Appraising the Optimal Power Flow and Generation Capacity in Existing Power Grid Topology with Increase in Energy Demand

Abstract

Several socioeconomic factors such as industrialization, population growth, evolution of modern technologies, urbanization and other social activities do heavily influence the increase in energy demand. A thorough understanding of the effects of energy demand to power grid is highly essential for effective planning and operation of a power system network in terms of the available generation and transmission line capacities. This paper presents an optimal power flow (OPF) with the aim to determine the exact nodes through which the network capacities can be increased. The problem is formulated as a Direct Current (DC) OPF model, which is a linearized version of an Alternating Current (AC) OPF model. The DC-OPF model was solved as a single period OPF problem. The model was tested in several case studies using the topology of the IEEE test systems, and the computation speeds of the different cases were compared. The results suggested dual variables of the problem’s constraints as an extra tool for the network designer to see where to increase the network capacities.

1. Introduction

Power system networks are vast and a highly sophisticated system with numerous branches interconnected together in web-like form, which in most cases are being supervised by one network operator [1,2]. Such numerous web-like branches of the system, which can also be viewed as power grid topology or energy internet, are commonly called power transmission lines in which the energy in-flow and out-flow must be in accordance with Kirchhoff’s current and voltage laws. However, the determination of the best operating levels of the network in terms of efficient energy delivery from generation nodes to demand nodes calls for an optimal power flow (OPF) approach.

The OPF in the power system was first introduced in 1962 by Carpentier [3,4]. It has a major aim of minimizing the total operating cost of energy generation in the power system subject to the system’s resource constraints. Such constraints include: the power flow limits in each branch of the network, generator capacity limits, voltage magnitudes and their angles, and, in most cases, can also consider the active transmission losses in either part of the network or the whole network system [5].

The OPF model majorly depends on the static optimization method for minimization of a scalar optimization function. The OPF for minimization purpose was introduced in 1968 by Dommel and Tinney [6], where the first-order gradient algorithm is subject to inequality and also equality constraints [7]. Other conventional methods applied to solve the OPF problem include the Newton method network flow programming, quadratic programming, the interior point algorithm, linear programming, nonlinear programming and the emerging nature inspired algorithms [8].

Furthermore, the OPF tends to optimize the static operating condition of a power generation–transmission system with several benefits such as: ensuring static security of quality of service by imposing limits on generation and transmission system’s operation; the optimization of reactive-power/voltage scheduling; improvements in the economy of operation through the full utilization of the system’s feasible operating range and the accurate coordination of transmission losses in the scheduling process [9]. For convergence purposes, the generator operating cost functions are usually approximated by either quadratic or linear functions. In most cases, the operating cost is a function of both the generation active and reactive power of the network [10].

Moreover, it is highly imperative and ideal to carry out OPF in either an existing transmission system or newly planned transmission system in order to obtain the network information in terms of the generation and line capacities, voltage levels, and their phases. The OPF does show exactly where there are bottlenecks in the system and hence the network reinforcements, addition of new lines and exploration of new corridors, which are known as Transmission Network Expansion Planning (TNEP), can be conducted with ease if the OPF results are readily available.

The OPF can be formulated as DC or AC OPF problems. The AC OPF formulation uses the actual AC power flow equations with the incorporation of the actual voltage magnitudes, active and reactive power management. This generates nonlinear terms in the expression of the power equations, which makes it difficult to solve by using the classical optimization techniques [11], and the accuracy of the trending nature inspired algorithms in handling the nonlinear terms has not been properly verified in a real situation.

A novel linear-programming approximation of AC power flow that handles the reactive power and voltage magnitudes was proposed in [12]. The model was developed in terms of a polyhedral relaxation of the cosine terms in the AC equations and Taylor series expansion was used in handling the rest of the nonlinear terms. A convex formulation for the OPF in radial power grids, for which the AC OPF equations, including the transverse parameters, are considered can be found in [13].

However, the establishment of a robust solution for AC OPF remains a challenge, despite the numerous nonlinear algorithms developed in literature.

The DC OPF model formulation uses the linearized version of AC power flow, without the consideration of the aspects of voltage support and reactive power management [14]. This brings convexity, which allows for faster computation time [15].

The linear OPF models, for which the DC model is the major representative, are widely adopted in the electricity market as a simplified version of the AC OPF model [16]. Almost every study of the OPF model for market related purposes adopts the DC OPF model [14].

The linear approximation of AC OPF that considers the accuracy of the transmission losses and reactive load flows was presented in [11]. The analysis of the assumptions of the DC OPF model and the attempts to quantify the assumptions’ degree of accuracy are the work done in [14]. Meanwhile, the technique for solving the DC OPF with the minimum error is presented in [16]. The work of [17] handled the formulation of a linear Mega-Watts-only power flow algorithm with losses for multi-terminal Voltage Source Converter (VSC) AC/DC systems and consistency in estimation of losses in the converters was guaranteed.

An OPF with an integrated approach of geographically indexed production, demand and grid modelling for large-area power systems was proposed in [18] to account for variance in the loading of transmission lines with the scenario for the development of wind power in Switzerland. The result showed that, in prospective regions of wind power development, Switzerland’s grid is capable of providing congestion-free dispatch in its current state.

Energy management problem in the power network system, with inter- disciplinary techniques, was investigated in [19]. The energy regulation issue was considered based on the operational principles of energy internet (EI). The problem was formulated as a constrained optimal control problem with multiple targets. A model-free deep reinforcement learning (DRL) approach was applied with real-world power data with no explicit mathematical model for renewable power generation devices and loads. The results were compared with that of the conventional OPF method, and they showed better performance than the OPF solution.

However, the results of the model-free DRL approach still need further comparison analysis with a model-based DRL approach, to further consolidate the better performance claim. This is due to the fact that model-free DRL uses trial and error approaches without taking into consideration the actual system’s model [19,20,21]. However, the peculiar feature of model-based DRL is that its objectives are finite-horizon based objectives [22,23,24,25], and it often yields improved sample efficiency [26].

A modified Successive Linear Programming (SLP) algorithm was applied in [27] to solve a relaxed AC OPF problem. The algorithm was tested in several large networks specified by the U.S. Department of Energy (DOE) ranging from 500-bus to 30,000-bus systems. The results show tractability of the algorithm, and up to 80% of the test cases were solved faster than an Interior Point Method (IPM) with less number of iterations.

A novel method to approximate the AC OPF into tractable linear/quadratic programming (LP/QP) based OPF problems to be used for power system planning and operation was proposed in [28]. The result showed that the methods drastically reduced the computational complexity compared to the nonlinear AC OPF, thereby making them a good choice for power system planning purposes.

The previous work of the author, Hamam [29], affirms that a well planned power supply system for any given planning horizon should have the total installed capacity exceeding the peak demand by a certain amount within the specified horizon. Such extra capacity is known as reserve.

In this paper, a novel DC OPF model that optimizes how much energy is produced, realizes how many economic benefits are saved in the process and predicts where to increase the capacity in an existing grid topology is proposed.

The major contribution of this paper is in the area of

• the utilization of the dual variables of the constraints, which is known as the per unit cost of constraint relaxation to form an extra tool for the network designer to see where to increase the network capacities.

Such approach has not been covered in the literature for OPF analysis to the best of our knowledge. The chosen approach also has a primary concern of computational speed as required in real-time economic dispatch.

Moreover, the paper presents the matrix expansions of the developed model in terms of each decision variable’s relationship with other parameters of the network for proper representation of the model in any suitable optimization software.

The rest of the sections of the paper are organized as follows: Section 2 defines the OPF problem; the AC and DC formulations of the OPF problems are described in Section 2.1 and Section 2.2, respectively. The matrix expansion of the DC OPF Model is expressed in Section 2.3. The Duality Principle in Linear Programming is highlighted in Section 3, while Section 4 contains the results and discussions of the three different test cases, followed by conclusions, acknowledgements, appendices and references.

2. Optimal Power Flow (OPF) Problem

The determination of optimal power flow (OPF) is critical for proper power system planning and operation, owing to the fact that it is utilised in finding the optimal operating state of power network systems under certain constraints such as active and reactive power limits, loss of load limits and as well as thermal limits of the transmission network with other constraints posed by integration of renewable energy sources such as the wind and solar energy [7].

Moreover, it is highly imperative and ideal to carry out OPF in either an existing transmission system or a newly planned transmission system in order to obtain the network information in terms of the generation and line capacities.

However, the determination of the exact network nodes where the capacity of the network can be increased to satisfy any further increment in demand with minimum cost of energy production is eminent.

2.1. AC Optimal Power Flow Problem Formulation

The formulation of AC power flow problem adopts the exact power flow equations. The description of the problem takes into account the voltage magnitude and phase information at each bus for a particular load scenario with regard to the voltage level of each generator, active power output and real and reactive power at all available loads.

With such information, the active and reactive AC power flows in each branch of the network can be determined by finding the feasible solution to a set of nonlinear nodal balance equations as shown in (1) and (2), respectively:

$$\sum _{g\in {\Omega }_g}P{G}_g-V_i\sum _{i,j\in \Omega }{V}_j(G_{k}cos{\delta }_k+B_{k}sin{\delta }_k)=\sum _{d\in {\Omega }_d}P{D}_d$$

(1)

$$\sum _{g\in {\Omega }_g}Q{G}_g-V_i\sum _{i,j\in \Omega }{V}_j(G_{k}sin{\delta }_k-B_{k}cos{\delta }_k)=\sum _{d\in {\Omega }_d}Q{D}_d$$

(2)

where $P{G}_g$ and $Q{G}_g$ are active and reactive power generated at generation bus g. $P{D}_d$ and $Q{D}_d$ are the real and reactive demands at load bus d. $V_i$ and $V_j$ are the voltages at buses i and j. $G_k$, $B_k$ and ${\delta }_k$ are the line conductance, suspectance and phase, respectively.

Based on the assumption of a quadratic cost curve of the generators, the AC OPF model can be represented as a quadratic programming model, which is easier to solve due to its convex nature [30]. The model’s objective function tends to minimize the total energy cost without the consideration of the unit commitment problem:

$$\begin{gathered} min\sum _{g\in {\Omega }_g}(\gamma P{G}_g^2+\beta P{G}_g+\alpha ) \\ \mathrm{Subject}\mathrm{to}: \end{gathered}$$

(3)

$$\sum _{g\in {\Omega }_g}P{G}_g+\sum _{k\in {\Omega }_k}{P}_k^{-}-\sum _{k\in {\Omega }_k}{P}_k^{+}=\sum _{d\in {\Omega }_d}P{D}_d$$

(4)

$$\sum _{g\in {\Omega }_g}Q{G}_g+\sum _{k\in {\Omega }_k}{Q}_k^{-}-\sum _{k\in {\Omega }_k}{Q}_k^{+}=\sum _{d\in {\Omega }_d}Q{D}_d$$

(5)

$$\sum _{k\in {\Omega }_k}{P}_k^{-}+V_i\sum _{i,j\in \Omega }{V}_j(G_{k}cos{\delta }_k+B_{k}sin{\delta }_k)=0$$

(6)

$$\sum _{k\in {\Omega }_k}{P}_k^{+}-V_i\sum _{i,j\in \Omega }{V}_j(G_{k}cos{\delta }_k+B_{k}sin{\delta }_k)=0$$

(7)

$$\sum _{k\in {\Omega }_k}{Q}_k^{-}-V_i\sum _{i,j\in \Omega }{V}_j(G_{k}sin{\delta }_k-B_{k}cos{\delta }_k)=0$$

(8)

$$\sum _{k\in {\Omega }_k}{Q}_k^{+}+V_i\sum _{i,j\in \Omega }{V}_j(G_{k}sin{\delta }_k-B_{k}cos{\delta }_k)=0$$

(9)

$$∥P_k^2+Q_k^2∥\le S_k^{max}$$

(10)

$$P{G}_g^{min}\le P{G}_g\le P{G}_g^{max}$$

(11)

$$Q{G}_g^{min}\le Q{G}_g\le Q{G}_g^{max}$$

(12)

$$V^{min}\le V_i,V_j\le V^{max}$$

(13)

$$-{\delta }_k^{min}\le {\delta }_k\le {\delta }_k^{max}$$

(14)

where $\gamma ,\beta$ and $\alpha$ are the constant coefficients of the generation cost function. $P_k$ and $Q_k$ are real and reactive power flows of the line. $S_k^{max}$ is the maximum flow limit of each branch. The forward and the reverse direction of flow of the real and reactive power flows are represented as $P_k^{+}$, $Q_k^{+}$ and $P_k^{-}$$Q_k^{-}$, respectively. The bus-to-bus phase angle difference ${\delta }_k$ is kept small for security purposes.

2.2. DC Optimal Power Flow Problem Formulation

Despite the tremendous improvements in nonlinear programming (NLP) in terms of convex relaxations and reactive power handling, the AC OPF linear approximation still remains the major practice all over the world by system operators due to its computational simplicity and market benefits [31].

Such market benefits evolved from the fact that traditional AC OPF incorporates both active and reactive power flows in its process and management. However, the present day liberalized and deregulated electricity markets regard active power as a tradeable commodity and the reactive power as a subsidiary service from the system operator, and the costs are the responsibility of the system users. The decoupling idea of these two products enhances the high interest in active power flow compared to the reactive power flow [14].

Moreover, in terms of computational speed, the linear approximation of AC OPF is frequently in use to speed up multi-scenario steady state analysis [17]. This is mainly due to the superiority of linear programming (LP) in terms of computational efficiency and real-time reliability in market applications, where its capability to provide LMP at each node dominates [31].

Hence, the DC OPF, which is the parent linear approximation of the AC OPF model among others (such as transportation, disjunctive and hybrid models [2,32,33,34,35,36]), is the linearized version of AC power flow with some key assumptions as follows [37]:

• The bus voltage magnitudes must be set to 1.0 pu.
• The phase angle difference of the bus voltage is so small that $sin{\delta }_k\approx {\delta }_k$
• The algebraic sum of branch flow has to be zero ($P_k^{+}+P_k^{-}=0$) hence, $G_k$ is negligible.
• The reactive power flow has to be zero ($Q_k=0$) and
• The reactive generation has to be zero ($Q{G}_g=0$).

Therefore, DC OPF undertakes fixed bus voltage magnitudes and negligible reactive power and network losses [37].

The AC OPF model’s forward and reverse active power flows constraints shown in (6) and (7), respectively, can be reformulated in terms of DC OPF model’s power flow (by neglecting the effect of the branch conductance, $G_k$ and considering $sin{\delta }_k\approx {\delta }_k$) as shown below:

$$\sum _{k\in {\Omega }_k}{P}_k^{+}+\sum _{k\in {\Omega }_k}{B}_k{\delta }_k=0$$

(15)

$$\sum _{k\in {\Omega }_k}{P}_k^{-}-\sum _{k\in {\Omega }_k}{B}_k{\delta }_k=0$$

(16)

Similarly, the DC power flow nodal balance equation can be represented as follows:

$$\sum _{g\in {\Omega }_g}P{G}_g-\sum _{k\in \Omega }{B}_k{\delta }_k=\sum _{d\in {\Omega }_d}P{D}_d$$

(17)

In the alternative DC power flow model, the voltage magnitudes can be set to real values other than 1.0 p.u., with active power losses included. However, the most commonly used model is the ideal lossless DC power flow model, which aims to find the solution to a set of linear equations. Hence, it is usually formulated as a linear programming problem shown below:

$$\begin{gathered} min\sum _{g\in {\Omega }_g}\beta P{G}_g \\ \mathrm{Subject}\mathrm{to}: \end{gathered}$$

(18)

$$\sum _{g\in {\Omega }_g}P{G}_g-\sum _{k\in {\Omega }_l}{P}_k=\sum _{d\in {\Omega }_d}P{D}_d$$

(19)

$$P_k=-B_k{\delta }_k$$

(20)

$$P{G}_g^{min}\le P{G}_g\le P{G}_g^{max}$$

(21)

$$-{\delta }^{min}\le {\delta }_k\le {\delta }^{max}$$

(22)

The DC OPF model represents only the linear term of the original quadratic model of the AC OPF and that brings convexity, which allows for faster computation time [15]. However, the effects of the system’s reactive power and losses remain non-salient in that case.

2.3. Matrix Expansion of the DC OPF Model

The expansion of the developed model in terms of matrices is essential for proper representation of the model in any suitable optimization software. The DC-OPF model in generic matrix form is represented in (23) to (29):

$$\begin{gathered} min\beta P{G}_g \\ \mathrm{Subject}\mathrm{to}: \end{gathered}$$

(23)

$$C_{g}P{G}_g-C^t{P}_k=P{D}_d$$

(24)

$$P_k+B_k{\delta }_k=0$$

(25)

$$-{\delta }_k+C{\theta }_i=0$$

(26)

$$P{G}_g^{min}\le P{G}_g\le P{G}_g^{max}$$

(27)

$$-2\pi \le {\delta }_k\le 2\pi$$

(28)

$$-\pi \le {\theta }_i\le \pi$$

(29)

where $C_g$ is the node incident matrix of the generator buses, $C^t$ is node-branch incident matrix, C is the branch-node incident matrix and ${\theta }_i$ is the angle at bus i. The angles’ limit factor, $\pi$, is in degrees.

The summary of the matrix expression of the model is shown in

Table 1. The summary of the DC-OPF Matrix Model.

${\mathit{PG}}_{\mathit{g}}$	${\mathit{P}}_{\mathit{k}}$	${\mathit{\delta }}_{\mathit{k}}$	${\mathit{\theta }}_{\mathit{i}}$	b
$C_g$	$-C^t$	0	0	$P{D}_d$
0	0	$-I$	C	0
0	I	$B_k$	0	0

, and the detailed illustration of the formula derivation and calculation process using a 4-bus test system can be found in Appendix A.1, where I (in Table 1) represents the identity matrix corresponding to the sizes of the respective decision variables’ matrices.

3. Duality Principle in Linear Programming

Duality in optimization techniques is the principle that views one optimization problem in two perspectives viz: the primal problem and the dual problem. The main problem is usually called the primal problem while the sub-problem is the dual counterpart of the primal problem [38]. The primary purpose of the dual problem, which is always a convex LP problem [39], is to provide a lower bound to the solution of the primal minimization problem [40]. Its convexity implies that it can easily find a lower bound of the primal problem’s optimal solution [39].

Such lower bound can provide extra analysis for the correct interpretation of the optimal solution of the dual problem, which can strengthen the sensitivity of the model in question [38].

Consider the general notation of LP minimization problem formulation:

$$\begin{gathered} min{c}^{t}x \\ \mathrm{Subject}\mathrm{to}: \end{gathered}$$

(30)

$$Ax=b$$

(31)

$$x\ge a$$

(32)

where $c^t$ is the transpose of row vector of the cost matrix, x is the decision variables, A is the equality constraint coefficients matrix, b is the right-hand side of the equality constraints and a is the lower bound of the decision variables.

This is the general representation of an LP type of problem with equality constraints only and that looks very similar to our defined problem in (23) to (29).

The dual problem on the other hand is formulated by maximizing the right-hand side of the primal problem’s constraints, b with a new cost coefficients denoted as u subject to a new constraints where the primal costs become the right-hand side of the constraints and u must be greater or equal to the lower bound of the primal problem (a) as follows:

$$\begin{gathered} maxub \\ \mathrm{Subject}\mathrm{to}: \end{gathered}$$

(33)

$$Au=c^t$$

(34)

$$u\ge a$$

(35)

The purpose of including the concept of duality in this work is due to the proposed dual variables of the DC OPF model’s constraints needed for obtaining the cost of constraint relaxation (CCR) for further analysis of the defined OPF problem. Hence, in terms of the developed DC OPF model:

$$\mathrm{x}=|{\begin{array}{cccc}\hfill P{G}_g\hfill & \hfill P_k\hfill & \hfill {\delta }_k\hfill & \hfill {\theta }_i\hfill \end{array}|}^t, \mathrm{A}=\left|\begin{array}{cccc}\hfill C_g\hfill & \hfill -C^t\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -I\hfill & \hfill C\hfill \\ \hfill 0\hfill & \hfill I\hfill & \hfill B_k\hfill & \hfill 0\hfill \end{array}\right| and \mathrm{b}=\left|\begin{array}{c}\hfill P{D}_d\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \end{array}\right|$$

The yielded optimal solution of the dual is the per unit cost of relaxing the respective constraints and that can form an extra tool for the system designer to see where to increase the capacities of the power network.

4. Results and Discussion of the Test Cases

In this section, the simulation results of the proposed model are reported. The performance of the proposed model was tested in IEEE 6-, 9-, 24-, 39- and 118-bus systems from matpower 7.0.

CPLEX 20.1.0 was used as a solver in MATLAB 2021b installed in an Intel(R) Core(TM) i3-2310M [email protected] GHz 4.00 GB RAM Computer with a 64-bit operating system.

The simulations were conducted in two scenarios of the power networks. The first simulation was conducted on the initial state of networks’ base year, while the second simulation was conducted with an increase in a demand scenario.

The upper and lower bounds of the generator capacities are set according to the maximum and minimum generation capacity of each generator in all the test cases.

4.1. The IEEE 6-Bus System

The IEEE 6-bus system test case as used in this paper has a total base-year energy demand of 210 MW located in buses 4, 5 and 6 as shown in

Table 2. The 6-bus system of the base year demand versus increment in energy demand.

Original Demand ($\mathit{MW}$)		Increased Demand ($\mathit{MW}$)
Bus	Load ($\mathit{MW}$)	Bus	Load ($\mathit{MW}$)
1	0	1	68
2	0	2	45
3	0	3	36
4	70	4	108
5	70	5	110
6	70	6	100
Total	210		257

with the expected 257 MW increase in a demand scenario. Figure 1 shows the comparison of base load profile and that of the increase in demand scenario. The system also has a total maximum generation capacity of 530 MW located in generator buses 1, 2 and 3 as shown in

Table 3. The results of the optimal power flow of the 6-bus system with increase in demand.

			Base Demand Scenario	Increased Demand Scenario
fb-tb	MPF($\mathit{MW}$)	${\mathit{B}}_{\mathit{k}}\left(\mathit{Mho}\right)$	OPF($\mathit{MW}$)	OPF($\mathit{MW}$)
1-2	40	5	0.78	22.3
1-4	60	5	25.52	52
1-5	40	3.33	23.70	40
2-3	40	4	10.52	−6.28
2-4	60	10	49.49	60
2-5	30	3.33	23.18	25.33
2-6	90	5	32.58	29.9
3-5	70	3.85	16.64	35.27
3-6	80	10	38.88	75.5
4-5	20	2.5	5.01	4
4-6	40	3.33	−1.46	−5.4

.

Table 2 shows the OPF results of the 6-bus system network with respect to the base and the increase in demand scenarios, respectively. It can be noticed that both OPF results of each line of the two cases remain below their respective limits as shown in Table 3.

The optimal generation capacities as well as their operating costs with respect to the base and the increase in demand scenarios are shown in

Table 4. Results of the base year—Optimal generation capacity versus optimal generation capacity with demand increase in a 6-bus system.

		Base Demand Scenario		Increased Demand Scenario
Bus	MGC ($\mathit{MW}$)	OGC ($\mathit{MW}$)	[OC   CCR] ($\mathit{M}\$$)	OGC ($\mathit{MW}$)	[OC   CCR] ($\mathit{M}\$$)
1	200	50	58.35   1.033	182	212.38   1.167
2	150	115	118.83   1.033	131.96	136.35   1.033
3	180	45	48.75   1.033	153.04	165.79   1.083
4	0	0	00.00   1.033	0	000.00   1.351
5	0	0	00.00   1.033	0	000.00   1.143
6	0	0	00.00   1.033	0	000.00   1.081

.

The novel feature in Table 4 is the CCR as discussed in Section 3 part of this work. It can be noticed that the CCR remains the same in each node of the network’s base year due to the low demand capacity. That indicates equal chances of increase in capacity at each bus. However, such increment in capacity heavily depends on the location and the quantity of electricity demanded, which does affect the unit price of energy production to satisfy such demand. Hence, the variations in the CCR in the increase in demand scenario show that the buses of the network do not have equal chances of increment in capacity to satisfy any further demand increase.

It can be noticed from Figure 2 that bus 4 has the highest per unit price of further increment in generation capacity, while bus 2 has the highest chance of further increment in capacity to satisfy any further increment in demand due its lowest CCR. Hence, such approach can become an extra tool for the system designer to know where to increase the capacities.

4.2. The IEEE 9-Bus System

The IEEE 9-bus system test case as used in this paper has a total base energy demand of 315 MW located in buses 5, 7 and 9 with the expected increase in demand of 310 MW as shown in

Table 5. The 9-bus system—the base year demand versus increment in energy demand.

Original Demand ($\mathit{MW}$)		Increased Demand ($\mathit{MW}$)
Bus	Load ($\mathit{MW}$)	Bus	Load ($\mathit{MW}$)
1	0	1	50
2	0	2	20
3	0	3	0
4	0	4	60
5	90	5	50
6	0	6	50
7	100	7	30
8	0	8	50
9	125	9	0
Total	315		310

. Figure 3 shows the comparison of base load profile and that of the increase in a demand scenario, and the corresponding values are in Table 5. The system also has a total maximum generation capacity of 820 MW located in generator buses 1, 2 and 3 as shown in

Table 6. Results of the base year—optimal generation capacity versus optimal generation capacity with demand increase in a 9-bus system.

		Base Demand Scenario		Increased Demand Scenario
Bus	MGC ($\mathit{MW}$)	OGC ($\mathit{MW}$)	[OC   CCR] ($\mathit{M}\$$)	OGC ($\mathit{MW}$)	[OC   CCR] ($\mathit{M}\$$)
1	250	10	050   1.20	85.313	426.56   5.00
2	300	35	042   1.20	270	324.00   1.20
3	270	270	270   1.20	270	270.00   1.00
4	0	0	000   1.20	0	000.00   5.00
5	0	0	000   1.20	0	000.00   5.88
6	0	0	000   1.20	0	000.00   1.00
7	0	0	000   1.20	0	000.00   1.96
8	0	0	000   1.20	0	000.00   2.65
9	0	0	000   1.20	0	000.00   4.19

.

Table 7. The results of the optimal power flow of the 9-bus system with increase in demand.

			Base Demand Scenario	Increased Demand Scenario
fb-tb	MPF ($\mathit{MW}$)	${\mathit{B}}_{\mathit{k}}\left(\mathit{Mho}\right)$	OPF ($\mathit{MW}$)	OPF ($\mathit{MW}$)
1-4	250	17.361	10	35.313
4-5	250	10.87	−38.585	−10
5-6	150	5.882	−128.59	−150
3-6	300	17.065	270	270
6-7	150	9.920	141.41 6	69.687
7-8	250	13.889	41.415	-60.313
8-2	250	16	−35	−250
8-9	250	6.211	76.415	139.69
9-4	250	11.765	−48.585	14.687

shows the OPF results of the 9-bus system network with respect to the base and the increase in demand scenarios, respectively. It can be noticed that both OPF results of each line of the two cases remain below their respective limits.

The optimal generation capacities as well as their operating costs with respect to the base and the increase in demand scenarios are shown in Table 6. Similarly, the CCR remains the same in each node of the network’s base year due to the low demand capacity, which also indicates equal chances of increase in capacity at each bus.

However, Figure 4 shows that bus 5 has the highest per unit price of further increment in capacity while buses 3, 6 and 2, respectively, have the highest chance of further increment in capacity to satisfy any further increment in demand due its lower CCR and that can form an extra tool for the system designer to know where to increase the capacities.

4.3. The IEEE 24-Bus System

The IEEE 24-bus system test case as used in this paper has a total base energy demand of 2003 MW and an expected increase in demand of 1947 MW as shown in

Table 8. The 24-bus system—the base year demand versus increment in energy demand.

Original Demand ($\mathit{MW}$)				Increased Demand ($\mathit{MW}$)
Bus	Load ($\mathit{MW}$)	Bus	Load ($\mathit{MW}$)	Bus	Load ($\mathit{MW}$)	Bus	Load ($\mathit{MW}$)
1	0	13	0	1	0	13	0
2	97	14	194	2	0	14	100
3	180	15	217	3	0	15	100
4	74	16	100	4	0	16	100
5	71	17	0	5	0	17	340
6	0	18	0	6	45	18	350
7	0	19	181	7	225	19	100
8	171	20	128	8	9	20	60
9	175	21	0	9	0	21	67
10	195	22	0	10	0	22	56
11	0	23	0	11	125	23	234
12	220	24	0	12	0	24	45
Total			2003				1947

.

Figure 5, shows the comparison of base load profile and that of the increase in demand scenario, which the corresponding values in Table 8.

The system also has a total maximum generation capacity of 4575 MW located in their respective generator buses as shown in

Table 9. Result of the base year optimal generation capacity versus optimal generation capacity with demand increase in a 24-bus system.

		Base Demand Scenario		Increased Demand Scenario
Bus	MGC ($\mathit{MW}$)	OGC ($\mathit{MW}$)	[OC   CCR] ($\mathit{M}\$$)	OGC ($\mathit{MW}$)	[OC   CCR] ($\mathit{M}\$$)
1	192	16	2080   1.61	192	24,960   13.0
2	192	16	2080   1.61	192	24,960   13.0
3	800	175	2814.2   1.61	400	6432.4   13.0
13	591	591	9503.9   1.61	591	9503.9   13.0
14	160	0	0000.0   1.61	0	000.00   13.0
15	225	2.4	312.0   1.61	160	20,800   13.0
16	655	655	10,533   1.61	655	10,533   13.0
18	400	383.3	6163.9   1.61	400	6432.4   13.0
21	400	100	4366.61   1.61	400	17,465   13.0
21	300	10	436.61   1.61	300	13,098   13.0
22	660	54.3	2370.8   1.61	660	28,817   13.0

. The results of the optimal generation capacities as well as their operating costs with respect to the base and the increase in demand scenarios are shown in Table 9. Similarly, the CCR remains the same in each node of the network’s base year due to the low demand capacity, which also indicates equal chances of increase in capacity at each bus.

However, Figure 6 shows that the 24-bus system’s nodes tend to have equal CRR tendency in all loading scenarios except in bus 7, which indicates viability in additional capacity supply for any further increment in demand due its low CCR.

Table 10. Results of the optimal power flow of the 24-bus system with increase in demand.

			Base Demand Scenario	Increased Demand Scenario
fb-tb	MPF ($\mathit{MW}$)	${\mathit{B}}_{\mathit{k}}\left(\mathit{Mho}\right)$	OPF ($\mathit{MW}$)	OPF ($\mathit{MW}$)
1-2	175	71.941	50.027	35.68
1-3	175	4.735	−33.334	58.5
1-5	175	11.834	−0.693	97.82
2-4	175	7.893	−2.92	79.744
2-6	175	5.208	−28.06	50.935
3-9	175	8.403	−5.13	−9.745
3-24	400	11.92	−208.21	−111.76
4-9	175	9.643	−76.92	5.744
5-10	175	11.325	−71.693	26.823
6-10	175	16.529	175	5.94
7-8	175	16.53	−1.82	175
8-10	175	6.06	5.82	3.698
9-11	400	11.92	−158.21	0.302
9-12	400	11.92	−100.65	−81.95
10-11	400	11.925	−173.25	−93.35
10-12	400	11.925	−115.68	−75.271
11-13	500	21.01	−220.25	−86.668

shows some of the OPF results of the 24-bus system network with respect to the base and the increase in demand scenarios, respectively. It can be noticed that both OPF results of each line of the two cases remain below their respective limits.

4.4. The IEEE 39-Bus System

The IEEE 39-bus system test case as used in this paper has a total energy demand of 7381.23 MW located in respective load buses as shown in

Table 11. The 39-bus system increment in energy demand versus the base year demand.

Original Demand ($\mathit{MW}$)				Increased Demand ($\mathit{MW}$)
Bus	Load ($\mathit{MW}$)	Bus	Load ($\mathit{MW}$)	Bus	Load ($\mathit{MW}$)
1	97.6	21	274	2	50
3	322	23	247.5	5	120
4	500	24	308.6	6	80
7	233.8	25	224	9	13
8	522	26	139	12	10
9	6.5	27	281	13	72
12	8.53	28	206	14	85
15	320	29	283.5	17	97
16	329	31	9.2	19	50
18	158	39	1104	30	550
20	680
Total			6254.23		1127

.

Figure 7 shows the comparison of base load profile and that of the increase in demand scenario, and the corresponding values are in Table 11. The system also has a total maximum generation capacity of 7398.87 MW located in the respective generator buses as shown in

Table 12. Result of the base year optimal generation capacity versus optimal generation capacity with demand increase in a 39-bus system.

		Base Demand Scenario		Increased Demand Scenario
Bus	MGC ($\mathit{MW}$)	OGC ($\mathit{MW}$)	[OC   CCR] ($\mathit{M}\$$)	OGC ($\mathit{MW}$)	[OC   CCR] ($\mathit{M}\$$)
1	192	16	2080.0   1.61	192	24960   13.0
2	192	16	2080.0   1.61	192	24960   13.0
3	800	175	2814.2   1.61	400	6432.4   13.0
13	591	591	9503.9   1.61	591	9503.9   13.0
14	160	0	0000.0   1.61	0	0000.0   13.0
15	225	2.4	312.00   1.61	160	20,800   13.0
16	655	655	10,533   1.61	655	10,533   13.0
18	400	383.3	6163.9   1.61	400	6432.4   13.0
21	400	100	4366.6   1.61	400	17,465   13.0
21	300	10	436.61   1.61	300	13,098   13.0
22	660	54.3	2370.8   1.61	660	28,817   13.0

. The result of the optimal generation capacities as well as their operating costs with respect to the base and the increase in demand scenarios are also shown in Table 12.

The CCR in this case (as plotted in Figure 8) tends to vary in each node of the network’s base year due to the low demand capacity, which indicates unequal chances of increase in capacity at each bus by default up to any level in load increments. However, the designer still has the choice of choosing an appropriate point of capacity increase based on the CCR’s information.

Moreover,

Table 13. The results of the optimal power flow of the 39-bus system with increase in demand.

			Base Demand Scenario	Increased Demand Scenario
fb-tb	MPF ($\mathit{MW}$)	${\mathit{B}}_{\mathit{k}}\left(\mathit{Mho}\right)$	OPF ($\mathit{MW}$)	OPF ($\mathit{MW}$)
1-2	175	71.941	50.027	35.68
1-3	175	4.735	−33.334	58.5
1-5	175	11.834	−0.693	97.82
2-4	175	7.893	−2.92	79.744
2-6	175	5.208	−28.06	50.935
3-9	175	8.403	−5.13	−9.745
3-24	400	11.92	−208.21	−111.76
4-9	175	9.643	−76.92	5.744
5-10	175	11.325	−71.693	26.823
6-10	175	16.529	175	5.94
7-8	175	16.53	−1.82	175
8-10	175	6.06	5.82	3.698
9-11	400	11.92	−158.21	0.302
9-12	400	11.92	−100.65	−81.95
10-11	400	11.925	−173.25	−93.35
10-12	400	11.925	−115.68	−75.271
11-13	500	21.01	−220.25	−86.668

shows the OPF results of the 39-bus system network with respect to the base and the increase in demand scenarios, respectively. It can be noticed that both OPF results of each line of the two cases remain below their respective limits.

4.5. The IEEE 118-Bus System

The IEEE 118-bus system test case as used in this paper has a total energy demand of 14,126.55 MW located in respective load buses. Figure 9, shows the base year energy demand versus the expected increase in energy demand scenarios in different buses of the 118-bus system.

The system also has a total maximum generation capacity of 14550 MW located in the respective generator buses.

The CCR remain the same for all the buses in a base demand scenario as shown in Figure 10, while increase in demand scenario indicates unequal chances of increase in capacity in major parts of the network. Hence, the designer can also vividly see an appropriate point of capacity increase based on the CCR’s information in this case.

The CPU computation times in terms of the different test cases of the IEEE test systems were recorded and compared as shown in

Table 14. The comparison of the computation times of the 5 different network sizes.

Computation Time (s)
Network Size	Base Demand Scenario	Increased Demand Scenario
6	0.015625	0.046875
9	0.015625	0.09375
24	0.046875	0.015625
39	0.078125	0.125
118	0.109375	0.1875

and the graphical representation is shown in Figure 11, which shows that the model can be applied in real and large network systems with acceptable computation times.

5. Conclusions and Future Work

An OPF model that can predict the exact nodes through which the network capacities can be increased in order to optimize how much energy is produced in the process has been presented. The problem was formulated as a Direct Current (DC) OPF model, which is a linearized version of the Alternating Current (AC) OPF model. The DC-OPF model was solved as a single period OPF problem, which was tested in several case studies using the fixed topology of the IEEE test systems and the computation speeds of the different cases were compared, which shows that the model can be applied in real and large network systems with acceptable computation times. In addition, the results also suggested dual variables of the problem constraints as an extra tool for the network designer to see where to increase the network capacities.

Therefore, this work has demonstrated the efficacy of the dual variables of the constraints in locating the exact nodes where the capacity expansion could be needed in a fixed network topology to satisfy already known increase in energy demand. However, the area of uncertainties in energy demand was not included in the work.

Hence, the recommendation for future work will be the inclusion of uncertainties and incorporation of stochastic and multi-period objectives in the proposed DC OPF analysis due to the current high penetration of renewable energy sources in the grid system.