribution networks.
The rest of this paper is organized as follows: in Section 2, the methodology will be stated, then in Section 3, the proposed method will be studied numerically, and finally, in Section 4, the conclusion will be expressed.
2. Methodology
The basic idea of this paper is to compute the price similar to the voltage and current, based on the local information. Interpreting price as a new state variable introduces a new concept into the context of load flow analysis in distribution networks. For this purpose, the flowchart of the conventional load flow algorithm is modified by adding the shaded box to calculate the price at each bus, as shown in Figure 3.
By developing this new price-based model, there is more insight to analyze the interaction between the price and conventional state variables. Figure 4 typically shows the relation between the price at different buses and variables such as voltage, power, and losses. Analyzing the locational price of electricity in different conditions of the distribution network can effectively improve the knowledge of the network and provide information for better planning and operation in the network.
In the price-based load flow method, as mentioned earlier, each bus only needs the information of its adjacent bus, and the computation is performed locally in a decentralized environment. Therefore, the formulation of the method is expressed for each bus according to other information and signals from adjacent bus. First, it is necessary to number the buses in the radial network. Bus numbering is performed so that for each feeder connecting two buses, the end bus number is greater than the beginning bus number. Numbering starts from 1, and the last number is equal to the number of buses, i.e., . The relation between two adjacent buses is shown in Figure 5.
In the first step, the network data are given as input to the price-based load flow computation function. According to Figure 5, the initial voltage as a complex variable has a magnitude equal to 1 per unit and an angle equal to 0. In addition, similar to voltage, the same price is considered for all buses as an initial guess. The computational procedure of the price-based load flow algorithm consists of a backward sweep and a forward sweep, which are described as follows.
2.1. Backward Sweep Computation
In the backward sweep, by moving from the bus with a bigger number to the bus with a smaller number, the generation and demand at buses and consequently the power and current in feeders are computed. The power of distributed generation and price-sensitive load at each bus are determined based on the locational price. It is evident that with increasing prices, the tendency of PSLs to buy energy will decrease and the tendency of DGs to sell energy will increase. For example, their linear relations are represented by equations (3) and (4), which are popular in power systems due to the quadratic form of the cost function. is the bus with DG, is the bus with PSL, is the bus price, is the power generation, and is the power consumption.
are the fixed numbers to form a linear relation (submitted by generators/consumers).
When the price is known, the power generation at buses with DG as well as the power consumption at buses with PSL is obtained from equations (3) and (4). The generation and consumption values of the buses must be within a feasible range.
For adjacent buses and , bus can be considered the beginning bus. After calculating the generation and consumption at buses, the power and current in feeders are computed using equations (5) and (6). In these equations, the initial losses are considered to be zero. is the power flowing through the connecting feeder of buses and , is the net power at bus , represents the losses in the connecting feeder of two buses and , represents all buses connected to bus that have numbers greater than bus , is the power flowing through the connecting feeder of two buses and , is the current flowing through the connecting feeder of two buses and , and is the voltage at bus .
2.2. Forward Sweep Computation
In the forward sweep, by moving from the bus with a smaller number to the bus with a bigger number, the voltage at buses is obtained according to equation (7). When the voltage at buses and the current in feeders are known, losses are computed according to equation (8) and the bus prices are updated. As mentioned earlier, these equations are written for the adjacent buses as follows:where is the voltage at bus and is the impedance of the feeder connecting two buses and .
As the price goes higher in a network, the profit of generators (energy sellers) increases and the profit of consumers (energy buyers) decreases. It is shown according to equations (9)–(11). These equations present a problem aimed at maximizing social welfare. Since congestion rarely occurs in the distribution network, it is not considered in the following equations: and are the reactive power generation and consumption at bus , respectively, and and are the total active and reactive losses of the network, respectively.
Considering equation (2) for active power and similarly for reactive power, the problem can be shown according to equations (12)–(14).
is the net reactive power at bus , is the Lagrange multiplier of active power balance equation, and is the Lagrange multiplier of reactive power balance equation.
Therefore, according to equations (15) and (16), the problem is solved using KKT conditions as follows: represents Lagrangian.
As a result, the price at bus is shown according to equation (17) as follows:
Since there are no losses or congestion at bus 1 (PSP), the price at bus 1 is equal to the energy component of DLMP, i.e., the Lagrange multiplier of the power balance equation. In addition, according to the radial structure of the distribution network, bus can be considered the first bus of an imaginary distribution network. Therefore, the relation of the price of active power between two adjacent buses can be shown according to equation (18). By performing similar calculations, the relation of the price of reactive power between two adjacent buses is obtained according to equation (19) as follows: is the price at bus , is the price at bus , is the price of reactive power at bus , is the price of reactive power at bus , is the total active losses after bus , represents total reactive losses after bus , and is the net active power at bus .
Due to the losses equations ( and ) and since the derivatives of and with respect to variables and are important, the only current term after that depends on and is the current in the feeder connecting two buses and . So, other current terms are independent of and . Therefore, the total active and reactive losses after bus can be shown according to equations (20) and (21) as follows: and are the resistance and reactance of the feeder connecting buses and , respectively, represents the total active losses after bus except for active losses between buses and , and is the total reactive losses after bus except for reactive losses between buses and .
Equations (24)–(27) result from the combination of equations (20)–(23).
Therefore, the price update is shown according to equations (28) and (29) as follows:
To compute the price of active and reactive power at bus , it is necessary to obtain and , which are presented as follows. First, is computed according to equation (30).
is the current injected into bus , and is the current flowing through the feeder connecting two buses and .
Since is related to after bus and independent of and , only is written in terms of and , according to equation (31) as follows: is the net reactive power at bus .
Therefore, and can be obtained by the by the combination of equations (22), (23), (30), and (31), which is shown according to the following equations: is the real part of voltage at bus , is the real part of current flowing through the feeder connecting buses and , is the imaginary part of voltage at bus , and is the imaginary part of current flowing through the feeder connecting buses and .
The price at the first bus can be considered according to the transmission LMP or according to the total generation. In addition, the total power injected into the distribution network is determined according to the power balance constraint.
To check the convergence condition, the differences between voltages and prices compared to the previous iteration are computed according to inequality (34). If this inequality is not satisfied, the next iteration must be run. In case of satisfaction at all buses, the algorithm converges and the price-based load flow will end. is the magnitude of difference between the new and the previous voltage at bus , is the new voltage at bus , is the previous voltage at bus , is the accuracy limit of the voltage value, is the magnitude of difference between the new price and the previous price at bus , is the new price at bus , is the previous price at bus , and is the accuracy limit of the price value.
The information exchange between two adjacent buses as well as backward/forward computation is shown in Figure 6. In addition, Figure 7 shows the flowchart of the proposed algorithm. As shown in Figures 6 and 7, the backward or forward computation is performed at each bus according to the backward/forward mode received from the adjacent bus. The backward sweep is performed from bus to bus 1. In the backward sweep, the power and current in feeder are obtained from the information of bus . Next, the sweep direction is changed at bus 1, and the forward sweep is performed from bus 1 to bus . In the forward sweep, the voltage and price at bus are computed using the information of bus . Then, if the convergence condition is not satisfied, the backward sweep starts from bus in the next iteration. When the algorithm converges, the results of all buses will be displayed. In this situation, the computation process is performed without requiring the values of variables from other buses. In other words, the computation is performed for each bus separately, which indicates the decentralization of the proposed method.
3. Case Study
In this section, the impact of PSLs and DGs on the voltage and price at buses is analyzed using price-based load flow. Hence, the proposed algorithm has been tested on the IEEE 33-bus distribution network, as shown in Figure 8. The network data are presented in Table 1.
First, to verify the accuracy of the proposed load flow algorithm, no PSL and DG are considered and the results are compared with the conventional ACOPF method. Table 2 shows that the voltage results of the proposed method are completely the same as those of the conventional method. Also, the values of prices, i.e., DLMPs, are slightly different from those of the conventional method, as shown in Table 3.
In the next step, some buses are considered with PSL or DG, as shown in Tables 4 and 5. In this case, the transmission LMP at the first bus is assumed to be equal to 10 cents/kWh. Moreover, the price of reactive power at the first bus is assumed to be 0. As a result, the generation of DGs and the consumption of PSLs are presented in the last column of Tables 4 and 5. In addition, the voltage at buses is shown in Table 6. It can be seen that the possibility of voltage improvement is increased by the incresed presence of PSLs and DGs in the network. For example, in the network and with the mentioned conditions, price-dependent generation and consumption help improve the voltage by an average of 4.8%.
The price at buses in different cases is also shown in Table 7. Comparing the results shows that the increased presence of PSLs and DGs in the network increases the possibility of price improvement. For example, in the network and with the mentioned conditions, price-dependent generation and consumption help improve the price by an average of 8.2%.
As seen in Tables 4 and 5, bus 30 has DG and PSL. Hence, Figure 9 uses supply and demand bid functions and shows how the generation and consumption values at this bus are obtained.
The ascending graph in Figure 9 is related to the power generation at bus 30, and the descending graph is related to the power consumption at bus 30. The mentioned graphs are obtained from equations (4) and (5). Moreover, the horizontal graph shows the price at bus 30. By intersecting the graphs, the generation and consumption values at each bus are obtained. The trend of convergence of the algorithm for bus price is also shown in Figure 10.
Figure 11 shows bus voltages/prices in cases with/without DGs and PSLs. As seen, the price-dependent generation and consumption significantly improve bus voltages/prices.
(a)
(b)
As a result, the efficiency of the proposed price-based load flow method in the presence of DGs and PSLs is shown. In this way, the effect of price-dependent generation and consumption on the voltage and price at buses can also be achieved.
4. Conclusion
Due to the changes in the distribution system and the emergence of distributed generation and price-sensitive loads, the electricity price has a vital role in the network. Hence, this paper introduces a new method for the load flow analysis of a distribution network that can be properly implemented in a decentralized environment based on local information. In this distributed methodology, the computation is performed for each bus using the information of the adjacent bus and without using all network variables to clear the market by DSO.
In the proposed method, each bus price, as a state variable, and the price-dependent generation and consumption are entered directly into the context of the load flow equations. The values are obtained without the need to calculate the market equilibrium point and without the need to solve the local marginal price optimization problem.
In this paper, the procedure of calculating bus prices in each iteration of load flow analysis is explained and the proposed algorithm for price-based load flow is presented. A numerical study is performed on the IEEE 33-bus distribution network to demonstrate the validity of the proposed method. It is observed that the proposed method yields accurate results. This method can consider price-dependent generation and consumption without encountering the limitations of the previous load flow methods, such as the need for all network information, the need to solve the central optimization problem, and the possibility of nonconvergence in radial networks. It is also observed that the voltage and price at buses are improved by the increased presence of PSLs and DGs in the network.
This paper presents the price-based load flow in decentralized networks and demonstrates the efficiency of the proposed method. This method can be used in the analysis of various network parameters and the study of network conditions in operation and planning applications, especially the effect of PSLs and DGs on the operation performance of the network.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.