13.1 Introduction

A power system is a highly dynamic system whose stability has to be ascertained in all cases; moreover, load centres expect a secure and reliable system. Voltage stability assessment is one of the major concerns in power system planning and secure operation as power grids span over several regions and sometimes even countries [1]. A direct link between the voltage and the reactive power makes it possible to control the voltage to desired values by control of the reactive power. The operator of the power system is responsible for controlling the transmission system voltage, which means having enough reactive power available to handle voltage violation conditions [2]. In normal conditions, the reactive power of the system not only dictates the voltage profile but also leads to losses. Loss minimization is an indispensable objective that must be considered in efficient power system operation [2–4]. Hence, to achieve certain global control objectives (e.g. N-1 secure operation, reactive power planning, minimization of losses, etc.) it is necessary to coordinate control actions among regional operators, but at the same time avoiding exposing local system data pertaining to regional infrastructure [5]. This chapter considers loss minimization as the objective function and maintaining voltage profile within safe limits as the constraint.

Optimization approaches based on power flow calculations often provide accurate results, but calculating nonlinear equations requires high computational time and resources, hence it is not suitable for real-time applications [6]. This chapter proposes an optimization approach in which the objective function is augmented to incorporate global optimization of a linearized large scale multi-agent power system using the Lagrangian decomposition algorithm [7, 8]. The aim is to maintain centralized coordination among agents via a master agent, leaving loss minimization as the only distributed optimization, which is analysed while protecting the local sensitive data. The efficiency of the local objective function stems from the use of active power loss sensitivity with respect to the control variables of the system. Control variables are defined as reactive power injection of generators and tap changer positions of transformers. The power loss sensitivities with respect to all control variables in the system are used in the first stages, which are calculated from a linearized model of the system achieved through various control schemes and stored over regular intervals (viz. state estimation, PMU measurements, etc.) [9, 10]. Ultimately, this chapter has a centralized convex optimization problem which is solved using the aforementioned decomposition algorithm.

Lagrangian decomposition is a classical approach for solving constrained optimization problems. The augmented Lagrangian method adds an additional term to the unconstrained objective which mimics a Lagrange multiplier [11]. This method has been extensively used for solving numerous engineering problems, especially in the power systems field [12–14]. The advantage of this algorithm is the fact that local grid data does not have to be made globally available, which is often of crucial concern in actual power system operation, albeit degrading the computational performance since it involves iterating many times to reach graceful optimization. The optimization is carried out until the agents negotiate with the neighbouring areas on their inter-area variables (i.e. voltages at the interlinked buses).

The remainder of the chapter is structured as follows. Section 13.2 formulates the control problem exploiting power loss sensitivities. Section 13.3 proposes the augmented Lagrange formulation and its implementation. In Section 13.4 the proposed algorithm is implemented on a modified IEEE 30-bus system, and the performance analysis of the control scheme for various scenarios is studied. Section 13.5 concludes.

13.2 System Model and Problem Formulation

13.2.1 Power System Model

Governing any system comprises measuring system states and identifying an objective function which has to be achieved while concurrently tackling system constraints. A power system can be expressed in general differential-algebraic form given by (13.1)–(13.3):

13.1

13.2

13.3

Here x corresponds to the dynamic state of generators and system loads, z corresponds to the algebraic voltages at buses, and u are the control variables (α tap changer set-points and q reactive power injection). Power system in real scenarios span several thousand kilometres, sometimes even countries and continents, mainly for utilization, efficiency, and management efficiency purposes.

If these regions are partitioned into N^a areas, each controlled by a local operator, this gives rise to an interconnected network of multiple operators with local supervisory control. Communication among these operators is imperative for an efficient and reliable system. Nevertheless, regional assets such as generation capacity, reactive power reserves, infrastructure details, and so on are to be protected from each other. To specify the interconnection among areas, Let be the voltage magnitudes at the interconnection buses connected to area i of neighbouring areas j which is expected by area i; v_ji is the voltage magnitude at these buses expected by their own area j. Consensus among these areas implies the negotiation of the voltage profile at the coupling bus given as in (13.7). As shown in Figure 13.1, the regions are connected via the interconnecting buses or coupling buses with voltages v_ij (the solutions of the above equations) for N^a areas governing the whole system. The equations can then be written as:

13.4

13.5

13.6

13.7

where j is a subset (neighbouring area) of the power system (N^a areas) and x_i, z_i, and u_i represent the state variables, dynamic states, and control variables of area i, respectively.

13.2.2 Optimal Reactive Control Problem Formulation

The purpose of an optimal reactive power dispatch is mainly to improve the voltage profile in the system and to minimize transmission losses. In this chapter, loss minimization is considered as the objective function J, while voltage profile is kept at values between 0.9 pu and 1.1 pu. The objective functions in (13.9) are introduced in a linearized form through sensitivities that are obtained by the linearization of the load flow equations around the nominal operating point as presented in the next section (Section 13.2.3). Changes of the control variables at time-instant k are achieved by minimizing the objective functions (13.9):

13.8

13.9

13.10

subject to

13.11

13.12

13.13

13.14

13.15

where and – that is, the loss sensitivities and voltage sensitivities of area i with respect to the control variables of area i and inter-area variables connected to area i, respectively – are calculated as in Section 13.2.3. and are the minimum and maximum acceptable voltages in area i, respectively. The constraints in the above equations are the result of linearization of the power system equations. They show the constraints of the voltage at the nodes, reactive power injections via devices, and transformer tap-positions. These parameters would be assessed and re-calibrated after every local optimization by the regional operators.

13.2.3 Multi-Agent Sensitivity Model

Sensitivities of a region are calculated based on sensitivity coefficients, the linearized factors arising from the nonlinear relationship among variables of the system. Load flow analysis is the backbone of sensitivity analysis, and through it the voltage magnitude and phase angle at each node and the complex power flowing in each transmission line can be obtained. The Inverse Jacobian matrix of the initial load flow analysis is manipulated to perform sensitivity analysis of the system, which in turn depends on the control parameters themselves; thus, the impact of change of magnitudes of control variables on control performance are significant and hence needs to be completely analysed for a wide range of changes. Since we have more than one control parameter, we need to develop an algorithm which studies the effect of concurrent effects of variation in multiple parameters. The calculation of the sensitivities in the multi-agent model has been split in two layers as seen in (13.16):

13.16

13.2.3.1 Calculation of the First Layer

The objective is to minimize real power losses during the operation and control of a network. The real power loss of area i is represented by

13.17

where G_k is the conductance of line k which is connected between buses h and l in area i, and N_br is the number of branches of area i. In (13.17), the losses are represented by a nonlinear function of the bus voltages phase angles.

Then, the losses function is linearized as follows:

13.18

13.19

For every transmission line, the partial derivatives of with respect to the voltages at buses h and l are calculated. Partial derivatives pertaining to a certain bus are summed to form the power loss sensitivities with respect to all bus voltages in the system.

13.2.3.2 Calculation of the Second Layer

The vector of the control variables w_i is combination of three different vectors of the following variables: the inter-area variable , reactive power injection of generators q_g,i, and tap ratio α_tap,i. Therefore, the second layer was specified by computing the three sensitivities , and .

It is clear that is a unity vector and is the inversion of the Jacobian matrix calculated below:

Reactive power injection at bus k

13.20

The Jacobian matrix is partly structured from partial derivatives of the reactive power injections as:

13.21

13.22

In this chapter, changing the tap ratio of the transformer is equivalent to the injection of two reactive power increments into buses which are connected to the transformer terminals. Thus the sensitivities are equivalent to two layers of sensitivities below:

13.23

The sensitivities is essentially a sub-matrix of the known . While is calculated from (13.35) and (13.36) for all branches equipped with tap changing transformers.

Transformer tap changing is more difficult to model since two buses are directly involved in the tap changing process. Let us consider a transformer connecting buses h and l with tap α, as shown in Figure 13.2. This branch can be represented by an equivalent π circuit.

The admittance of the branch is:

13.24

From Figure 13.2, the complex power injection to bus h is

13.25

where ^* indicates the complex conjugate of the variable. So

13.26

Similarly, the complex power injection to bus l is represented as

13.27

From (13.26) and (13.27), the equations for q_h and q_l are

13.28

13.29

If is the increment of q_h with respect to voltage and tap position changes, then

13.30

However, for the power flow in Figure 13.2, we have

13.31

So, differentiating (13.28) with respect to v_h and α, we have

13.32

Similarly, differentiating (13.29) with respect to V_l and α, we have

13.33

Moreover, (13.32) can be rewritten as follows

13.34

Since the value of α is close to unity, and and are small, we have:

13.35

Similarly, from (13.33):

13.36

13.3 Multi-Agent Based Approach

13.3.1 Augmented Lagrange Formulation

It can be seen that the overall control problem (13.8)–(13.15) is not separable into sub-problems using the local variables of one agent i alone due to the interconnecting constraints (13.11). Therefore, a distributed algorithm based multi-agent system is introduced and presented in this section in order to achieve a global optimum of the whole system by separately solving local sub-problems.

In a multi-agent system, global optimization must be achieved while protecting regional data, hence we make use of the augmented Lagrangian method to integrate the interconnecting constraints of (13.11) into the global objective function (13.8) in the form of additional linear cost in terms of Lagrangian multipliers . The global objective now along with the losses and the inter-area constraints is as seen in (13.37):

13.37

with the constraints being the ones from (13.12)–(13.15), and coefficient c is a positive scalar penalizing interconnecting constraint violations.

Then (13.37) can be decomposed into separate sub-problems so that they can be tackled and solved independently with i = 1,..,N^a as:

13.38

where and v_{ji. prev} are and v_ji, respectively, computed at the previous iteration for the other agents.

13.3.2 Implementation Algorithm

The implementation algorithm proposed in [11] is described in Figure 13.3. Firstly, each agent in turn updates its sensitivities and then minimizes its problem (13.38) to determine its optimal local and inter-area variables, while the variables of the other agents stay fixed. After the last agent completes its optimization sub-problem, the inter-area variable is transmitted to the master agent where termination conditions such as agreement on the inter-area variables in (13.41) or maximum allowable number of iterations in (13.42) are checked. If the conditions are satisfied, the determined actions are implemented. Otherwise, the Lagrangian multipliers are updated following the strategy described by (13.39) and (13.40), and the whole process is then repeated in a further iteration:

13.39

13.40

13.41

13.42

with

13.4 Case Studies and Simulation Results

13.4.1 Case Studies

Firstly, it is worth mentioning that the proposed approach can be extended to any number of interconnected areas with an arbitrary number of interconnection lines without conceptual modification to its structure.

The proposed method has been implemented on a modified IEEE 30-bus system taken from [15]. The test system is partitioned into three areas as shown in Figure 13.4 comprising ten transformers in total, equipped with tap-changers, each having two generators and several loads. There are 7 interconnection lines in total, corresponding to 14 inter-area variables.

13.4.2 Simulation Results

Note that the simulation results below are given with the initial control parameters set up as presented in Table 12.1. They are categorized into two groups: namely general parameters referring to the global control formulation, and those being relevant to the multi-agent based approach.

General parameters				Parameters of multi-agent system
(MW)	& (p.u.)	& (MVAR)	& (p.u.)	ϵv (p.u.)	c	itermax
0.01	0.03	0.02	0.002	0.0005	9.5	300

Control circle No.	Multi-agent				Single-agent
	Area 1(MW)	Area 2(MW)	Area 3(MW)	Total(MW)	Total(MW)
1	4.6341	1.6414	1.1029	7.3784	7.3784
2	4.3574	1.6158	1.0899	7.0631	6.679
3	4.1376	1.6066	1.0886	6.8327	6.2077
4	3.7285	1.5814	1.0581	6.368	5.83
5	3.4451	1.554	1.043	6.0421	5.5356
6	3.2059	1.5366	1.0328	5.7754	5.3163
7	3.0039	1.5293	1.0269	5.5601	5.1591
8	2.8541	1.5345	1.0204	5.409	5.0318
9	2.7493	1.5516	1.0136	5.3144	4.9593
10	2.6798	1.5637	1.0101	5.2537	4.9148
11	2.6479	1.5123	1.003	5.1633	4.8647
12	2.6202	1.5035	1.0016	5.1253	4.8545
13	2.5958	1.4718	1.008	5.0756	4.8055
14	2.5787	1.441	1.0135	5.0332	4.7998
15	2.5565	1.4132	1.0256	4.9953
16	2.5408	1.3877	1.0342	4.9628
17	2.5308	1.3587	1.0492	4.9387
18	2.5283	1.3439	1.0705	4.9427
19	2.5126	1.3258	1.0853	4.9237

N_step indicates the trade-off between controller performance and computational expense since a larger N_step translates to both higher performance and a higher computational burden. In this fashion, is intuitively selected in this example. In addition, it is also assumed that the calculation time of the controller to give control actions is 2 s.

13.4.2.1 Performance Comparison Between Multi-Agent Based and Single-Agent Based System

It can be seen from Table 13.2 that the single-agent system provides a better convergence value of the losses compared to multi-agent. This likely stems from the fact that the higher values ϵ^v of the agreement on inter-area variables in (13.41) are, the worse the performance in terms of losses is. Moreover, the convergence speed of the single-agent system is faster as well.

In addition, the proposed control algorithm shows its capability in establishing cooperation between agents to achieve the global objective. This can be seen from Table 13.1, in that from control circle 12, the losses of area 3 show a trend pointing to an increase; in the meantime, losses of other areas decrease and the whole system loss decreases in response.

13.4.2.2 Impacts of General Parameters on the Proposed Control Scheme's Performance

In order to study the impacts of control parameters on performance, each simulation experiment below is carried out varying only one of the parameters while keeping the others unchanged. The parameters are set up at the values presented in Table 12.1.

Figure 13.6 depicts loss convergence with different change limits for selected voltages. As can be seen in the following, the narrower limits provide a better convergence value but with slower convergence speed within initial control circles that are far from the optimum operating state. The large limits often lead to fluctuation of convergence values, since the algorithm performance depends on sensitivities accuracy, which is inversely proportional to the limits; thus there is higher risk of triggering the termination condition (13.41).

Again, algorithm performance is adversely influenced by a larger change magnitude of control variables as presented in Figure 13.5 and Figure 13.7. This is clear because the sensitivities calculated are based on small changes of control variables.

13.4.2.3 Impacts of Multi-Agent Parameters on the Proposed Control Scheme's Performance

From Figure 13.8, it is likely that determining a correct value of the coefficient c in order to achieve the best performance will be difficult. Moreover, it should be highlighted that the coefficient c has a strong association with the control algorithm's performance.

13.5 Conclusions

The need for coordination while protecting sensitive data in large interconnected grids for reliable operation of power system prompted the idea of a multi-agent based system. An effective augmented Lagrange decomposition algorithm was implemented and analysed by considering loss minimization as the objective function. This approach not only protects local data, but also gives a close performance to the conventional single-agent based system as seen in the simulation results.

On the other hand, improper selection of the general and multi-agent system parameters degrades the performance as shown in the control scheme's performance plots. Selection of these parameters for now was based on trial and error, but for future work, proper estimation of these parameters will play a vital role in the reduction of computation time and will also enrich the performance of the algorithm. In addition, further investigation is required to develop faster and more robust algorithms, and minimization of communication overheads should be included in the objective function.

The augmented Lagrange decomposition algorithm in this chapter has a disadvantage of no longer being separable across subsystems because of its dependence on the master agent. To achieve both separability and robustness for distributed optimization, we can instead use the alternating direction method of multipliers (ADMM) [16], which has been widely adopted in various industries due to its ease of decomposition and its convergence guarantees on a wide range of problems.

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